THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING P-v-T DATA FOR NEON AND HELIUM AT TEMPERATURES FROM 70~ K TO 120~ K AND PRESSURES TO 690 ATMOSPHERES John A. Sullivan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Mechanical Engineering 1966 February, 1966 IP-730

Doctoral Committee: Associate Professor Richard E. Sonntag, Chairman Associate Professor Edward R. Lady Professor Joseph J. Martin Assistant Professor Gene E. Smith Professor Gordon Jo Van Wylen

ACKNOWLEDGEMENTS The author would like to express thanks to those persons and organizations who have, by their support and interest, made the completion of this research possible. Thanks are extended to the National Science Foundation, the Ford Foundation and the Mechanical Engineering Department for personal financial support of the author and his family for the duration of the research. Recognition for funding the research is extended to the National Science Foundation and the Mechanical Engineering Department. Recognition for donation of the test gases is extended to the Linde Division of Union Carbide Corporation and to the Bureau of Mines. Linde Company is also to be commended for direct support of the research with contributions of liquid argon. Special recognition for continued assistance and interest is extended to Professor Sonntag and Professor Van Wylen. Thanks are extended to the other members of the doctoral committee for their constructive criticism and helpful advice. Special thanks are also given to Professor Herman Merte, who, although he was not directly connected with the project, was always willing to give special attention to problems which arose. The author extends thanks to his colleague, Richard W. Crain, who served as a critic for new ideas and a sounding board for ironing out experimental difficulites. Thanks are also extended to James Schairer for technical advice in the construction of electronic apparatus and to Jack Brigham and his staff for technical assistance in the construction and maintainance of the experimental equipment. Finally special thanks are extended to my wife and family for sustaining the years of sacrifice and tight-budget living conditions forced upon them by the author's decision to pursue the Doctor of Philosophy degree. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.......................................ii TABLE OF CONTENTS................................................ iv LIST OF TABLES..............................................vi LIST OF FIGURES................................................ viii NOMENCLATURE..................................................... x I. INTRODUCTION............................................ 1 II. LITERATURE SURVEY....................................... 4 A. Neon............................................... 4 B. Helium...............................7.............. C. Burnett Method...................................... 9 D. Related Literature........................... 10 III. Theory.................................................. 11 A. Burnett Method...................................11 B. Treatment of Partial Runs........................... 17 C. Discussion of Errors in the Burnett Method.......... 18 D. Review of Present Data Treatment Techniques......... 20 E. The Exact Method for Determining No and Po/Z..... 22 F. The Virial Equation of State.................... 31 IV. EXPERIMENTAL APPARATUS AND PROCEDURES.................. 39 A. Experimental Apparatus.................... 39 B. Experimental Procedures............................. 66 C. Difficulties Encountered......................... 70 V. DATA REDUCTION AND EXPERIMENTAL RESULTS................. 75 A. Data Reduction...................................... 75 B. Error Analysis...................................... 84 VI. EXPERIMENTAL RESULTS.................................... 98 A. Experimental Results for Helium..................... 98 B. Experimental Results for Neon....................... 109 iv

TABLE OF CONTENTS (CONT'D) Page VII. THEORETICAL AND EMPIRICAL CORRELATIONS............. Ito 119 A, Determination of Intermolecular Parameters for the Lennard-Jones 6-12 Potential..................... 119 B, Fit of the Experimental Data to the Leiden Expansion........................................... 119 C, Fit of the Experimental Data to the Berlin Expansion...........................................< 121 D, Values of Compressibility for Even Values of Pressure......................................, o 126 APPENDIX A - LIST OF PHYSICAL CONSTANTS...,.................. 139 BIBLIOGRAZPHY..................,.,..,.................E,..., e. 140 r

LIST OF TABLES Table Page 1. HELIUM PURITY............................................. 98 2. EXPERIMENTAL RESULTS FOR HELIUM...................... 100 3. COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR HELIUM TO PUBLISHED VALUES.......................... 104 4. EXPERIMENTAL SECOND AND THIRD VIRIAL COEFFICIENTS FOR HELIUM............................................ 106 5. EXPERIMENTAL RESULTS FOR NEON............................ 110 6. COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR NEON TO PUBLISHED VALUES........................................ 113 7. EXPERIMENTAL SECOND AND THIRD VIRIAL COEFFICIENTS FOR NEON.................................................. 115 8. RESULTS OF THE DETERMINATION OF a AND c FOR HELIUM WITH QUANTUM CORRECTIONS................................ 120 9. RESULTS OF THE DETERMINATION OF a AND E FOR NEON WITH QUANTUM CORRECTIONS................................ 120 10. LEIDEN COEFFICIENTS FOR HELIUM............................ 122 11. LEIDEN COEFFICIENTS FOR NEON......................... 123 12. BERLIN COEFFICIENTS FOR HELIUM............................ 124 13. BERLIN COEFFICIENTS FOR NEON........................... 125 14. FIT OF THE 700 K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION................................................ 127 15. FIT OF THE 80~ K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION......................................... 128 16. FIT OF THE 100~ K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION.............................. 129 17. FIT OF THE 120~ K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION.......................................... 130 vi

LIST OF TABLES (CONT'D) Table Page 18, FIT OF THE 70~ K ISOTHERM OF NEON TO THE BERLIN EXPANSION W e...W......... a a..., o................. 131 19. FIT OF THE 80~ K ISOTHERM OF NEON TO THE BERLIN EXPANSION,,............... o o............. o a.. o a aaa a a o 132 20. FIT OF THE 100~ K ISOTHERM OF NEON TO THE BERLIN EXPANSION..SION...................d0 o.. o.... e o. 133 21, FIT OF THE 120~ K ISOTHERM OF NEON TO THE BERLIN EXPANSIONo..................................................... 134 22, VALUES OF COMPRESSIBILITY FOR HELIUM FOR EVEN VALUES OF PRESSURE. e................................. e............ 135 23, VALUES OF COMPRESSIBILITY FOR NEON FOR EVEN VALUES OF PRESSURE............................................ 138 vii

LIST OF FIGURES Figure Page 1. Isotherms of Helium Investigated in This Research......... 3 2. Isotherms of Neon Investigated in This Research........... 3 3. Schematic Illustrating the Burnett Method................. 12 4. Effect of Varying No According To (10).................... 21 5. P-Plot For Helium........................................ 25 6. P-Plot For A Typical Substance For Isotherms With A Region Where Z < 0.5................................... 25 7. P-Plot For 100~ K Helium Isotherm (6 in atm).............. 28 8. p-Plot For 80~ K Neon Isotherm.......................... 29 9. Cross-section Drawing of the Burnett Cells............... 45 10. Photograph of The Burnett Cells........................... 46 11. Photograph of The Dead Weight Gage........................ 48 12. View of The Mueller Bridge and Wenner Potentiometer and Associated Galvanometers............................ 51 13. Schematic Diagram of the Test Gas System.............. 53 14. Circuit Diagram of the Heater System................... 56 15. View of The Burnett Cells With The Heaters and Stirrer in Place.......................................... 57 16. View of The Control Panel................................ 59 17. Schematic Diagram of the Vapor Pressure Control System,... 60 18. Circuit Diagram For The Solenoid Control System,.......... 61 19. Schematic Drawing of the Cryostat....................... 64 20. View of The Cryostat Showing The Lifting Mechanism...... 67 viii

LIST OF FIGURES (CONT'D) Figure Page 21. Compressibility Diagram For Helium Showing The Experimental Points..,.,.,...................... 102 22. Compressibility Diagram For Helium Showing The Low Pressure Experimental Points..................... 103 23. Comparison of Experimental Values of the Second Virial Coefficient of Helium..,..................,...... 107 24~ Compressibility Diagram For Neon Showing The Experimental Points..,,....,............................ 112 25. Comparison of This Work To The Compilation of McCarty, Stewart and Timmerhaus(59)....................,,. 117 26. Comparison of the Experimental Values of the Second Virial Coefficient of Neon.,..,,,......................... 118 ix

NOMENCLATURE Arabic Symbols A first virial coefficient. A' first coefficient in the Berlin expansion. Ae effective area of the dead weight gage piston. Ao area of dead weight gage piston at zero pressure and 25~ C. B second virial coefficient. B* reduced second virial coefficient defined by Equation (57). B' second coefficient in the Berlin expansion. Bcl second virial coefficient obtained from corrected Boltzmann statistics. B* reduced classical second virial coefficient. cl Bo second virial coefficient for Fermi-Dirac and Bose-Einstein gases at low pressure. B* reduced ideal gas second virial coefficient for Fermi-Dirac and Bose-Einstein gases. B first quantum correction to second virial coefficient. BII second quantum correction to second virial coefficient. B* reduced first quantum correction to the second virial coefficient. B*I reduced second quantum correction to the second virial coefficient. bo reducing parameter 2/3 n N 03 bJ, bJ, bJ defined by Equations (62), (63) and (64). I II C third virial coefficientandthird coefficient in Leiden expansion. C' third coefficient in the Berlin expansion. D fourth coefficient in the Leiden expansion. D' fourth coefficient in the Berlin expansion. x

NOMENCLATURE (CONT'D) E fifth coefficient in Leiden expansion and Young's modulus. E' fifth coefficient in the Berlin expansion, Ev error introduced into compressibilities due to incomplete evacuation of VII. F sixth coefficient in the Leiden expansion. F' sixth coefficient in the Berlin expansion, G seventh coefficient in the Leiden expansion, G' seventh coefficient in the Berlin expansion, g gravitational acceleration, gc unit conversion factor for ft. lbm, lbf, sec system, h Planck's constant and also height. 10 ~ intercept on a plot of Pi ITI Ni against Pj.'j i=l K defined by Equation (24), k ratio of outside to inside radius of the Burnett cells, Boltzmann's constant, L defined by Equation (83). I number of lead space volumes prior to expansion and subscript for last measured point of a partial run. M intercept on a plot of Pjl /Pj against Pj, m mass, number of dead space volumes after expansion, and molecular mass. n number of moles. N Avogadro s number, No equipment constant. N. defined by Equation (15). xi

NOMENCLATURE (CONT'D) P Pressure. Pm pressure due to mass on dead weight gage. R, R Universal gas constant. Dead weight gage resolution. r molecular separation. S slope on a plot of Po/Zo against No T temperature. T* reduced temperature, VI volume of large Burnett cell. VTI volume of small Burnett cell. Vcj The volume the gas in the extraneous volume would occupy if at the test temperature. V total volume of Burnett cells. V' Corrected dead space volume at zero pressure after expansion. Vo Corrected dead space volume at zero pressure prior to expansion. v specific volume. W weight of mass on dead weight gage. Z compressibility factor. Greek Symbols Pi ~ error in vj(Zj - 1) A (Po/Zo) error in (Po/Zo) due to error in No also total error in (Po/Zo) in Equations (20) and (118). A Vii volume of VI at pressure Pj minus volume of VI at zero pressure. 6 extrapolation error in Po/ZO e xii

NOMENCLATURE (CONT D) CE ~ depth of potential well. A deBroglie wavelength of relative motion. A* defined by Equation (56). Poisson s ratio. intermolecular potential. P density. ar molecular separation at which the intermolecular potential becomes zero, xiii

I. INTRODUCTION In the past fifty years much attention has been devoted to the determination of thermodynamic properties of pure substances, and to a lesser extent of mixtures. Such investigations are indispensable to the engineer designing separation and refrigeration equipment, and in fact for any application requiring the use of a thermodynamic analysis. These basic measurements of P-v-T behavior in addition to specific heat measurements are used to fix the values of other thermodynamic variables such as enthalpy, entropy, and internal energy. This can be done graphically, but the normal procedure is to fit an appropriate equation of state to the experimental data and proceed analytically. The virial equation of state, which is derived theoretically by employing statistical mechanics, makes it possible to determine some characteristics of the force field between molecules with the aid of accurate P-v-T data. Thus, in addition to serving the engineer, the collection of accurate P-v-T data also serves the scientist who is attempting to more fully understand the basic nature of molecular interaction and thus improve existing theories. Presently helium is used to cool magnets and in the low temperature operation of masers in the range from about 4 to 13~K and is being considered as a prime refrigerant at temperatures to 40K(49). As technology advances the construction of such refrigeration devices becomes more complex and in order to determine and improve their performance it is necessary to have an accurate and consistent set of properties for helium in the temperature range from 3~K to room temperature, -1

-2Neon, as it becomes more readily available, will become an important cryogenic fluid since it has a boiling point which lies between those of nitrogen and hydrogen. Oxygen falls partially in this range but due to the precautions that must be observed to achieve safe operation it is not suitable for many applications. On the other hand, neon is suitable for such applications as magnet cooling and cooling of infra-red detectors. Neon has also been used as an intermediate fluid in hydrogen liquification. Helium, like neon, is inert and is thus safe to work with. For both of these substances the existance of high pressure P-v-T data is very sparceo The discovery of superfluidity, the X-point and associated phenomena tended to divert attention from the collection of P-v-T data. Also the recent work of Do White(l08) indicates that the low pressure data collected for helium and used to determine the second virial coefficients is subject to question in the region covered by this research. In view of these considerations the research carried out by the author was initiated to extend the range of existing P-v-T data on neon and helium. In carrying out the research the Burnett Method, which is a relatively new method of collecting P-v-T data, was used for the first time at liquid nitrogen temperatures. A comparison of the data range of this research to existing data for helium is given in Figure 1o A similar comparison for neon is given in Figure 2. Figures 1 and 2 show that experimental data for neon and helium in the range from 700K to 1200K was quite limited prior to this worko

-3700 NEW DATA 600 _ EXISTING DATA 500400E a. 300200100 I 0 40 80 120 160 T(OK) Figure 1. Isotherms of Helium Investigated in This Research 300NEW DATA EXISTING DATA —200 410000 40 80 120 160 T (~K) Figure 2. Isotherms of Neon Investigated in This Research.

II. LITERATURE SURVEY A. Neon Neon was discovered by Sir William Ramsay and Morris William Travers in 1898 when they passed an electric discharge through a sample of gas prepared by distilling argon and observed a "blaze of crimson light'. Neon is a rare gas and only recently has the supply of neon exceeded the demand. As a result of the difficulties of purifying neon it is also expensive, which accounts for the relatively small amount of P-v-T data available for this substance. Since most of the P-v-T data available for the inert gases was collected at well known laboratories, it seems appropriate to cite the references consulted under the name of the laboratory where the data was collected. In quoting the pressure ranges for the references the highest and lowest pressures measuredare used, For any one particular isotherm the pressure range will lie between the limits quoted. 1. The Kamerlingh Onnes Laboratory, Leiden, Holland. Isotherms for neon for temperatures from -217.5 to 23~ C and pressures from 23 to 93 atm. were measured by H. K. Onnes and C. A. Crommelin(68) in 1915. This work was continued in 1919 by H. K. Onnes, C. A. Crommelin and J. P. Martinez(6) when isotherms of neon were measured for temperatures from -217.52 to 20~ C and pressures from. 21 to 93 atm. 2. The Physikalisch-Technische Reichsanstalt, Berlin, Germany. Work at this laboratory was carried out by Lo Holborn and J. Otto and consists of the following measurements: 4

-5a. Reference (32): Isotherms from 0 to 400~ C and pressures from 0 to 105 atm. b. Reference (31): Isotherms from -183 to 400~ C and pressures from 0 to 105 atm. c. Reference (30): One isotherm at -207~ C at pressures from 20 to 90 atm. Also V. W. Heuse and J. Otto(28) measured the 0~ C isotherm of neon for pressures from 0.4 to 1 atm. 3. The Van der Waals Laboratory, Amsterdam, Holland. Isotherms for neon were measured at temperatures from 0 to 100~ C and pressures from 20 to 500 atm. by A. Michels and R. 0 Gibson(60) in 1928. Thirty years later A. Michels, T. Wassenaar, and P. Louwerse measured isotherms for neon at temperatures form 0 to 150~ C and pressures from 24 to 2900 atm. 4). The National Research Council Laboratories, Ottawa, Canada. G. A. Nicholson and W. G. Schneider(63) measured isotherms for neon for temperatures from 0 to 7000 C and pressures form 10 to 80 atm. This reference is unique because the Burnett method was used to obtain P-v-T data whereas the previous references used either a piezometer or the constant volume method. The fact that the results agree within experimental accuracy shows that accurate P-v-T data can be obtained using the Burnett method, which is somewhat easier to use and eliminates the necessity of making accurate volume and mass measurements.

-6Other experimental P-v-T measurements for neon include the following a. Measurement of the 0~ C isotherm for pressures from 0.18 to 1.1 atm by F. P. Burt(9) in 1910. b. Measurement of the 0~ C and 100~ C isotherms at pressures less than 2 atm by Jo Oiski (67) in 1949. c. Measurements by W. Ramsey and M. S. Travers(78) of isotherms from 11.2 to 237.3~ C and pressures from 39 to 94 atm. The most recent papers concerning neon P-v-T data are theoretical in that an equation of state was fitted to existing neon data and used to compute the properties for neon in those regions where no data is available. The first of these works was carried out by Yendall(l02) in 1958 and consisted of fitting a twelve constant equation of state to the experimental data collected at Leiden in 1915 and 1919. The comparison of theoretically calculated values and experimental values was 1.055% maximum deviation for the Leiden points and 1.64% for those obtained by Michels and Gibson(60) in 1928. In 1962 McCarty, Stewart and Timmerhaus(59) presented a paper at the 1962 Cryogenic Engineering Conference giving compressibility factors for neon from 27 to 300~ K and pressures from 0.1 to 200 atm, The data were calculated using Yendall' s(02) equation of state which was fitted to existing neon data and additional reduced values calculated from data on nitrogen and argon using the theory of corresponding states An empirical correction based on the differences in the argon-nitrogen P-v-T surfaces was made for similar differences in the P-v-T surfaces of neon and nitrogen.

-7A later paper by McCarty and Stewart(58) repeats the results of the above mentioned paper using the Strobridge equation of state. The accuracy to which the equation of state fitted the experimental data was comparable to the accuracy of the data. This paper extends the results of the previous work to the liquid region and gives more consistent thermodynamic properties for the entire range. The above survey presents all the work done on P-v-T behavior of the vapor phase of pure neon. Other investigators have measured the critical constants of neon and its vapor pressure but these results are not of interest here. B. Helium The inert element "helium" was first discovered in 1868 when scientists observed a solar eclipse with spectrographs and found a bright yellow line near the sodium D line. Since that time helium has been the subject of many experiments and is the most widely studied inert gas. In 1942 Keesom(40) made an exhaustive literature survey on helium and published a book which discussed in detail the behavioral characteristics of helium. It would be to lengthy to cite all of the references available on helium P-v-T behavior and only those which overlap into the region of this work are given. As in the case of neon the major references will be cited under the name of the laboratory where the experimental observations were made. 1. The Kamerlingh Onnes Laboratory, Leiden, Holland. Isotherms of helium for temperatures from -258.82~ C to 100.55~ C and pressures from 10 to 67 atmospheres were measured by H. K. Onnes(69), (70)

-8in 1907 and 1908. Later in 1924 Boks and K. Onnes(73) measured isotherms of helium for temperatures from -258.78 to 20~ C and pressures from 19 to 59 atm. Isotherms for temperatures from 16.65 to 69.86~ K for pressures less than 1 atm were measured by van Agt and K. Onnes(93) in 1925. Keesom and Nijhoff5) measured isotherms of helium for temperatures of -183o07~ C and -201.52~ C for pressures from 3 to 8 atm in 1927. Also Nijhoff, Keesom and Illum(64) measured isotherms from -259 to -103.6~ C at pressures of 1.5 to 14 atm in 1927. These latter measurements by Keesom and co-workers were made at low pressures in an attempt to more accurately determine the second virial coefficient for helium, 2, The Physikalisch-Technische Reichsanstalt, Berlin, Germany. Measurements were made at this laboratory by Holborn and Otto over a wide range of temperature. These measurements include the following: a. Reference (33)'Measurements for temperatures from -183 to 50o C and pressures from 0 to 105 atm in 1924. b. Reference (34): Measurements for temperatures from -183 to 400~ C and pressures from 0 to 105 atm in 1925. c. Reference (35) ~ Measurements for temperatures from -258 to -183~ C and pressures from 0 to 105 atm in 19263 Of particular interest to this work is the work of D. White(l08) of Ohio State University. White measured the second virial coefficient of helium over a temperature range from 20 to 300~ K. The results of his work agree favorably with older work except in the temperature range from about 60 to 150~ K where his results are considerably higher than

-9the older values. A comparison between the results of the present work, that of White, and that of previous investigators is given in Chapter 5, section C. Many other references on helium could be cited, but the range of P-v-T measurements for these is outside of the range of temperatures of this work. A complete listing of P-v-T references to 1942 can be found in Keesom(40) and to 1961 in Cook(14) C. The Burnett Method The Burnett method of obtaining P-v-T data for gases employs a technique whereby no accurate volume or mass measurements are required. The method itself will be described later in more detail. E. S. Burnett(8) first introduced his method in 1936 and from 19536 to 1949 many references appeared in the literature (see for instance(1)' (11) (89) (104)) describing the use of the method by natural gas companies. The method was first used as a research tool by Schneider and co-workers13)'(48)(82)'(83)(99) (103) at the National Research Council Laboratories in Canada. Their studies include work on helium. carbon dioxide, argon and neon. All of their investigations are above 0~ C. Considerable improvements over the industrial models of the Burnett apparatus were introduced and the results obtained are as accurate as those obtainable by other methods. The Burnett method has also (26),(27),(76),(77), (97) been used extensively at the University of Pennsylvania 6,27)'(76 (77)' (96) where the primary emphasis has been on the study of gaseous mixtures. Other investigators who have used the Burnett method successfully are Silberberg, Kobe and McKetta(85),(86) (87); Watsen(97); Crain(l5); and Canfield( )and Mueller at Rice University.

-10D. Related Literature Many references were consulted during the course of this research regarding design, theory, temperature control etc. These references are listed in the bibliography and will not be discussed here.

III. THEORY A. Burnett Method The Burnett Method is an experimental technique introduced by E. S. Burnett8 in 1936 which makes it possible to obtain accurate P-v-T data for pure gases and gas mixtures without having to determine the volume and mass of the test gas at each experimental point. This method of collecting (10>,(1~>,(~5>,( 44), (62) data has been used by several investigators ),(1),(5)(44),(62)(82) (85) in recent years over a wide range of pressures and temperatures. In the ideal case all of the gas under observation is held at the same fixed temperature and is expanded in a series of steps from some initial pressure to some final pressure. After each expansion the pressure is measured and with the aid of analytical techniques to be discussed below it is possible to compute the compressibility factor P.v. z. - j (1) R T for each of the measured pressures. Knowing Z and P for a series of points along an isotherm makes it possible to determine any other thermodynamic property for that temperature by fitting the experimental data to an appropriate equation of state and calculating the desired property. In practice it is not always possible to maintain all of the gas at the same temperature and corrections to the method for such "dead space volumes" must be introduced. To implement these corrections it is necessary to accurately measure the dead space volumes and to have a fairly accurate -11

-12determination of the total volume of the cells. In addition, if the method is extended to high pressures, corrections for variations of the volumes of the Burnett cells with pressure must be included. In the following derivation the existence of dead space volumes is assumed and the correction for variation of the cell volumes with pressure is included. INLET VALVE EXHAUST VALVE EXPANSION VALVE I - CI lI Figure 3. Schematic Illustrating the Burnett Method. Consider the two volumes VI and VII shown in Figure 3. To initiate a series of measurements V is evacuated and then charged I

-13with the test gas to some initial pressure P. After measuring this pressure VII is evacuated, the test gas is expanded from VI to VII and the new pressure P1 is measured. After closing the expansion valve, VII is again evacuated and a second expansion to a pressure P2 is carried out. This process of evacuation, expansion and measurement of the pressure is continued until a low pressure is reached. The experimental data consists of the isothermally measured pressures generated by the above procedure. To show how the experimental compressibility factors are obtained from such data let it be assumed that the Burnett cells and the portions of the inlet and exhaust lines within the dashed boundary in Figure 3 are held at the test temperature. The volume of gas in the valves and connecting lines outside of the dashed boundary is assumed to be at some temperature different from the test temperature. To simplify the analysis it will be assumed that the volumes VI and VII include the volumes of the lines within the dashed boundary. Because of its frequent appearance the quantity, VI + VII VT = - (2) VI VI evaluated at zero pressure, is given the symbol No and called the equipment constant. Equation (1) written for the condition with VI having a pressure P.j- and VIi evacuated is P (VI + A V ( ) + V( l)) () nA~n-1 ~- R T nj"- RT

-14where A VI(j_l) is the change in VI due to pressurization and Vc(jl) is the volume the amount of gas in the lines and valves up to the expansion valve would occupy if it were at the test temperature. After expansion to the new pressure Pj, Equation (1) written for the same amount of gas is P (VI + VA + AV + V () Z. = r I II ______ ~2 ______ ^ ^ (4) nj-l RT Dividing Equation (4) by Equation (5) and rearranging gives VI + VII A Vij VII A V II V ) (V I. ~Q+ VII CJ Z P. VI V V VI V V zJ1 Pj- + I(1-l) /+ c(j- ) VI VI For the purposes of analysis, it is assumed that the Burnett cells are thick walled cylinders with the same ratio (k) of outside to inside radius. For such cylinders the Lame formula for volume dilatation can be used A VI! P. [ k2+ 1 - 2 — I J [12( -+l ) [\ + -l2 (6) VI E k2- 1 k2 - where E is the modulus of elasticity of the cell material and p. is Poisson's ratio. For cells of different volume but with the same value of k and made of the same material Equation (6) shows that A VI A VII () AV I11 (7) I II

-15for the same change in pressure. Making use of (7) and (2) in (5) gives the result 1 + +V- \ z = 1 _______________________ (8) Z.l PJ A V VTc (8) Zj-1 Pj-1 + +) VI VT For m dead space volumes each at a different termperature, the corrected volume is m Vci cj = ZjT Z (9) i=l Zji Ti where Vci is the volume of the i-th dead space, Z. is the compressibility factor of the test gas at pressure Pj and temperature T, and Zji is the compressibility of the test gas at pressure Pj and temperature Ti. Rearranging (8) and taking the limit as the pressure approaches zero gives 1 + v lim P. T lim j-1 = No lim (lo) p -O P j p - No Vc(jl) J 1 +. VT A Vi_) since A VIj/VI AV - V( ) --- 0 and Z -> 1 as P - 0. From (9) it can VI be seen that as the pressure approaches zero m T cj T z Vci/Ti V (11) i=l

-16and Vc(jl) Vci/Ti = Vo (12) i=l In view of (11) and (12) Equation (10) can be written as Vco / 1 + — lim P = N- (13) urn P* M (13) P ->O Pj N V 3 0 1+ o o c \ VT / From (13) it can be seen that a plot of P j_/Pj against P. when extrapolated to zero pressure gives the numerical value of M. Knowing M makes it possible to compute No from M N0 + — M —-- - (14) V V 1 + -' M co. VT VT Defining the quantity Nj as L + V\ 1 + + - VI V N. =N i I (15) 1 A VI(jl) + NVc(.-l) VI VT and substituting into (8) gives Z. P. j-1 j-l

-17By applying Equation (16) repeatedly for the first, second, etc. expansions the following result is obtained z zz. p Zi Z2 Z Pi P2 Pj Zo Z1 Zj-1 o PI Pj-l 1 2 o 1 J-l o 1 j-1 or i PO P. [ N. = z. (17) J Z=1 1 Taking the limit of (17) as the pressure approaches zero gives J P lim Pj NI Ni = (18) P -X 0 i=l Zo J Equation (18) shows that a plot of Pj I Ni against Pj i=l when extrapolated to zero pressure yields the quantity P/ZO, which is called the fill constant for a run. Knowing Po/Zo and No the values of the compressibility factor at each of the other experimental points can be calculated from Equations (6), (9), (15) and (17). It should be noted that the procedure described above implies a prior knowledge of the compressibilities to be determined. To circumvent this problem an initial guess at the values of the unknown compressibilities is made and the error is then iterated out by repeating the calculations several times. B. Treatment of Partial Runs Because of the nature of the Burnett method the high pressure points are widely spaced and to more accurately determine the isotherm at high pressures a partial run is desirable. To reduce the data for such a run

-18it is necessary to assume that the value of No as determined from a prior complete run at the same temperature is the same for the partial run. In addition, the data for the complete run must be used to determine the value of Z for the last measured pressure of the partial run. With this information the fill constant for the partial run can be determined from P P a o = IE Ni (19) Zo Z i=l1 and the compressibilities can be determined in the same way as for a complete run. It is apparent that any error in the values for the complete run is carried over into the determination of the values for the partial run. C. Discussion of Errors in the Burnett Method It can be shown (see Chapter V section C) that the total error in the compressibilities calculated by the method outlined above is given by A Z. A P. A N. A(Po/Z) Z\ -= X+j + + A T (20) Z Pj Nj Po/Zo T Temperature and pressure can be very accurately measured so that the first and last terms on the right in Equation (20) are not critical. The second term in Equation (20) shows that the accuracy in the compressibility factors decreases as the number of expansions increases. It is obvious that the error in N. must be kept to as small a value as possible. The error in N. is a result of errors introduced by inaccurate correction J

-19for volume dilatation and dead space corrections and the error in No For a properly designed system the errors introduced by inaccurate correction for volume dilatation and dead space corrections are small compared to the error that can be introduced into N. due to error in N. The j o error in No in turn is primarily due to the error in M. The magnitude of the error in M depends on the lower limit of the pressures measured and on the accuracy of the extrapolation to zero pressure on a plot of P j_/Pj against Pj. For a system with no dead space volumes M is equal to No and can be determined quite accurately by making a calibration run using helium. For a system with dead space volumes an accurate determination of No by the above method is possible only if the total volume and dead space volumes are very accurately known and their temperatures are accurately measured. The third term in Equation (20) shows that the accuracy of the compressibilities are directly proportional to the accuracy of the determination of the intercept on a plot of Pj il Ni against Pj To accurately determine Po/Zo the value of No must be precisely known, the low pressure data must be taken with great care, and care must be exercised in choosing the degree of the equation used to fit the low pressure data. Incomplete evacuation of VII also introduces error into Z but this error can be shown to be negligible for the normal condition of evacuating VII to pressures less than 50 microns of mercury.

-20It is extremely fortunate that the errors in No and PO/ZO can be detected and eliminated by the technique developed in Chapter III, section E. Thus, the prime sources of error in the Burnett Method can be restricted to errors in pressure, temperature, correction for volume dilatation, and corrections for dead space volumes. D. Review of Present Data Treatment Techniques Data collected using the Burnett method can be reduced by employing analytical approaches as described by Silberberg, Kobe and McKetta(86) However for systems with dead space volumes such analytic techniques would be difficult to apply. Recently Canfield, Leland and Kobayashi proposed a method of determining N using only the experimental results for the gas under investigation, thus eliminating the necessity of making calibration runs. The principal assumptions of this method are that a plot of v.(Zj - 1) against l/vj will be linear over some density range pi to p2 that experimental points for densities less than p exhibit large non-random deviations from linearity due to the physical nature of the dead weight gage and that the error introduced in extrapolating a plot of Pj il Ni against P. to zero pressure does not significantly affect the value of No. References (10) and (37) indicate that an incorrect value of No will cause a plot of vj(Zj - 1) against l/v. (virial plot) to become non-linear after the fashion indicated in Figure 4.

-21CORRECT No. No TOO SMALL ~I l vj(Zj-l) / / No TOO LARGE l/vj Figure 4. Effect of Varying No According To (10). In attempting to employ this method it was found that it was not possible to select a unique No and that the extrapolation error in PO/ZO influenced the choice of N. This discovery lead to the realization that the correct No could not be found unless the extrapolation error in PO/ZO could be eliminated. It is also apparent from the nature of the equations for reduction of the data that a small error in No creates an additional error in Po/ZO of considerable magnitude. The mathematical analysis presented below shows that errors in Po/Zo and No both influence the nature of the non-linearities on a virial plot. The analysis also shows that the error in No and the extrapolation error in Po/Zo combine in such a manner that a virial

-22plot exhibits non-linearities very much different from those indicated in Figure 4. An important conclusion that can be drawn from the analysis is that the non-linearities in the low pressure points are not due to the physical behavior of the dead weight gage. In fact, the method proposed here assumes that the percent error in the pressure measurements is essentially constant with pressure. The method shows that the nonlinearities exhibited by the low pressure data on a virial plot can be used as a powerful tool for eliminating small errors in No and Po/ZO E, The Exact Method for Determining No and Po/ZO To demonstrate the assertions made in section D above, let it be assumed that the true values of No and Po/ZO are known so that the true values of the compressibilities are given by Equation (17). Let No be varied by an amount A No causing PO/Zo to change an amount A (Po/Zo). Let J it also be assumed that the intercept I on a plot of Pj il Ni against Pj differs from PO/Zo by an amount A (Po/Zo) + 6 where I = Po/Zo + + A (Po/Zo) (21) The error in Zj due to the above errors can be computed from Equation (17). To a first approximation this error can be derived from Pi + j N j P. ( j -) I N+ No / i=l - --- +P= Zj + A Zj (22) P/Z + + + A (Po/Zo)

-23Solving for A Zj in Equation (22) yields A Zj = K Zj (23) where A No 5 + A (Po/Zo) No Po/Zo K = (24) 8 + A (Po/Zo) 1 + Po/Zo From experience it is known that PO/Zo varies approximately linearly with No, so that A (Po/Zo) = S A No (25) where S is the slope on a plot of Po/Zo against No. The slope S is always positive and has a magnitude which depends on the number of points taken for a given run. Substituting (25) into (24) gives A No 5 + S A No i\oo Po/Zo K = (26) 5 + S A No 1 + Po/Zo The error in vj brought about by error in Zj is given by A v.= Kvj (27) J a so that the error in vj(Zj - 1) can be shown to be = vj (1 + K)[Zj(l + K) -1] - vj(Zj-l) = K vj(2Zj - 1) (28) a~~~~~~~~~~~~~(8

-24where it has been noted that K < < 2K for small errors. The true nature of the effect of errors in No and PO/ZO on a virial plot can be demonstrated by plotting P against l/v. (3 - plot). The effect of errors in l/vj will not be considered here. These errors are small and would shift the abscissa of the points slightly but would not significantly change the shape of the curves on a ( - plot. The consideration of errors in No and Po/Zo can be divided into the following cases: Case 1. No error in No For no error in No Equation (26) becomes K _ (29) Po/Zo + 5 Equation (29) in conjunction with Equation (28) shows that P is positive for 5 negative (I < Po/Zo) and negative for 5 positive (I > Po/ZO) with the exception of the region for which Zj is less than 0.5. For a substance such as helium (or a typical substance for temperatures greater than the Boyle temperature), the variation of P with l/vJ is shown by the curves labeled C1 in Figure 5. A P - plot for a typical substance for isotherms with a region where Zj is less than 0.5 is shown in Figure 6. Case 2. No Extrapolation Error in Po/Zo For the case where 5 = 0, Equation (26) becomes S No A No K =- (30) 1 + S /A P0 /Zo

C3 -25 - C3 C3 Figure 5. B-Plot For Helium. C3 02 P g So Figure 6. -Plot For A Typical Substance For Isotherms With A Region Where Z < 0.5.

-26Examination of (30) and (28) indicates that there are three places where the error will be approximately zero. For isotherms which have values of Zj < 0.5, the error will be close to zero at Zj = 0.5, which occurs at two places and for all isotherms the error will be near zero for S No j po/Z0 (31) Po/Zo S No Equation (30) shows that for j < / and A No negative o/Zo (No too small), P will be positive except for the range where Zj is less than 0.5. For j > S No/Po/Zo and A No negative, the sign of P is reversed. A similar argument applies for A No positive (No too large). The curves labeled C2 in Figures 5 and 6 show the variation of P with l/vj for the case where 8 = 0. Case 3. Error in both Po/ZO and No For this case, which is the normal situation, a 3 - plot takes on appearances which result from the four possible combinations of the curves for Cases 1 and 2. The true nature of the curves will depend on the relative magnitudes of 6 and A No. The general appearance of the curves is shown in Figures 5 and 6 for the same 6 and A No which gave rise to the Case 1 and Case 2 curves. It should be noted that Figures 5 and 6 are drawn to indicate the trends predicted by the equations derived in this section and thus may be distorted somewhat from the appearance of actual experimental data. The position of the Case 5 curves is a function of the relative magnitude of A No and 6. In general they will be positioned above and below

-27the axis as indicated in Figures 5 and 6, but for the cases where 6 is the governing error the generalization does not hold. A 5 - plot for the 100~ K isotherm of helium as measured in this research is shown in Figure 7. The plot was generated by imposing the errors indicated in the figure on the final experimental results which deviated from linearity with a standard deviation of 0.647 x 10-3 in vj(Zj -1) for densities less than half the critical density. Figure 7 shows the same trends as Figure 5 and thus demonstrates that the theory presented here gives a close approximation to the true nature of the effects of errors in Po/Zo and No Figure 8 is a P - plot for the 80~ K isotherm of Neon. This - plot for neon shows that the isotherms of a substance for temperatures less than the Boyle temperature will be S-shaped on a virial plot if incorrect values of No and Po/Zo are used. The development presented here shows that to obtain the true values of Po/Zo and No the deviation of the low pressure points from linearity must be minimized with respect to these parameters. This is equivalent to finding the minimum point of a surface generated by plotting the deviation from linearity as the elevation above the No - Po/Zo plane. Due to the amount of work required to find such a minimum point for Burnett data the tsk can be performed readily only with the aid of a digital computer.

0.02 =-0.03, ANo =-0.00003 0.0|1~ S -- ~8=o, ANo= —0.00003 0.01 — o- ~ 8=0.03, ANo =0.00003 I ~3 E 8=-0.03, ANo=0 0 ~ 0 |0.01 0.02 0.03 0.04: | _ —--— ~o: S=0.03, ANo=0 - -o~. Ol ^ -: 8=-0.03, ANo=0.00003 -0.01 -0.0.... S=0, ANo=0.00003 0~~ 0..o._e._. —O~* 8=0o03, aNo=0.00003 -0.02 I/Zj ( gm/cc) Figure 7. P-Plot For 100~ K Helium Isotherm (6 in atm).

12 ~ Q-SOC3 8=0.02 atm C3 9- ~ ANo= 0. 00005 C2 1 -O — I I ____ I 0.02 0.03 00.03 0.2 0.3 0.4 0.5 F K Neon Isotherm. 6 _ C 2 C3 912I 0.02 0.03 0.04 0.05 0.10 0.2 0.3 0.4 0.5 1/'~ (gm/cc) Figure 8. b-Plot For 80~ K Neon Isotherm.

-30As opposed to the technique introduced by Canfield, Leland and Kobayashi(lO)the technique introduced here permits the reduction of the experimental data with no chance of bias on the part of the investigator. The values of No and Po/Z are unique using this method and the values of the second virial coefficient are obtained from the low density points. This technique eliminates the necessity of resorting to the theory presented in Reference (37) to obtain accurate virial coefficients from Burnett data. Finally the overall accuracy of the final data is greatly improved due to the simultaneous minimization of error in No and PO/ZO In using the technique proposed here for eliminating errors in N and Po/Z the natural curvature of an isotherm on a virial plot plays an important role. If experimental points above the linear section of an isotherm are included in the linearization erroneous values of No and Po/Zo will result. In addition it is apparent that the value of No will be affected by any other errors which introduce non-linearities into the isotherms on a virial plot. Thus, for a system with dead space volumes, exact repeatability of No cannot be expected. In the light of the above remarks it is apparent that for a system with dead space volumes No is not strictly an equipment constant as defined by Equation (2). For systems with dead space volumes No assumes the additional role of a linearizing factor for the low pressure data points.

-31F. The Virial Equation of State The routine of fitting experimental P-v-T data in terms of a series expansion of the form - ^A + _+....) _ = A + B + C +,,,, (32) RT v v2 was first introduced by H. K. Onnes(71) in 1901. The number of terms necessary in (32) to obtain a satisfactory fit to experimental data increases as the density increases. This form of equation of state attained special significance when developments in statistical thermodynamics made it possible to arrive at (32) from a purely theoretical standpoint. Statistical thermodynamics shows that (32) can be obtained from the relation between the pressure and the partition function P =k T n ZN k T bd () where P is pressure, k is the Boltzmann constant, T is temperature, Z is the partition function and V is the volume. The factor d is called the active number density and is equal to the product of the activity and the number density d = -a (34) V where N is the number of molecules of the system. The activity as used here is the ratio of the fugacity at a given state to the fugacity at the standard state. As shown in Chapter 3 of Hirschfelder, Curtiss and Bird(29)

-32d can also be expressed as 00 N - z P. - (55) NX (35) d = V e The b~ in (33) are cluster integrals defined by b = (rl, r2r)drldr2...)d 2, dri (36) V. The U functions are in turn defined in terms of the Boltzmann factor by WN(rN) = n U (rX) (37) (Z ~mp = N) The summation in (37) is over all possible divisions of N molecules into mi groups of I molecules with the restriction that Z Im =- N. The superscripts indicate sets of N and X molecules respectively. The Boltzmann factor can also be written as WN(r N) = e - (rN)/kT (38) where 0 is the potential energy of the system of N molecules. Expansion of (38) yields the following relations Wl(r) u(Pi) == 1 (39) W2(ri,.) = U2(rrj) + U(ri) U(r) (40)'3~~~~~~~~~~~~~~(0

-33which gives the Boltzmann factor for one, two, etc., molecules alone in volume V. Inverse relations from (59) and (40) expressing the U functions in terms of the Boltzmann factors are easily obtained. Thus Ui(ri) = W(ri) = 1 (41) U2(rrj) =W2(rirj) - Wlr) W(rj) (42) etc. From (36), (38), (41) and (42) the cluster integrals can be written as 1 b= f d r = 1 (43) V 2 - ^ /I ^f r 2f 1 - W2 W2(72 d 1d rl 2 00 -Tp(r)/kT f. (e (/ - 1) 4 I r2dr (44) 0 etc. Returning now to (35) and noting that the Pi are functions of the cluster integrals given by PI = 2 b2 P2 = 2b3 -6b (45) etc.

-34Equations (35) and (33) can be combined to give the following form for the virial equation of state. = - i Pi ( (46) NkT i=l i +1 V or 1Pv = 1 i N ~i (47) RT i=l (i+l)vi where N is Avogadro's number. Expansion of (47) gives P - N P- 2 N _-.., (48) RT 2 v 3 v2 Equation (48) is normally written in the form of Equation (32) and the coefficients B, C, etc., which represent the deviation from ideal behavior when interactions between two, three, etc. molecules become important, are called virial coefficients. By direct comparison of (32) and (48) and with the aid of (43), (44) and (45) the virial coefficients A and B are A = 1 B N- = - 2 = - -- (e -( / 1)41r2dr (49) 2 2 o In order to be able to evaluate (49) the intermolecular potential cp(r) must be known. The most widely used intermolecular potential is the Lennard-Jones 6-12 potential, which is p(r) = 4 e (- )] (50) _r V

-35In Equation (50), a is the intermolecular distance at which the potential passes through zero and e is the depth of the potential well. These quantities are determined from experimental data on the second virial coefficient, from viscosity data, or heat of sublimation data. The determination of the intermolecular force parameters from P-v-T data requires that the data be very accurate. Since helium and neon both exhibit quantum deviations in the range of the experimental data covered in this research, the classical development of the virial equation as presented above must be modified. In order to make the previous development applicable in quantum mechanics it is necessary to replace the Boltzmann factor by the Slater sum and the partition function by the quantum mechanical partition function as shown in Chapter 16 of Hirschfelder, Curtiss, and Bird(29). The Slater sum is given by WN(rN) = N: x5N z pq qN)*(r ) N) eH/kT (51) q where the functions pq(r N) are a complete orthonormal set of eigenfunctions, 2 = h2/2TmkT and H is the quantum mechanical Hamiltonian operator. By replacing W2(r1,r2 ) in the equation just prior to Equation (44) by the Slater sum and noting that Wl(rl ) W2(r2 ) = 1 the second virial coefficient using quantum corrections becomes N - B =- - - [W2(rl,r2 )- 1] d rld r2 (52) 2V

-36where use has also been made of Equation (49). After subtracting the second virial for ideal behavior from (52) and writing the Slater sum for two particles and substituting this into (52) the following expression for B is obtained B = cBc+ h Bi+( BI+ + h2 +/2 Bo(T) (53) ci m m B in which Bc~ is given by (49) and ('\oo -cp(r)/kT 2 2 BI = 21 N. 1 e dfp e dJr2dr (54) k4812 k3 T3 dr dr2'<J ( + 0 -T(r)/kT 2 W 2 B II 2H N\1920AH4 T4 \er 2 3 4 + - + 2 — 2- r2dr (55) r dr/ 9kTr dr 36k2T2 dr The term (h2/m) 3/ Bo(T) which is the second virial coefficient for ideal behavior is added for Fermi Dirac gases and substracted for Bose Einstein gases. The solution to (49), (54) and (55) has been carried out by J. DeBoer and A. Michels(21) for the Lennard-Jones potential. The method used involves the expansion of B, B and BII in rapidly converging series in powers of A* where A* = - = -h (56) a a ^me

-37The quantity A is the deBroglie wavelength of relative motion of two molecules at a temperature where the mean relative kinetic energy is c, and is an indication of the significance of quantum effects. The result is B* = [B A* B + B* + A B* +...]- A*3 B* (57) ~ I II + o wherein 00 B* = Z bj T* -( + 6j)/12 (58) cQ j=o B = Z bj T* -(13 +6j)/12 (59) I j=o I B* = bj T* -(23 + 6)/12 (6) II Ij=o TI B* = T* -/2 (61) and the coefficients b, bj and bJ are given by T ~ II i = - 2 + r 6_ 3 (62) 4 j 12 b 2J ( 36._ 1 2 3/ (6) 768 n2 / j \ 12 bj 767 + 4728 j + 3024 j 2 26 + 1() (64 91520 I / 12 491520 H4 j' 12

-38The reduced virial coefficient given by (57) is related to that of (53) by B* = B /bo (65) where b = 2/5 1I N a5. The quantity T* is the dimensionless temperature given by T* k T (66) Use is made of the developements of this section in Chapter VII, section A where the determination of the intermolecular force parameters using the results of this research is discussed.

IV. EXPERIMENTAL APPARATUS AND PROCEDURES A. Experimental Apparatus 1. General Description The experimental apparatus used consisted of the Burnett Cells suspended in the lower portion of a cryostat insulated with multilayer insulation. The temperature of the cells was determined with a platinum resistance thermometer which was embeded in the Burnett cell block between the two gas chambers. The pressure of the test gas in the cells was measured by means of a Ruska dead weight gage with the null point determined with a differential pressure diaphragm. The Burnett cells were connected by means of appropriate valves and tubing to a piston type pressure intensifier and to a vacuum system. A differential thermocouple referenced to the location of the resistance thermometer was located in each cell. In addition, two other differential thermocouples were placed on the cells, one with its junctions on the top and the side and the other with its junctions on the bottom and the side of the cells. Resistance heaters were located around the cell and connected to an automatic temperature controller which was actuated by a wafer type resistance thermometer attached to the cells. Cooling of the cells when required was achieved by controlling the vapor pressure of the bath fluid. Uniformity of the temperature of the bath was achieved by means of a magnetic stirring arrangement. -39 -

-402. Design of the Burnett Cells The design of the Burnett cells was dictated by the following criteria a. Ease of construction. b. Temperature range of the data. c. Pressure range of the data. d. Desired accuracy. These considerations will be discussed below. a. Ease of Construction. Some consideration was initially given to making the cells spherical, but due to the difficulty of machining spherical cells this idea was discarded in favor of cylindrical cells. This decision caused the cells to have greater mass for the same total volume which is detrimental in terms of temperature response. The choice of thick walled cylinders makes it possible to determine the volume change of the cells with pressure by means of the Lame formula for thick walled cylinders. The Lame formula is AV P [ k2+ 1 ) 1 - 2p A ( -- -E- 2 ++ + - (67) V E k2- 1 k2 1 [jk2+l k2~l}

-41With the values of P, E and [i being dictated by the type of material and. the pressure range of the data the only variable which can be chosen to help the design is the ratio of the outer to the inner radius. The larger the value of k the less will be the volume dilatation for a given pressure. A plot of AV/V/P/E against k showed that a value of k = 3 was sufficiently large to minimize volume dilatation and a value of k = 2 was as small as one should go. To minimize the mass of the cells and thus achieve better temperature response the design value k = 2 was chosen. The length of the cells was dictated by the dimensions of the cryostat and by necessity of achieving uniform temperature. With the total volume determined by consideration d, the inside radius was arbitrarily set at 3/4 inches and it was found that the length was not excessive. The wall thickness of the cells was checked and found to conform with the ASA code for pressure piping. Each of the three stresses in the cylinder wall was checked and the tangential stress at a pressure of 1.5 times the working pressure of 680 atm was found to be about half the yield strength of type 347 stainless steel. Type 347 stainless was chosen for its high yield strength and because of the necessity of avoiding structural metals which become brittle at low temperatures.

-42b. Temperature Range of the Data, At the outset of the research it was proposed that the system be designed to cover the temperature range from 20 to 100~ K with the range from 70 to 100~ K being obtained by immersing the cells in liquid nitrogen and the lower range being obtained by using an adiabatic calorimeter technique with the system immersed in liquid hydrogen. The upper range offers no problems other than the mass be kept small enough so that the system will come to temperature equilibrium with the bath in a reasonable amount of time. For the lower range it is necessary to keep the cell mass to as small a value as possible to minimize the amount of liquid hydrogen to be used and to achieve good temperature response between the heater shield and the cells. It was not possible to satisfy this consideration and at the same time satisfy considerations c and d.

-43c. Pressure Range of the Data. The maximum desired pressure of 10,000 psi necessarily forces the cells to be quite bulky for a given total volume. Also a margin of safety is necessary to assure that the cells will not yield at some point and make the use of the Lame formula invalid, or rupture creating unknown hazards because of the large quantity of liquid hydrogen surrounding the cells. One of the serious shortcomings of the Burnett method is the spacing of the data points for a given isotherm. At high pressures the data points are far apart and at low pressures they become very dense. The larger the ratio of the total volume to the volume of the larger cell the greater the separation of the points at high pressures. On the other hand a small cell constant forces one to take a large number of data points which reduces the accuracy of the final results of the method, and lengthens the time necessary for completion of an isotherm. It was found after cal-eful consideration that a cell constant of 1.2 would be suitable for the proposed pressure range. Even with this value it was necessary to make a short run on the high pressure end of the isotherms to fill in the gaps.

-44d. Desired Accuracy. Because it was found necessary to locate the valves and the differential pressure diaphragm, which separates the test gas from the oil of the dead weight gage, outside of the cryostat it was impossible to construct a system with no dead space volume. Since the dead space volume is one of the governing errors when using the Burnett method it is essential to keep it as small as possible relative to the total volume of the system. In view of the work of previous investigators it was decided to hold the dead space volume to about 0.2% of the total volume. After careful design of the valves and selection of tubing size it was found that the dead space volume would be about 0.5 cm3. This made the design value of the total cell volume 250 cm3 which was arbitrarily increased to 260 cm3. The initial pressure measurement system which involved the use of a mercury U-tube proved to be unfeasible and had to be replaced by a differential pressure diaphragm. This increased the dead space volume to about 0.8 cm3 which is 0.3% of the total volume. With all of the above considerations the Burnett cells shown in Figure 9 were arrived at. Later it was decided to loosely clad the cells with 1/4 inch copper to help in the attainment of uniformity of temperature. A view of the exterior of the cells suspended in place is shown in Figure 10. The volumes of the cells after assembly and insertion of the thermocouples turned out to be VI = 211.5 cm3 V = 41.1 cm3 II

RESISTANCE THERMOMETER HOLE \J1 Figure 9. Cross-section Drawing of the Burnett Cells Figure 9. Cross-section Drawing of the Burnett Cells.

.......... ~ ~ ~ ~ rl::~all: a.o: ~~........................:j::::::::~ ~.....................~ ~ ~ ~ ~ ~ ~~ j.~F.............................S~a ~.......................I....................: -::: i: j::::-::::.......~i..........~~~~~~~~~~~~~~~~~~~~~~~~~ili.......................................................~ ~~~~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~i.........ii liii:XiX~.~i::X'':X~~~~~~~~~~~~~~~~~~~~~~~i 1~~~~~~~~~~~~~~~18~~~~~~~~~~:itm Modifi::ca:ti::::: r:! Mina n. iBI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.....:i::::::,~~~~~~~,:it~~~~l.....:,x......... ~~~:a:::::~~~~~~~~~~is::::::a:.:- ~ ~ ~ ~ ~ ~...... ~~~~~~~i~~~~~~~~~~~~.......... Figure Tt.O, ~~~~~~~~~e~~o~~l~~ogsaljh 09 tX~~~~~~~~~ze B~~~~tsneZ;~I; Cel~~~~.......

-47Thus the total volume is 252.6 cm3 and the cell constant is roughly 1.194 for the temperature range of thisresearch. 3. Pressure Measurement System The pressure was measured with a type 2400.1 Ruska dead weight gage. The gage was calibrated by intercomparison with Ruska Instrument Corporation's master gage No. 7544 which in turn was calibrated by the National Bureau of Standards. The gage is equipped with two pistons, one covering the range from 30 to 12,240 psi and the other the range from 6 to 2428 psi. The high range piston was used to measure pressures from 2000 to 10,000 psi and the low range piston was used for pressures below 2000 psi. The change of pistons was made to take advantage of the higher degree of accuracy attainable using the low range piston for pressures below 2000 psi. The oil of the dead weight gage was separated from the test gas by a Ruska model 2416.1 differential pressure indicator. The location of the differential pressure indicator (DPI) resulted in an oil leg of 4.882 inches, which was determined with a K & E precision level. The gas side of the DPI was connected to the inlet valve to the large cell with 100.75 inches of.006 I. D. capillary tubing. This connection was made just below the seat so that the large cell was at all times in communication with the DPI. The pressure as determined by the mass placed on the dead weight gage and the oil leg is subject to several corrections which are discussed in Chapter V, section A. The dead weight gage is shown in Figure 11l

.........8. 41P) bi) P-1....&.....N. ~~~~:~~~i~~ ~Lrr::niB......~~~~~~~~~............. S ~~ j....................t../ ~~~~: ~~ ~ i ('1~i::i ~ ~ ~ ft v&~ ff~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~:::: j::: j::.:::-:::::::::::: —::::::::~:: ~:::: I-:::i::.:::::::::::~:::::::::::::::::::::::::::::-:'I Ij ~j:1::::::::-:::-:::::ij:;:::::::-::a:::::::::i -:-:~::~:::::::::::::::::::::::::::::::::::::::::::::i:::'3)::I:-:li-'-::: i:::l_:::::-:-:: —::'::::::::::':::l:::-::::::::::::i-~1 < I:~~ ~ -::i:::i:i_::'cii-'::::::::::~,:'l,':::::::::-i::~: i::::: —i -:: -:::I::I:I:::~~~~~~~~~~~~~~~~~~~~~~~~~~~:i:::::-::::- -::::~ ~ ~ ~ ~ ~ ~ ~ ~ ~~0 i: i:::::::::: —: n j- ~ ~ ~ ~ ~ ~ ~~~~~~~a~~~

-494. Temperature Measurement The temperature was measured with a Leeds & Northrup No. 8164 platinum resistance thermometer which was placed in a 7/32 diameter hole drilled in the cell material between the two chambers as indicated in Figure 9. The thermometer was placed at a depth such that only the wire leads showed outside of the cell. Number thirty copper wire was attached to the thermometer leads and wound once around the cell before being joined to No. 24 copper wire which was used as lead out wire to a terminal strip at the top of the cryostat. To connect the leads to the Leeds & Northrup type G-2 Mueller bridge, which was used to measure the resistance, about 20 feet of No. 14 copper wire was used. The thermometer was calibrated by the National Bureau of Standards to five significant figures. The Mueller bridge is capable of reading a change of 0.0001 ohms which is roughly 1/1000 of a degree Kelvin. All temperature measurements were made with the resistance within +.0003 ohms of the value determined by the calibration for the given temperature. Four copper-constantan difference thermocouples were used to detect temperature gradients on the cells. A difference thermocouple referenced to the location of the resistance thermometer was located in each cell and two other thermocouples were used to detect temperature differences between the top and side and the bottom and side of the cells. On the thermocouples located in the cells one wrap of the lead wires was made around the cell before they were led out through the neck tube to a Conax gland in the gland holder at the top of the cryostat. In addition

-50to the four difference thermocouples on the cell four additional copperconstantan thermocouples were used to measure dead space volume temperatures. One thermocouple with its reference junction in the bath fluid was located four inches down from the gland chamber at the top of the cryostat in the tube containing all of the lead-in wires and the inlet and exhaust lines to the cells. Another thermocouple was located on the exhaust tube at the point where it left the gland holder. The remaining two were located on the valves and the DPI. Thus, on the inlet and exhaust tubes where the greatest gradient in temperature occurs, four known temperatures are available for the computation of a meaningful average tube temperature. To be able to check the liquid level in the cryostat three difference thermocouples with their reference junctions at the bottom of the lead-in tube on the cryostat were located at 3-3/4, 9-1/2 and 12 inches from the bottom of the cryostat lid. The emf output of the thermocouples was measured with a Leeds & Northrup Wenner Potentiometer. The smallest reading possible with the Wenner Potentiometer is one-tenth of a microvolt. For copper-constantan at 70o K the output of a thermocouple is about 16 microvolts per degree Kelvin. Thus it was possible to detect gradients on the cell of the order of + 0.005~ K. The difference thermocouples normally produced readings in the range from zero to +.3 microvolts. The Mueller bridge and Wenner potentiometer in conjunction wth the indicating suspension tye galvanometers are shown in Figure 12.

fail~~~~~~~~~~~~ D 31L:. 4: ~~~~~~~~~~~~~~~~~~~~~j jj~~~~~~~~~ I~ ~ ~I Figtnr 1~, Viewr~ or Thle M~uo~.eflr Bvid~e Wnd'Elexnerc~ i,"otentin ieter~ilo~t~ c~r OXn As see~ated. Gatl.:vmnernetc's.

-525. Test Gas System. The test gas system is shown schematically in Figure 13. A Ruska piston type pressure intensifier was used to attain the initial pressure of 10,000 psi used in taking the data for helium. This unit consists of a single piston having different diameter heads on each end with the space separating the two gas chambers vented to the atmosphere. The ratio of the areas of the piston heads is nine to one. This made it possible to achieve output pressures of 10,000 psi by using bottled nitrogen gas at 2200 psi as the working gas. The test gas system is set up so that expansion of the gas from V can be directed back to the intensifier and recompressed into a storage chamber. This feature made it possible to cut the waste of valuable neon gas to a tolerable amount. It can be seen from Figure 13 that the entire test gas system except the volume in the intensifier can be evacuated. This made it relatively easy to clear the system of any foreign gases prior to filling the large Burnett chamber. All lines except the line leading from the exhaust valve on Cell II to the vacuum pump and the lines connecting the cells to the valves and the DPI were 1/8 0. D. x 0.040 I.D. stainless steel. The tubing connecting the valves and the cells was 1/16 0. D. x 0.020 I. D. stainless and the tube running to the DPI was 1/16 0. D. x 0.006 I. D. stainless. The vacuum line running from the exhaust valve was 1/4 0. D. x 1/8 I. D. for the high pressure portion and 1/4 0. D. copper line for the low pressure section.

11000 Psi o, -15000| 1 RUPTURE DISC X t CHECK VALVE DPI D-l ^ (0- 2000 FILTER STORAGE m I I I X 1 (a) I I X INTE NSIFIER DWG J OIL ) 3 — VENT INJECTOR VENT c... DUMPU TEST NITROGEN Mc LEOD GAS GAGE VENT Figure 13. Schematic Diagram of the Test Gas System.

-54In order to hold down the dead space volume, the expansion, inlet and exhaust valves had to be specially designed. The inlet and exhaust valves utilized the bonnets from Dragon model 808 extended stem valves with the stem tip modified to a 22 degree included angle. The stem material on these valves is 17-4 PH and the bodies were made from type 316 stainless steel. The expansion valve was designed similar to High Pressure Equipment Companies midget series valves, with the stem length increased to accommodate additional packing and with the stem tip angle reduced to 22 degrees. The expansion valve was of the non-rotating stem type, whereas the inlet and exhaust valves had rotating stems. Since the valves were operated at room temperature teflon packing served quite well and no problems with leakage were encountered. The noxious volumes introduced by the connecting lines, the DPI, and the valves were measured using the technique described by Kaminsky and (l05) (15) Blaisdell and Crain. The vacuum pump used to evacuate VTI was a Welch Scientific Company model 1402 duo seal vacuum pump. With this unit the pressure in VII could be reduced to 20 microns of mercury in 20 minutes. Vacuum readings were made with a Folsdorf type Stokes Mcleod gage. 6. Temperature Control System. The temperature of the cells was maintained constant by setting the vapor pressure of the bath fluid such that a very slight cooling tendency prevailed and then offsetting the cooling with automatically controlled heaters. The bath was stirred constantly with a magnetic stirring arrangement.

-55a. Heater Design A schematic diagram of the heater circuit is shown in Figure 14. The neck heater was made by winding bare No. 28 Chromel-A wire on a 1 inch O. D. wooden dowel. The output of this heater was from zero to a maximum of about 4 watts. The inner, middle, and outer heaters were made by winding No. 28 Chromel-A resistance wire on a cage constructed from three formica rings and six 3/8 0. D. x 13-1/2 inch long wooden dowels. The middle heater was wound around the cage with 8 turns per inch from top to bottom and the inner and outer heaters were made by passing the resistance wire from the top formica ring to the bottom ring and back in intervals of 30 degrees. The output of the middle heater ranged from zero to 20 watts depending on the control setting. The outer and inner heaters had miximum outputs of about 5 watts. By means of the controls indicated in Figure 14 the output of each heater could be controlled. The total heat input was controlled with the variac. The clearance between the heater cage and the cells was 1/2 inch. The heater system was originally designed for control of the cell temperature under adiabatic conditions and was modified by making the heaters described above. Figure 15 shows the physical appearance of the heaters described above and also shows their location in relation to the Burnett cells. The input to the heaters came from a Hallikainen Instrument Company Thermotrol temperature controller. The temperature controller was actuated with a wafer type resistance thermometer attached to the side

0.8 AMP TEMPERATURE 0.2 AMP 0.2 AMP 0.2 AMP. CONTROLLER VARIAC 25 WATT 25 WATT 25 WATT 25 WATT 110 VOLT^, 0' 60 CYCLE loon 110 500. soo lQ NECK INNER MIDDLE OUTER 0.8 AMP Figure 14. Circuit Diagram of the Heater System.

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-58of the Burnett cells. The width of the temperature band to which the controller would control could be varied by adjusting the gain in the instrument. For the system employed it was found that the controller operated best with the gain set at 8 and the reset dial set at 60 seconds. This setting corresponded to an on-off temperature band of about 0.005~ C. The temperature controller and heater control unit are visible in the lower center of Figure 16. b. Vapor Pressure Control System. The vapor pressure control system is shown schematically in Figure 17. In operation the gas from the cryostat was throttled by means of the metering valve and bypass valve until the desired operating pressure was reached. The valve across the small mercury U-tube was then closed causing the cryostat pressure to communicate with the pressure in the insulated volume by means of the small U-tube. The U-tube is equipped with mercury contacts and serves as a switch to operate the solenoid valve on the panel which is normally closed. Gas was either introduced into or taken out of the control volume in small slugs until the desired operating temperature was reached. In operation the controls were set so that the mercury rose very slowly in the left leg of the U-tube switch. When the mercury reached the upper contact the solenoid opened and allowed a small slug of gas to escape, thus dropping the pressure slightly and causing the switch to break contact. The circuit diagram for control of the solenoid is shown in Figure 18.

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VENT TO VACUUM PUMP I —-------— I --- \- - 0 —MANOMETER VACUUM BY PASS METERING VALVE VALVE SOLENOID 0-200 VALVE VALVE HEISE GAG SURGE CONTROL VALVE 0 FROM CRYOSTAT Hg Hg TRAP r TRAP Hg CONTACT INSULATED _Hg CONTACT REFERENCE ~VOLUME U- TUBE VO~LUME~ SWITCH N2 SUPPLY Figure 17. Schenmatic Diagram of the Vapor Pressure Control System.

SOLENOID VALVE RELAY F6 Hg U-TUBE B SWITCH ce~ y^ o; s f 2AMP F6 0 F6|A | 20 KPST Figre 1.3 Solo oo VAC ~ K ~ VA C -| ~( —- ~ oj o ~LAMP F6 1 7p d F6 (DIODE) SPST Figure 18. Circuit Diagram For The Solenoid Control System.

-62The mercury switch operates with a d. c. voltage of from 4 to 6 volts. The system was designed to give a minimum spark in the U-tube switch since it was originally intended to operate the system with hydrogen as the bath fluid. In the actual taking of the data it was found that the heating effect upon expansion was severe enough to cause the mercury to be blown from the U-tube switch and the system could not be operated as described above under such unsteady state conditions. However, by carefully manipulating the metering valve very satisfactory vapor pressure control could be attained and the cell temperature could be maintained within 0.005~ K of the desired value for extended periods of time without using the U-tube switch and solenoid. c. Stirring Mechanism The bath stirrer consisted of three aluminum rings, 7-1/2 inch outside diameter 5/16 inch wide, connected together with 3/16 0. D. aluminum tubing and was located outside of the heater assembly in the lower portion of the cryostat. A 1/4 0. D. thin walled stainless steel tube was attached to the top ring and passed up through the cryostat lid through the outlet pipe. The upper end of the stirrer rod was silver soldered to a soft iron core 5/8 0. D. x 2 inches long. The core was housed in a 3/4 0. D. stainless steel tube which was counterbored to produce a seat for the core. A 115 volt 60 cycle coil was mounted on the outside of the stainless steel core housing in such a position as to cause the core to rise about 1 inch when the coil was actuated. The coil was housed in a copper cylinder wound with 1/4 inch copper tubing. Water was continuously

-63circulated around the coil to keep it cool during operation. The lower portion of the stirrer is visible in Figurel5. A timer motor and micro switch actuated the coil approximately once every second causing the stirrer to pass through the bath twice per second. In addition to the three large rings on the lower section of the stirrer three 1-1/2 inch diameter aluminum discs were placed on the stirrer rod in the upper chamber of the cryostat and three teflon guides were mounted on the center ring of the lower section. The magnetic stirring arrangement was chosen to eliminate the necessity of sealing a rotating shaft in the cryostat lid and for its simplicity in design, construction, and maintainance. With the stirring arrangement described no gradients in the bath could be detected except near the top of the liquid where the vapor, which is at a much higher temperature than the liquid, causes heating of the liquid. The stirrer controls are visible in Figure 16. 7. Cryostat The cryostat was designed by the author and contracted for construction to the Linde Company. A simple schematic outline of the cryostat appears in Figure 19. The unit is capable of holding about 50 liters of bath fluid with about 15 liters in the lower chamber when the cells are in place. The inner shell is made from 3/16 inch thick 304 stainless and the outer shell is made from 1/8 inch 304 stainless. The space separating the two shells was filled with Linde super-insulation. The cryostat was designed with the small diameter at the bottom to keep from wasting expensive cryogenic fluids since the fluid in the bottom section

-64l LEV ] -UPPER PORTION OF LIQUID LEVEL CHECK TUBE STIRRER FILL BOSS -\ F l VENT LINE 1PRESSURE RELIEF TUBE ----- /~ GLAND HOLDER oiiDQBURSTING DISC.,,, -EVACUATION OUTLET NECK HEATEREC TUBE s |0 ~ STIRRER HEATERS o 10-CELLS o o0 MULTILAYER O INSULATION, - aI o 0Figure 1 0 0 Figure 19. Schematic Drawing of the Cryostat.

-65is unused when the unit is shut down. A cylindrical 6 inch 0. D. x 1-5/8 inch deep can mounted on a 1 inch 0. D. tube which extends through the lid serves as a gland holder. All tubes and lead out wires are brought up through the neck tube and passed through conax glands to the outside. The cover plate is a 1/2 inch thick 14 inch 0. D. stainless plate secured in place by ten 3/4 0. D. bolts. The unit is sealed with a United Aircraft Products teflon coated self energized 0-ring. The cover on the gland holder is a 6 inch 0. D. 5/16 inch thick stainless plate secured by eight 3/8 inch bolts and sealed witha United Aircraft O-ring. The outer shell of the cryostat extends about 7 inches above the cryostat top forming a large reservoir which can be filled with insulation or with liquid nitrogen if hydrogen is placed in the cryostat. In the temperature range from 70 to 120~ K the top reservoir was filled with a vermiculite insulation. A plywood housing was constructed around the cryostat. The space between the plywood and the outer shell was insulated with styrofoam to a depth of 10 inches from the top. Below that foam glass insulation was used down to the small diameter of the cyrostat. The space surrounding the lower portion of the cryostat was filled with Zonolite insulation. The plywood housing was screwed to two 2 by 8 by 22 inch planks which were in turn bolted to a Unistrut framework. The cryostat lid with all the connecting lines and the cells mounted in place was suspended from a unistrut framework with the lid about 6 ft. from the floor. A system of pulleys and a small wench were attached to the framework and four nylon ropes were tied to the framework around the cryostat. With this lifting

-66arrangement the cryostat could be raised up to the cover plate and bolted in place. The above arrangement made it possible to make repairs on the experimental package without having to disconnect the wires and lines attached to the test cells and cryostat lid. The cryostat and lifting arrangement are visible in Figure 20. Four stainless steel tubes were arc-welded to the cryostat lid. These consisted of a 1/2 inch 0. D. fill tube, a 1/4 0. D. tube available for insertion of a liquid level probe, a 1 inch 0. D. tube for exhaust, and a 1/2 inch 0. D. tube connected to a 200 psi rupture disc. The 1 inch 0, D. tube was equipped with a cross with the stirrer mounted on the upper port, a Conax gland containing the lead out wires from the liquid level thermocouples mounted in one side port, and the 3/4 0. D. copper vent line connected to the other side port. The cryostat was designed for a maximum operating pressure of 200 psi and tested to 300 psi by Linde Company. B. Experimental Procedures 1. Cleaning the Cells Before putting the cells into use they were cleaned by flushing them repeatedly with trichloroethylene, benzene, and acetone. The cells were first flushed with trichloroethylene until no discoloration of the solvent was apparent. Then the same flushing procefure was carried out using benzene. Finally the cells were flushed repeatedly with acetone, Care was taken when connecting the lines to the cells and inserting the thermocouples to assure that no foreign matter entered the cells.

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-682. Charging the Burnett Cell. Prior to making the initial run the electronic null detector on the DPI was zeroed by opening both sides of the diaphragm to the atmosphere. The test gas system was then evacuated to 10 microns of mercury or less and purged with the test gas. This procedure was repeated again and then the cells were re-evacuated and pressurized to 200 to 300 psig. After attaining the proper temperature the large cell was then charged to the desired initial pressure with the pressure intensifier. On subsequent runs the electronic null detector was normally rezeroed prior to initiating the run. However, it was found from experience that the null position on the indicator changed only slightly over long periods of time and on runs which were made close together the rezeroing procedure was not deemed necessary. 3. Pressure Measurement. A typical pressure measurement was made by first obtaining a close guess of the pressure to be measured and then placing enough mass on the dead weight gage to create a pressure slightly above this. This was done to assure that the DPI was always overpressurized from the same side. The manufacturer recommended this in view of the possibility of causing the null point of the DPI to shift under frequent changes of direction of overpressure. After placing enough mass on the gage to create a pressure exceeding the pressure to be measured the valve isolating the weight table was opened and weights were removed until the null detector indicated too little mass. Then small weights were added and removed as required until a rough balance was attained. The rough balance was maintained until

-69the cells had been at the proper temperature for 10 to 15 minutes. After this waiting period the pressure was balanced and the expansion valve was closed. After closing the expansion valve the pressure was re-balanced within the limits of the accuracy of the gage. 4. A Typical Data Point The temperature of the cells was frequently checked and the controls were manipulated so as to force the cells to remain at the desired temperature during the time required to take a point (roughly 20 to 30 minutes after expansion). The difference thermocouples were also monitored and when it was decided that equilibrium had been achieved the expansion valve was closed and the final null of the dead weight gage was carried out. Then the valve isolating the weight table and oil injector from the DPI was closed and the pressure on the weight table was reduced to zero. Immediately following this the exact temperature reading was taken, the difference thermocouples on the cells were read and then the thermocouples on the noxious volumes were read. Following this the barometer reading was recorded and the gas was expanded from VII and evacuation initiated. The room temperature and the temperature of the oil in the piston housing on the dead weight gage were then recorded. Finally to complete the data necessary for a single point the weights used on the dead weight gage were recorded. The average total time for one data point was about 1 hour and 10 minutes.

-70C. Difficulties Encountered. 1. Design of the Cells. The problems encountered in designing the cells were previously pointed out. Several unforseen problems arose due to the geometry of the cells. First it was difficult to tighten the fittings connecting the inlet and outlet lines to the cell and a special wrench had to be made for that purpose. The cells were heavy and difficult to put in place due to the necessity of threading the inlet and exhaust lines up through the neck tube before the cells could be raised to their mounting position. The cells were also difficult to clean since no large openings were provided. 2. Valves. Initially the valves were tested in liquid nitrogen and after breaking one specially designed stem in the expansion valve and noting that the freeze-up problem was very severe in the inlet and exhaust valves it was decided to discard the idea of placing the valves in a liquid nitrogen bath. Since the valves could not be placed in the bath the dead space volume they introduced had to be kept small. No commerical valves which had small enough dead space volumes were available9 so the valve bodies were specially designed. Some difficulty in machining the valves was encountered because of the small holes and close tolerences. The seats on the valves were made with a 22~ cone which was coated with lapping compound and repeatedly forced against the seat. The stem points were

-71polished to a mirror finish before installing them in the valves. 3. Initial Pressure Measurement System. Initially it was intended that a mercury U-tube would be used to separate the test gas from the oil of the dead weight gage. This unit was constructed of 1/4 0. D. high pressure stainless steel tubing and fitted with mercury contacts at the upper end of each leg and in the center of one leg. It was possible by closing a valve at the bottom of the leg on the gas side to isolate the oil side from the gas side. The gas side leg was in continuous communication through a Ruska gage separator with a 15,000 psi Heise gage filled with oil. The intended method of taking a pressure reading was to note the pressure on the gas side with the above mentioned gage and then impose a pressure slightly smaller on the oil side by means of the oil injector on the dead weight gage. Then the valve on the gas side leg was opened and the pressure was balanced with a differential pressure diaphragm located between the U-tube and the dead weight gage. Weights were then placed on the dead weight gage that would correspond to a pressure equal to that indicated by the gages. The valve isolating the weight table was then opened and weights were added or subtracted until the differential pressure diaphragm again indicated null. Finally a bypass valve around the DPI was opened and the level of the mercury in the U-tube was adjusted until the level reached one of the upper contacts. Weights were to be placed on or taken off of the weight table and the mercury injector manipulated until a balance was reached. Balance was to be determined by the smallest weight that would cause one contact to make when placed on the weight table and the other to make when taken off.

-72While the system described above offered the advantage of very small dead space volume it was very cumbersome and hard to operate. Also it was found that upon evacuation of the cells mercury was drawn up through the capillary line leading to the cells and the entire assembly had to be disassembled and cleaned. This system was discarded in favor of the more simple and direct method of using a single DPI with an electronic null indicator even though this caused a doubling of the dead space volume. 4. Stirring Mechanism. The original stirrer coil was an 850 ohm d. c. coil powered by a 150 volt d. c. power supply. The coil was taken from a magnetic radio speaker. The unit functioned satisfactorily during the taking of data for two isotherms. Then when the system was tested with hydrogen enough moisture collected in the coil housing to short the unit out and a replacement coil could not be readily located. The coil was replaced with a 115 volt 60 cycle a. c. coil. This second coil overheated excessively and water cooling was necessary for continuous operation. Also the a. c. coil interfered with the temperature controller and had to be shielded. The capacity of the stirrer was increased at every opportunity, first by increasing the passes per second through the bath and the length of the stroke. In addition the three aluminum discs on the stirrer rod were added. In the final configuration used the stirring was adequate and no gradients could be detected in the lower chamber of the cryostat and only slight gradients were present in the upper chamber near the surface of the bath.

-735. Temperature Control. A problem that was encountered in the temperature control was the difficulty of finding a suitable setting for the gain and reset dials on the temperature controller. The unit would not always turn on the heaters at the same temperature and a small overshoot or undershoot sometimes occurred. It was found that the vapor pressure could be controlled very nicely without using the elaborate automatic system that was built for that purpose. In fact, manual control was desirable in view of the upset in vapor pressure that occurred when an expansion was made. 6. Thermocouples. The main difficulty encountered with thermocouples was the grounding out of the lead wires and the commercial Conax thermocouples used in the cells. The lead wires were No. 24 teflon coated copper wire. With the system at room temperature the thermocouple circuits could be checked and found to be ungrounded. A similar check with the cryostat full, however, showed the thermocouples to be grounded. It was found that this was due to moisture collecting on the wires at points where the insulation was damaged slightly when the wires were threaded through the porcelain guides on the Conax glands. It was finally necessary to reinsulate all of the thermocouple wires to eliminate the problem. The Conax thermocouples originally placed in the cells proved to be very unreliable and were finally replaced with units made by the author.

-74A Leeds & Northrup thermocouple selector switch was initially used to switch from one thermocouple to another. It was found that the switch produced a signal large enough to make very small readings (0.2 to 0.3 microvolts) meaningless. The switch was replaced with DPDT copper knife switches which eliminated the problem. Shielded leadwire was used to connect the knife switches to a terminal board on the back of the instrument panel, since this section of leadwire had to pass near the a. c. control units for the stirrer and heaters. It was also necessary to use shielded wire to connect the potentiometer and the galvanometer in order to eliminate a slight zero shift of the galvanometer. 7. Thermometer Leads. The experience of Crain(15) with the resistance thermometer leads indicated that some way of avoiding breakage of the platinum leads on the thermometer was to be desired. Accordingly, short pieces of teflon tubing were slipped over the leads and bound together with thread. A 90 degree bend was made in the leads about 1/4 inch from the butt of the thermometer and the thread winding was continued just beyond the bend. Finally the thread wound portion which extended from the bend up over the glass nipple to a point just behind where the leads enter the glass was coated with epoxy. This made a very sound unit which could be easily handled without fear of lead breakage and never offered any problems.

V. DATA REDUCTION AND EXPERIMENTAL RESULTS A. Data Reduction. 1. Pressure Calculation. Because of the nature of the dead weight gage the calculation of the numerical value of a measured pressure was quite involved. For accurate pressure determination an accurate value of the local gravitational acceleration is required. The value used was calculated using a formula taken from reference 106 since no actual measurement at the location of the laboratory was available. The formula was checked by comparing it to the value of local gravity published in reference 107 for Selfridge Air Force which is 66miles N.E. of Ann Arbor, at Mt. Clemens, Michigan, and found to agree within 0.006 cm/sec2. For a free floating piston the pressure exerted by a given mass is p _ W m g (68) P = ^ = ^ & (68) m A Ae gc e e c where m is the mass of the weights and Ae is the effective area of the piston. Since the mass of the stainless steel weights was determined by comparing them with brass standards, a correction for the difference between the air buoyancy force for brass and stainless steel must be introduced. The actual mass is given by m= Pa (1 _ a) (69) Pb — -75

-76where ma is the apparent mass obtained from the comparison with the brass standards, Pa is the air density of that location, and Pb is the density of brass. The effective area varies with pressure due to elastic distortion of the housing and piston and also varies with temperature. The correction for variation of the effective area with pressure and temperature results in the equation Ae = Ao(l + bP)(l + CAT) (70) where b and C are functions of the materials used in the construction of the gage and Ao is the area of the piston at zero pressure and a temperature of 25~ C. Substituting (69) and (70) into (68) gives Pa Pb nma ( a Pm = (71) Ao go (l+bP)(l+CAT) For the apparatus used in this investigation the DPI was lower than the reference mark on the dead weight gage and a small positive pressure on the test gas due to the oil leg was always present. In addition, the final balance of the gage involves the use of small weights that are calibrated by the manufacturer to indicate pressure directly. The pressure indicated by a given mass on the gage is thus given by P = Pm + Poil + Pwt + Pbar + Pdpi (72)

-77where Poil is the pressure due to the oil leg, Pwt is the pressure determined by the small weights used, Pbar is the barometric pressure and Pdpi is a small correction for the difference in pressure between the two chambers of the DPI. For calculation of the pressure due to the oil leg the density of the oil was assumed constant and the following formula was used. Poil = Poil hoil (73) gc The barometric pressure was determined from the equation bar = Phg h (74) where Phg is the density of mercury at 0~ C and h is corrected for temperatures other than 0~ C by means of a chart supplied by the manufacturer. At null of the DPI the pressure in the lower chamber (gas side) is greater than the pressure in the upper chamber by the amount dp = 3.56 x -6 P (7) where P is in lbf/in2. The values used in the above formulas were g = 32.1618 ft/sec2 gc = 32.1740 lbm-ft/lbf-sec2 = 0.0012 gm/cm3, = 8.4 gm/cm3

-78A0 = 0.0260417 in2 b = -3.6 x 10-8 /psi. High range piston C = 1.6 x 0 - / ~C Ao = 0.130220 in2 b = -5.4 x 10-8 /psi Low range piston C = 1.6 x 10o-5 / ~C Poil =.85 gm/cm3 P = 13.5951 gm/cm3 hg When the above values are substituted into (71), (73) and (74) the following equations are obtained. P0 = 0.999479 ma (76) Ae AeH = 0.0260417(1-3.6x10 8P)(1+1.6xlO-5T) (77) AeL = 0.130220(1-5.4xlO0 8P)(1+1.6x10 T) (78) Poil = 0.030696 hoil (79) Pbar = 0.0193295 h (80) where hoil is in inches and h is in mm of mercury. The area AeH is the effective area for the high range piston and AeL is the effective area for the low range piston. The oil head is 0.04 inches greater for the high range piston due to a slight shift in the balance reference for the gage. The measured value of the oil leg was 4.882 inches.

-79Using the above formulas a computer program was written to carry out the necessary computations. The value of pressure required in (77) and (78) was obtained from = Pm + Pwt + Pbar where Pm was computed without correcting the effective area for pressure and temperature variation. With the aid of the computer program the pressure for seventeen data points could be computed in about one second. The orders of magnitude of the terms in (5) are PM = 6 psi to 10,000 psi Poil = 0.15 psi Pwt = 0.002 psi to 0.595 psi ar = 143 psi Pdpi = 0.0005 psi to 0.036 psi An error analysis which discusses the magnitude of error in pressure measurement is presented in section C of this chapter. 2. Determination of Dead Space Temperatures The total dead space volume was broken into two parts. The valves and DPI and the connecting line between them were all at room temperature and constituted one dead space volume. The lines running from the valves to the cells constituted the second dead space volume. These volumes are slightly different prior to an expansion than they are after an expansion. In this respect there are four separate dead space volumes to be considered; two prior to an expansion and two after expansion. The temperature of the

-80DPI and the valves was measured for each point on an isotherm. The microvolt readings of these temperatures was averaged for the entire run and the average temperatures of the units was determined from this. These average temperatures were normally within 0.5~ K of each other. The temperature of the dead space volume in the valves and DPI and the line joining them was obtained by averaging the average temperature of the valves and of the DPI. This temperature was used for all points of the run. Four known temperatures were available for computing the average temperature of the volume in the lines connecting the valves and the cells. The microvolt readings of the thermocouples on the lines were first averaged for the entire run and the average temperatures computed from these. Each line was divided into four sections of different length. The first section which ran from the cells to the lowest liquid level thermocouple was assumed to be at the test temperature and its volume was thus added to the total volume of the cells and not treated as a dead space volume. The uppermost ten inches of each line was at room temperature and was added into the dead space volume of the valves and DPI. A weighted average temperature was computed for each of the two remaining sections by averaging the temperatures at the ends of the section and multiplying the result by the length of the section divided by the total length of line from the lowest liquid level thermocouple to a point 10 inches from the valves. These weighted average temperatures were then added to obtain the average line temperature for a run.

-813. Dead Space Corrections. To carry out the correction for dead space volumes outlined in Chapter III, section A, it is necessary to know the compressibility factor for each measured pressure. To obtain these compressibilities for helium the data tables published by D. B. Mann(49) were used. A linear least squares fit of the data to an equation of the form z = C1 + C2 P (81) was made for each different average dead space temperature. These equations were then used to calculate compressibilities at the experimental pressures. Since the Tables mentioned only extend to 100 atm this involves linearly extrapolating to obtain compressibilities at the higher pressures. For helium at the temperatures of the dead space volumes (about 200~ K and 300~ K) the isotherms are so close to being linear that Equation (81) gives a very good value for Z even at the highest pressures measured. In fact, even for a linear fit of the data of this research which extends to 690 atm and is at much lower temperatures the standard deviation in Z was only 0.5 x 10-. For neon the dead space compressibilities were obtained by fitting the data of McCarty, Stewart and Timmerhaus(59) to an equation of the form Z = C + C2P + C3P2 + C4P3 + C5P4 (82)

-82The data in (59) extends to 200 atm and since the pressure range of this research extends to 300 atm for neon the compressibilities of the highest pressures were obtained by extrapolating Equation (82). The average standard deviation in Z using (82) was 0.2 x 10-4 for all dead space temperatures. The initial guess of the experimental compressibilities required to start the calculations was obtained by fitting existing data (19) and (59) to (81) and (82) for the test temperatures. 4. Correction for Volume Dilatation. The properties of 347 stainless steel required in the Lame formula were obtained through correspondence with F. Garfalo of U.S. Steel. Since this data was at room temperature and above it had to be linearly extrapolated to the temperatures covered in this research. This extrapolation yielded the following result 1 - k2 + + 1 2j 0767T L = E 2 - (k2 + -.[188 + O 67T x10-6 (83) L 1 E000 where T is in ~K and L is in atm. The values of L for each isotherm were obtained from (83). 5. Calculation of Compressibilities. A computer program based on the equations presented in Chapter III, section A, and on the information in section E was written to handle the calculation of the compressibility factors. The computer program was set up to obtain an initial value of No and Po/Zo by proceeding exactly as outlined in Chapter III, section A. Starting with these initial values

-83PO/Zo was held fixed and No was varied until a minimum deviation from linearity was obtained for points below a predetermined density on a virial plot. For helium points below half the critical density were used and for neon points below four-tenths of the critical density were used for the minimization. For each small change of No the calculation of the compressibilities was repeated three times using the results of the previous computation each time. This was done to iterate out the error in the dead space corrections that would arise if the values of Z for the initial No were used. This procedure of finding an No which gives a minimum deviation from linearity on a virial plot for points below four tenths of the critical density was repeated for two values of PO/Zo below the initial value and two values of Po/Zo above the initial value. The above procedure produces five sets of No and PO/Zo. The deviation from linearity for these sets of No and P/Zo was compared and the pair with the smallest deviation was taken as the initial values to be used for a repetition of the above procedure. If the value of PO/Zo for this new pair fell inside the extreme of PO/ZO the step intervals of PO/Zo and of No were halved. If the value of PO/Zo for the new pair was at one of the extremes the step intervals were left unchanged. In either event the above method of varying Po/Zo and No was continued until the step size in PO/Zo reached 0.003 atm. The initial step sizes in PO/Zo and No were chosen such that the step size in No was nearly zero when the step size in PO/ZO reached

-840.003 atm. The values of Po/ZO and No obtained from the above minimization procedure were taken as the final answers and used to calculate the compressibilities. The maximum total variation in No using this method was 0.06% of the initial value. B. Error Analysis 1. Temperature The prime error in temperature measurement is the uncertainty in the uniformity of temperature of the cells. The Leeds & Northrup Mueller bridge used to determine the resisiance reading of the resistance thermometer is capable of indicating changes as small as + 0.0001 ohms which corresponds to temperature changes of + 0.001~ K. All experimental points were taken with the resistance reading within + 0.0003 ohms of the calibrated value for a given temperature. Thus the temperature readings are within + 0.003~ K of the calibrated values. Nonuniformity of temperature of the cells was indicated by the difference thermocouples located on them. The output of the difference thermocouples was read with a Leeds & Northrup Wenner potentiometer which is capable of detecting emf's as small as + 0.1 microvolts. All experimental points were taken with difference thermocouples 1 and 2 which are located in the cells and referenced to the resistance thermometer showing values of 0.25 microvolts or less. In most cases the readings were 0.2 microvolts or less. The output of a copper-constantan thermocouple at the lowest measured temperature (70~ K) is about 16 microvolts per degree Kelvin. Thus the maximum temperature gradient which could have existed

-85for a point is + 0.015 o K. In view of the above considerations it is felt that the temperature measurement is good to better than + 0.0150 K. The international temperature scale is known to the nearest + 0.01~ K so the above maximum possible temperature error is quite satisfactory. 2. Pressure Of the various factors that affects the accuracy of the pressure measurement, the most critical is the accuracy to which the area of the dead weight gage piston can be determined. This will become apparent in the following discussion. a. Gage Resolution. The resolution of the dead weight gage, which is the smallest change in mass on the weight table that will produce a measurable change in the condition of equilibrium is quoted by the manufacturer as being less than 5 ppm for both the high and low range pistons for all calibrating test points. Through experience with the gage it was found that a more meaningful resolution would be 10 ppm. The numerical value of the resolution is expressed as the ratio of the change in mass to the total mass R = = + 10-5 (84) Expressing (84) in terms of pressure yields R = P = + 10-5 (85) P

-86Equation (85) represents the resolution of the gage for both pistons for the range 6 to 12,000 psi. b. Oil Leg. Due to the location of the DPI relative to the dead weight gage a small oil leg was present in the system. Errors in the correction applied for the oil leg arise from inaccuracy of determination of the oil leg and from fluctuations in the room temperature. The height of the oil leg was measured to + 0.015 inches, making the uncertainty in the correction + 0.0005 psi. For maximum room temperature fluctuations of 3~ C the error in the oil leg correction is + 0.0009 psi. The total error due to error in oil leg correction is then about + 0.0014 psi. c. Error in Reading the Gage. The point at which the line on the weight table and the reference line on the gage exactly line up is determined by eye and could be in error by as much as + 0.015 inches. This introduces an error of +.0015 psi into the pressure measurement. d. Gas Leg. Due to the physical arrangement of the DPI relative to the cells, the line connecting the two has a considerable temperature gradient imposed upon it, with the result that the actual pressure in the cells is slightly higher than the pressure in the DPI. From calculations made on the gas leg, which was not corrected for, the error introduced was found to be - 2 x 10-6 (86)

-87e. Error in Correction for DPI Pressure Differential. Since an estimated value of the pressure was used to determine the correction for pressure difference between the two chambers of the DPI, a slight error was introduced. The estimated pressure was found by calculating, the pressure without correcting the piston area for temperature and pressure and adding the barometric pressure and the pressure due to the small weights. The maximum possible error in the estimated pressure is e = 4 x 10-4 (87) P The error in the correction for difference in pressure in the DPI is then dpi = 3.56 x 10-6 e =1.5 x 10-9 (88) P P f. Error in Local Gravity. The effect of error in local gravity on the pressure can be estimated by estimating the maximum error in the weight of the mass on the weight pan of the gage. The equation relating mass and weight is w = g (89) gc The error in W due to an error in g is given by AW AW g s(90) W g

-88and the resulting error in the gage pressure is Zk m Ag (91 P g The estimated accuracy in g is + 0.006 cm/sec so that the resulting error in Pm is Pm = 6 x 10-6 (92) P g. Error in Barometric Pressure. The barometric pressure was read to the nearest 0.1 mm of mercury. The total error in the barometric pressure due to reading error and to error in local gravity is given by APatm T h) Patm = + 0.002 psi (93) h. Error in Effective Piston Area. The uncertainty in the area of the pistons for the dead weight gage is reported as 104 in.2 In addition to this constant uncertainty, errors in the gage temperature and errors in the estimated pressure used to calculate the final pressure affect the precise determination of the effective area. The effective area of the piston can be very closely approximated by the equation Ae = Ao (1 + bPe + CAT) (94)

-89where b and C are constants for the gage, Ao is the piston area at zero pressure and 25~ C and AT is the gage temperature minus 25~ C. The total error in the effective area due to errors in estimated pressure, temperature, and measured piston area is sAe =Ae 0 + AbPe + AoC A(AT) (95) Ao Pe Assuming Ae A Ao and Pe = P Equation (95) becomes AAe AAo ZPe - = - + b P + C A(AT) (96) Ae Ao P The error in pressure due to error in Ae can be estimated from the equation'P - (97) Pm - Ae m=, e --- (98) P A m e Using AAo/Ao = + 104 and Equation (87) for APe/P the error in Pm becomes Pm = 1.08 x 10-4 + 1.44 x 10"11 P (99) Pm

-90Since the second term in this equation is negligible a2m X aP =1.08 x 10-4 (100) Pm P i. Total Error in Measured Pressure. The total error in the pressure which is computed from the equation P. Pm + Patm + Poil + Pdpi + Pwt + (others) is given by AP = APm + APatm + APoil + APdpi + (others) (101) Using the values obtained in the previous discussion in Equation (101) gives AP = 1.26 x 104 P + 0.004 psi (102) From the previous discussion it is seen that the maximum uncertainty in the pressure results from the uncertainty in the piston area. It should also be noted that Equation (102) does not represent the limits of repeatability of the pressure measurement, but the degree to which the measured pressure might deviate from the exact pressure if it could be measured. The repeatability of the pressure measurements can be computed from the above analysis by neglecting the uncertainty in the piston area. 3. Error in Total Volume and Dead Space Volumes. The maximum possible error in the total volume is estimated to be no greater than 6 cm3. The volume was experimentally measured at 80~ K.

-91The volume measurement was repeatable to + 2 cm3 This makes AVT 6 VT 25 = 0.024 (103) VT 252.6 The dead space volumes were measured to within 0.002 cm3. 4. Error in Nj. The value of Nj is influenced by the value of No, the dead space volume correction and the correction for volume dilatation due to pressurization. The equation for N. is AVVj V 1 + +.A \ (104) NjK,~~ = N o ~~~ --— ^No GO VV1 V N G VI(j-1) Vc(j-1) G + No " VI VT From (104) the total error in Nj can be shown to be IJ~ (:....' J L ) -NX 1) +J VI-' cT +c1jV j-l'G j-l N0 GG. AVIj ( + - _-c + No GjVT Gj-1 VT + N. (Vc(-l) Vcj VT (105) aVT / G.V VT ajT

-92as Error in Volume Dilatation Correction. Writing the Lame formula as AV L p. (106) VI where L is given by (83). The error in the correction is given by A/ ( —V = L A Pj + PjA L + E (107) In Equation (107) Ej is the error introduced due to neglecting end effects and is estimated to be about 1% of the total volume change. The value of L is believed to be good to better than 5%. Using L = 1.879 x 10-6 /atm and Equation (102) for A Pj, Equation (107) becomes A^ (.i = 7.68 x 109 P (108) where P is in lbf/in2o Similarly IVI( j 31)t9 P. (109) A ( ( l)j^ 7.68 x 10 P (109) Vi \ V~ j-1 In taking the difference in the errors for volume dilatation in Equation (105) the error in pressure due to area in piston area and the error in L and E cancel out leaving

-93(A/ - I A - l(j ) L (Pj-l - Pj) x 10-5= O (110) V (A VI /I j av,, ~(nV j-1 b. Error in Dead Space Volume Correction. The dead space volume correction is determined from m Vci Vj = Z T z ci (111) cj J i= Z.. T. Assuming no error in Z. and T the error in the corrected dead space volume is given by AL. m 1 A Z.. ZVc.j v = Z~i T i ZjiV + +j Vcj Zji m Vci A Ti ZiT Zl ( 2 (112) =lZi i Z T The maximum error in Zji is 1% and the maximum error in the dead space temperature for the dead space volume outside of the cryostat is + 1~ C. The uncertainty in the temperature of the tubes leading from the valves to the cells is probably no worse than + 20~ C. Substituting actual values into (112) and assuming Zj i Zji gives AV = 0.03 (113) cj

-94and _ (j-l) = 0.021 (114) c(j-1) By taking Vj = 0. 321 V ) = o.423 c(j-al) AVNT pi t= 0.000253 VI AV = 0.000332 VI and No = 1,194875, which are average values for the 80~ K isotherm of helium, the following values of Gj, Gj-1 and Nj are obtained Go = 1.00153 G = 1.00234 No = 1.193908 Estimating the maximum error in No to be 5 x 10-5 and using the approximate values given above in addition to Equations (103), (113), and (114) in Equation (105) gives the result a = (5 + 3.61 + 4.25 + 1.73) x 10-5 (115) N. a

-95The order of the appearance of the terms in (115) is the same as in (105) with the volume dilatation terms deleted due to (110). Equation (115) shows that the errors are all of the same order of magnitude. This result also shows that the accuracy of the Burnett method is highly dependent upon the accuracy to which No can be determined and to a large extent upon the accuracy of the determination of dead space corrections. For the case where N0 can be accurately determined the dead space corrections dominate as shown by (115). The error in NH due to error in total volume is about one- half of the combined errors due to dead space correction errors. When all of the terms in Equation (115) are added the final result is = \1.5 x 10-4 (116) 5. Total Error in Compressibility. The error in the compressibility factors will be determined from m Pi IT N. 0 i~l' Z. - -- (117) From Equation (117) the error in Zo is given by iZ. API ANiM. A(P0 /Z) /az = —P- +j -- + + T ~ + (118) ~Zj Pj Nj P0/Z \. AN AN2 AN where it has been assumed that - -.- -. The term Ev N N N.

-96has been added to account for incomplete evacuation of VI and is given by Zo Ev = P r ( 1Ni)- 1) (119) Normally VII was evacuated to pressures less than 30 microns of mercury so that Equation (119) gives Ev = 0.4 x 10-5 (120) for twenty successive expansions. For helium the rate of change of Z with temperature at constant pressure is approximated as I:z at -^ - 0.00174/oK (121) K/T The error in PO/ZO is not in excess of + 0.02 atm and the uncertainty in No is estimated to be + 5 x 105. Using these estimates and Equations (102), (116), and (121) in (118) gives - - (1.26 + 1.5 j + 0.6 + 0.26 + 0.04) x 10- (122) zj or i = (2.16 + 1.5 j) x 10-4 (123) Zj Equation (123) shows that for the first expansion of a run the error in Z is 3.66 x 10-4 or about 0.04% and for twenty expansions

-97the error is about 3.2 x 10-3 or 0.32%. These error estimates are based on the 80~ K isotherm for helium. It should be noted Equation (123) represents an estimate of the maximum error in the compressibilities. The actual errors in compressibilities are expected to be less than indicated by Equation (123).

VI. EXPERIMENTAL RESULTS A. Experimental Results for Helium 1. Compressibilities The isotherms for helium were investigated using high purity helium supplied by the Bureau of Mines helium plant at Amarillo, Texas. The purity of the helium conformed to the specifications listed in Table 1. TABLE 1 HELIUM PURITY Contaminant Amount (ppm) I2 < 0.4 CH4 0.0 H20 < 0.5 Ne < 14.0 N2 < 4.0 02 < 0.8 Ar Trace C02 < 0.1 Table 1 shows that the maximum amount of impurities in the helium did not exceed 20 ppm. The experimentally determined values of the compressibilities for helium are listed in Table 2 along with the computed values of the specific volumes. The number of experimental points for each isotherm is composed -98

-99of the points from a complete run from 680 atm down to about 17 atm and the points for a partial run (6 or 8 points) at high pressures. The compressibilities for the data points of a partial run were obtained by fitting the reduced data for the full run to a fifth or sixth order polynomial in pressure and computing the compressibility factor for the last measured pressure of the partial run and proceeding as outlined in Chapter III, section B. The consistency of the helium data is demonstrated by the compressibility diagrams presented as Figures 21 and 22. Table 3 is a comparison of the experimental compressibilities for helium with published values for selected values of pressure. The values of compressibility listed from the work of Holborn and Otto were obtained by linearly interpolating between the isotherms studied by them. The values published by Mann(49) were arrived at by fitting the Strobridge equation to existing data in the temperature range from 3 ~K to 300 ~K. The values of compressibility listed from this work were obtained by fitting each isotherm to a fifth or sixth degree polynomial in pressure and computing the compressibilities for the even values of pressure listed in Table 3. Table 3 shows that in the range from 0 to 100 atm the compressibilities from this work agree with the results of Holborn and Otto to within 0.3%. The agreement with the calculated results of Mann is somewhat better than 0.15%.

-100TABLE 2 EXPERIMENTAL RESULTS FOR HELIUM T 70~ K = 80~ K =8 P(atm) Z = Pv/RT v(cc/gm) P(atm) Z = Pv/RT v(cc/gm) 684.8179 2.38703 5.00180 693.3695 2.21331 5.23494 575.2362 2.17780 5.43269 601.2871 2.05939 5.61682 477.0025 1.98494 5.97132 492.1884 1.87656 6.25266 408.8116 1.84789 6.48630 433.8434 1.77490 6.70927 346.0660 1.71950 7.12996 362.1074 1.64925 7.46938 301.4177 1.62691 7.74528 323.3245 1.58022 8.01520 259.2827 1.53844 8.51432 274.2133 1.49211 8.92375 228.6351 1.47372 9.24944 247.1254 1.44302 9.57613 199.1316 1.41104 10.1682 212.1853 1.37949 10.6620 177.2556 1.36451 11.0463 192.5948 1.34369 11.4417 15.5.8527 1.31895 12.1438 166.9379 1.29679 12.7394 139.7372 1.28472 13.1928 152.3615 1.27013 13.6712 123.7492 1.25079 14.5038 133.0322 1.23480 15.2221 11.5619 1.22502 15.7569 121.9451 1.21469 16.3356 99.3449 1.19934 17.3229 107.0731 1.18756 18.1890 89.9440 1.17961 18.8196 86.8495 1.15102 21.7345 80.4433 1.15988 20.6902 70.8718 1.12237 25.9715 65.5593 1.12903 24.7123 58.1139 1.09975 31.0348 53.7108 1.10481 29.5167 47.8305 1.08162 37.0854 44.1820 1.08549 35.2552 39.4923 1.06718 44.3159 36.4628 1.07001 42.1097 32.6789 1.05524 52.9565 30.1718 1.05755 50.2972 27.0986 1.04566 63.2819 25.0185 1.04743 60.0767 22.5058 1.03777 75.6209 20.7789 1.03908 71.7578 18.7183 1.03142 90.3659 17.2845 1.03240 85.7105

-101TABLE 2 (CONT'D) EXPERIMENTAL RESULTS FOR HELIUM T = 100~ K T = 120~ K P(atm) Z = Pv/RT v(cm3/gm) P(atm) Z = Pv/RT v(cc/gm) 685.9706 1.96335 5.86729 692.8584 1.81597 6.44750 596.0191 1.84474 6.34483 518.3546 1.62214 7.69817 503.0904 1.71951 7.00653 396.7062 1.48243 9.19245 442.7845 1.63667 7.57728 309.0784 1.37927 10.9776 379.2859 1.54826 8.36801 244.2293 1.30160 13.1101 337.2588 1.48891 9.05005 195.1435 1.24208 15.6575 292.2335 1.42483 9.99488 157.3013 1.19579 18.7003 261.9459 1.38129 10.8098 127.6941 1.15939 22.3350 228.9879 1.33360 11.9387 104.2411 1.13043 26.67652 206.5452 1.30100 12.9125 99.1653 1.12411 27.8852 181.8324 1.26498 14.2612 85.4788 1.10716 31.8624 164.8037 1.24004 15.4246 81.3971 1.10206 33.3059 145.8905 1.21241 17.0360 70.3492 1.08834 38.0568 132.7315 1.19305 18.4259 67.0455 1.08422 39.7808 118.0002 1.17145 20.3510 58.0692 1.07302 45.4557 107.6887 1.15632 22.0117 55.3808 1.06970 47.5147 96.0702 1.13935 24.3116 45.8487 1.05776 56.7526 78.6122 1.11376 29.0434 38.0308 1.04798 67.7868 64.5923 1.09325 34.6964 31.5946 1.03990 80.9666 53.2492 1.07670 41.4500 26.2807 1.03319 96.7094 44.0156 1.06324 49.5186 21.8844 1.02764 115.513 36.4623 1.05223 59.1580 18.2393 1.02301 137.974 30.2599 1.04324 70.6741 15.2130 1.01918 164.802 25.1502 1.03587 84.4323 12.6974 1.01605 196.847 20.9281 1.02978 100.869 17.4326 1.02477 120.506 ~., L,

2.4 2.2 2ol0 0 FULL RUN 2.0 + PARTIAL RUN 70V S | / / yoo' K I 1.8 I I / y /- /120 0K co~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o N ft~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ LU. 1.4 10 Figure 21. PRESSURE p /20 - 00rm O- Hlim hoin Te xprienatl poi.s 21, omp ess b.~. Th agr For lie 1~ how1 Th D bperimeinta i Points.

1.14 1.12 0 FULL RUN + PARTIAL RUN 70~K 80/ 1.10 O0K +/ 120K K 1.08 1.060 I U) 0 o 1.04 1.02 10 20 30 40 50 60 70 80 90 PRESSURE P(atm) Figure 22. Compressibility Diagram For Helium Showing The Low Pressure Experimental Points.

-104TABLE 3 COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR HELIUM TO PUBLISHED VALUES T = 700 K Reference Pressure (atm) 20 50 100 This work 1.03788 1.09700 1.20060 Holborn and Otto 1.03520 1.09383 1.19690 (33), (34), (35) Mann (49) 1.03653 1. 09619 1.20002 (Compilation) T = 80 K Reference Pressure (atm) 20 50 100 This work 1.03440 1.08472 1.17445 Holborn and Otto 1.03199 1.08423 1.17654 (33), (34), (35) Mann (49) 1.03381 1.08771 1.18051 (Compilation)

-105TABLE 3 (CONT'D) COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR HELIUM TO PUBLISHED VALUES T = 100~ K Reference Pressure (atm) 20 50 100 This work 1.02841 1.07198 1.14508 Holborn and Otto 1.02682 1.06938 1.14460 (33), (34), (35) Mann (49) 1.02938 1.07489 1.15172 (Compilation) T = 120~ K Pressure (atm) 20 50 100 This work 1.02526 1.06293 1.12519 Holborn and Otto 1.02294 1.05906 1.12182 (33), (34), (35) Mann (49) 1.02579 1.06506 1.13092 (Compilation)

-o1062. Second and Third Virial Coefficients The values of the second and third virial coefficient for each isotherm were obtained by least squares fitting the points with densities less than half the critical density to the equation v(Z - 1) = B + C/v (124) The values obtained for the second and third virial coefficients of helium are listed in Table 4. TABLE 4 EXPERIMENTAL SECOND AND THIRD VIRIAL COEFFICIENTS FOR HELIUM Second Virial Third Virial Coefficient Coefficient Temperature K B (cc/gm) C (cc2/gm2) 70 2.608 14.32 80 2.690 12.56 100 2.883 12.26 120 3.106 9.809 Figure 23 shows a comparison between the values of the second virial coefficient from this work to values obtained by previous investigators. Figure 23 shows that the values of the second virial coefficient for the 80 and 100~ K isotherms of helium show satisfactory agreement with published values. The values for the 70 and 120~ K isotherms are slightly too large and do not form a smooth curve with the 80 and 100~ K values as they should. This inconsistancy is attributed to lack of low pressure

3.3 I I V CANFIELD,LELAND,KOBAYASHI (10) 3.2- A NIJOFF, KEESOM, ILLIIN (66) o BOKS AND ONNES (13) 0 3.1- EJ HOLBORN AND OTTO (33),(35) + 0 WHITE (108) O E V E 3.0 + THIS WORK o | —-THEORY (CHAPTER Err) O - - 2.9 2.8 - 2.7 + A ~0 — O 0 o 2.6 2.5 2.4 2.3 60 70 80 90 100 110 120 130 140 TEMPERATURE OK Figure 23. Comparison of Experimental Values of the Second Virial Coefficient of Helium.

-108data for the helium isotherms. The helium data was taken with the intention of using the method described in reference (10) to reduce the data. Since the low pressure data points are ignored when this method is used it was not considered necessary to take them. In using this method No was varied until a minimum deviation from linearity was obtained for points below half the critical density on a virial plot. In approaching the true value of No from below with this method the second virial coefficient for the 80~ K isotherm of helium came out to be about 3.1 cc/gm. On the other hand, if an initial value of No that was greater than the true value was used the minimum deviation in linearity occurred for No too large and the second virial coefficient came out to be about 2.44 (cc/gm) for the 80~ K isotherm of helium. The region between 2.44 and 3.1, where all of the published values lie, was inaccessable unless the density limit is increased to around 0.8 of the critical density. It was apparent upon observing the data for this higher density limit that considerable forcing of the data was present. It was this dilemma which ultimately led to the technique discussed in Chapter III for reduction of Burnett data. Unfortunately the new technique requires accurate low pressure data. The data on helium terminates at about the maximum point on a P-plot as can be seen in Figure 7. The fact that the data does not continue to lower densities makes it possible to obtain a nearly linear isotherm for erroneous values of Po/ZO and No. Thus even using the new technique it is possible to obtain a false answer although the degree of variation of the answer has been considerably reduced. This problem does not arrise in the

-109neon data where lower pressures were taken and about twice as many points are available below the density limit of four-tenths the critical density. In addition to the considerationers given above it should also be pointed out that the small value of the cell constant used in this work seriously limits the accuracy to which the virial coefficients can be determined due to the buildup of experimental error with the number of expansions. B. Experimental Results for Neon 1. Compressibilities The isotherms of neon were investigated using high purity neon supplied by Linde Company. Impurities in the neon were oxygen in the amount of 1 ppm and helium in the amount of 15 ppm. The experimentally determined values of the compressibilities for neon are listed in Table 5 along with the computed values of the specific volumes. Each isotherm for neon was determined by making a single run from 300 atmospheres down to about 7 atmospheres. Figure 24 shows the experimental points for neon and demonstrates that the data are consistent. Table 6 gives a comparison of selected values of the compressibility to published values. The values in Table 6 for the data of Holborn and Otto were obtained by plotting Z against T and drawing a smooth curve between the points. Only the 65, 90 and 123~ K isotherms were studied by them over the range of this research. Thus the interpolated values in Table 6 are not very accurate and serve only as a rough comparison. The average deviation of the compressibilities of this work from the values listed in Table 6 for the results of Holborn and Otto is about 2%.

-110TABLE 5 EXPERIMENTAL RESULTS FOR NEON T = 70~ K T = 80~ K P(atm) = Pv/RT v(cc/gm) P_(atm) Z = Pv/RT v(cc/gm) 302.5444 1.27578 1.20013 301.6140 1.25022 1.34825 201.1379 1.01210 1.43210 216.3159 1.07020 1.60921 149.0674 0.89524 1.70923 166,1444 0.98122 1.92095 118.1366 0.84686 2.04019 133.0937 0.93837 2.29324 97.2274 0.83197 2.43535 109.3310 0.92026 2,73780 81.6516 0.83404 2.90712 91.0234 0.91471 3.26861 69.2561 0.84448 3.47034 76.4044 0.91667 3.90240 58,9992 0.85879 4.14271 64.3973 0.92244 4,65912 50.3235 0.87444 4.94538 54.3665 0.92977 5.56261 42.9013 0.88991 5.90360 45.9233 0,93768 6.64133 36.5347 0.90468 7.04750 38.7807 o,94540 7.92927 31.0483 0.91780 8.41304 32.7298 0.95262 9.46698 26.3430 0.92960 10.0432 27.6038 0.95924 11,3029 22. 3139 09 3999 11.9892 2 3260 3.o96505 13.4949 18.8712 0.94900 14.3123 19.5856 0.97018 16,1120 15.9367 0.95672 17.0856 16.4790 0.97460 19.2367 13.4429 0.96338 20.3962 13.8561 0.97840 22.9674 11.3265 0.96899 24.3483 11.6434 0.98160 27.4215 9.5353 0.97382 29.0661 9.7805 0.98446 32.7395 8.0208 0.97787 34.6982 8.2123 0.98692 39.0888 6.7438 0.98149 41.4215 6.8930 o0.98903 46.6695

-111TABLE 5 (CONT'D) EXPERIMENTAL RESULTS FOR NEON T = 100 K T = 120~ K P(atm) Z = Pv/RT v(cc/gm) P(atm) Z = Pv/RT v(cc/gm) 304.2405 1.24211 1.65993 304.7889 1.23859 1.98269 232.6842 1.13411 1.98167 239.0719 1.15994 2.36718 184.1213 1.07144 2.36596 191.4889 1.10930 2.82640 148.9023 1.03457 2.82491 155.5557 1.07600 3.37483 122.0946 1.01290 3.37299 127.5737 1.05370 4.02978 100.9979 1.00047 4.02749 105.2952 1.03849 4.81191 84.0062 0.99363 4.80905 87.2889 1.02800 5.74593 70.1102 0.99020 5.74233 72.5757 1.02064 6.86130 58.6309 0.98879 6.85678 60.4697 1.01547 8.19324 149.0917 0.98859 8.18755 50.4512 1.01170 9.78378 41.1338 0.98911 9.77663 42.1355 1.00898 11.6831 34.4771 0.98995 11.6741 35.2140 1.00694 13.9513 28.9013 0.99091 13.9400 29.4441 1.00541 16.6598 24.2283 0.99192 16.6456 24.6269 1.00417 19.8941 20.3086 0.99282 19.8764 20.6036 1.00323 2 37565 17.0217 0.99365 2 37343 17.2411 1.00248 28.3687 14.2673 0.99451 28. 3410 14.4303 1.00195 33.8763 11.9583 0.99536 33.8418 12.0791 1.00153 40.4534 10.0211 0.99601 40.4104 10.1135 1.00136 48.3073 8.3973 0.99661 48.2539 I. P,...,,,..,.

1.30 I| 1.20 0 1.10 ~~~~~~~~~~~03:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CO 0^e^ ^)6~oe^.800K 0.90 70K 0.8I I I I I I 4 6 8 10 50 100 200 300 PRESSURE P(atm) Figure 24. Compressibility Diagram For Neon Showing The Experimental Points.

-113TABLE 6 COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR NEON TO PUBLISHED VALUES T = 70~ K Reference Pressure (atm) 20 50 100 200 This work 0.94611.87501 0.83251 1.00953 Holborn and Otto (31) 0.9402 0.8587 0.7590 -- McCarty, Stewart and 0.9400 o.8686 0.8251 0.9869 Timmerhaus (59) (Compilation) T =80O K Reference Pressure (atm) 20 50 100 200 This work 0.96977 0.93347 0.91650 1.03913 Holborn and Otto (31) 0.9647 0.9236 0.8880 - McCarty, Stewart and 0.9644 0.9285 0.9129 1.0334 Timmerhaus (59) (Compilation)

-114TABLE 6 (CONTWD) COMPARISON OF SELECTED VALUES OF COMPRESSIBILITY FOR NEON TO PUBLISHED VALUES T - 100~ K Reference Pressure (atm) 20 20 100 200 This work 0.99296 0.98857 1.00001 1.o9068 Holborn and Otto (31) o.9906 0.9825 0.9940 McCarty, Stewart and 0.9875 0.9833 0.9981 1.0964 Timmerhaus (59) (Compilation) T = 120~ K Reference Pressure (atm) 20o 100 200 This work 1.00310 1.01157 1.03524 1.11807 Holborn and Otto (31) 1.0010 1.0067 1.0300 McCarty, Stewart and 0.9973 1.0054 1.0319 1.1226 Timmerhaus (59) (Compilation)

-115The comparison of this work to the compilation of McCarty, Stewart and Timmerhaus shows an average deviation of about 0.6%. Figure 25 presents a graphical comparison of this work to the calculated results of McCarty, Stewart and Timmerhaus. It is clear from this comparison that their isotherms are displaced from those of this research by a constant factor. Also the zero pressure limits of their isotherms is not unity as they should be. 2. Second and Third Virial Coefficients The values of the second and third virial coefficients were obtained by least squares fitting the points with densities less than four-tenths the critical density to Equation (124). The values obtained for the second and third virial coefficients are listed in Table 7. TABLE 7 EXPERIMENTAL SECOND AND THIRD VIRIAL COEFFICIENTS FOR NEON Second Virial Third Virial Coefficient Coefficient Temperature ~K B (cc/gm) C (cc2/gm2) 70 -0.789 0.824 80 -0.531 0.783 100 -0.179 0.706 120 +0.049 0.647

-16Figure 26 presents a graphical comparison of the second virial coefficients obtained in this research to the results of previous investigators. The values of the second virial coefficient for neon from this research compare favorably with those of Holborn and Otto,

0 THIS WORK 1.01 -- Mc CARTY, STEWART AND TIMMERHAUS 1.00 -------- 20 K |.4~~~~~~~~~~ Ago~~~~~~ I -_-100 ~ K 0.99 0.98 7 O I Cn 0.97 C) - 0.96 0 0.95 0.94 0 2 4 6 8 10 12 14 16 18 PRESSURE P (atm) Figure 25. Comparison of This Work To The Compilation of McCarty, Stewart and Timmerhaus(59)

-0.20 - -0.40 - oE -0.60 in < >~ y/ ~ O~0 THIS WORK -0.80 - + HOLBORN AND OTTO (31) Z 0 ONNES, CROMMELIN AND MARTINEZ (16) O ) —THEORY (CHAPTER I) Cf) -1.00 - -1.20 -1.20 —--------- I I I I I 60 70 80 90 100 110 120 130 TEMPERATURE OK Figure 26. Comparison of the Experimental Values of the Second Virial Coefficient of Neon.

VII. THEORETICAL AND EMPIRICAL CORRELATIONS A. Determination of Intermolecular Parameters for the Lennard-Jones 6-12 Potential The theoretical expression for the second virial coefficient including quantum corrections is presented in Chapter III, section F. Equations (57) through (64) were programmed for solution on the IBM 7090 digital computer. Vaules of a and E based on the results of previous work were used as starting values. These values were then modified by determining the shift in the ordinate and abscissa on a plot of in Bexp against In T necessary to cause this plot to coincide with a plot of in B* against In T*. The variations in a and E were determined from Equations (65) and (66) which can be written as in Bexp = in B* + in b~ (125) and in T = in T* + in e/k (126) The values of e and a obtained from the above procedure for neon and helium are listed in Tables 8 and 9. The theoretical curves of the second virial coefficient as a function of temperature for neon and helium are shown in Figures 23 and 26. B. Fit of the Experimental Data to the Leiden Expansion 1. Helium Each isotherm of helium was least squares fitted to the Leiden expansion of Z in a polynomial in density. The coefficients resulting -119

-120TABLE 8 RESULTS OF THE DETERMINATION OF a AND FOR HELIUM WITH QJA.NTUM CORRECTIONS 0 a = 2.60 A c/k = 9.780 K A* = 2.69 T ~K T* B* B* B* B* B* B (cc/gm, c II calc 70 7.1576.4716.3828.01442 -.0001947.0002799 2.613 80 8.1801.4947.4190.01208 -.0001392.0002291 2.741 100 10.2252.5229.4647.009055 -.oooo0000804.0001639 2.897 120 12,2702.5379.4910.007191 -.0000519.0001247 2.980 TABLE 9 RESULTS OF THE DETERMINATION OF o AND FOR NEON WITH QJANTUM CORRECTIONS a = 2.782 A c/k = 34.82 ~K A- = 0.5935 T ~K T* B* B* B* B* B* B (cc/gm) cl I II calc 70 2.0103 -.5856 -.6193.09907 -.007025.oo00188 -.7884 80 2.2975 -.3944 -.4211.07845 -.oo004612.001539 -.5309 100 2.8719 -.1398 -.1583.05422 -.002347.001101 -.1881 120 3.4463 +.0207 +.00oo67.04076 -.001382.0008377 +.02782 - ~ ~ ~ ~~~~~~~ ~~~~~~~~~~~.. -

-121from the fit are listed in Table 10. In observing Table 10 it is apparent that the fit is well within the expected experimental error. It should be noted that the coefficients in Table 10 are not true virial coefficients, but empirical coefficients obtained by fitting the data to the Leiden expansion. 2. Neon Each isotherm of neon was least squares fitted to the Leiden expansion and the coefficients listed in Table 11 were obtained. As was the case for helium the fit is well within the expected experimental error. And as was stated for helium the coefficients in Table 11 are not true virial coefficients, but empirical coefficients necessary to fit the data to the Leiden expansion. C. Fit of the Experimental Data to the Berlin Expansion Each isotherm of neon and helium was least squares fitted to the Berlin expansion of Z in a polynomial in pressure. The coefficients which resulted for each isotherm are listed in Tables 12 and 13. Tables 14 through 17 show a point by point comparison of the experimental compressibilities for helium to those computed from the Berlin expansion. It is apparent that the 80~ K isotherm of helium shows a large variation compared to the other three isotherms. It is felt that this is due to the overall accuracy of the data for this isotherm. The 80~ K isotherm of helium was the first isotherm studied and a smooth repeatable technique of taking the data had not been fully developed.

TABLE 10 LEIDEN COEFFICIENTS FOR HELIUM Z = A + Bp + Cp2 + Dp3 + E + Fp5 + Gp6 Density in (gm/cc) Temperature Standard (~K) Deviation A B C D E F G 70 o405xlO4 0o 999637 2,65968 12 2893 18.9290 144o414 -5500248 - 80 o488x10o3 1o00037 2o64376 1406551 -56o3750 100o,16 -4889o81 9802.75 100 o367x10-4 Oo999684 2o93280 10 0783 26o0415 42 2053 45~45363 120 o203x10 0.999925 3013063 7083890 42 3823 -6353062 227480 - -

TABLE 11 IEIDEN COEFFICIENTS FOR NEON Z = A + Bp + Cp2 + Dp3 + Ep4 + Fp5 + Gp6 Density in (gm/cc) Temperature Standard u (~K) Deviation A B C D E F 70 o.115x10-3 0.998950 -.741799.241041 2.49833 -5.06805 5.29585 -1.54645 80 0.121x10-3 1.00015 -.555049.845095 -0.671967 3.01108 -3.61031 2.13445 -4 100 0.453x10- 0.999861 -.172746.674702 -0.453964 3.36785 -4.99988 3.21807 120 0.519x10 0.999479.075942.280106 1.69360 -2.97270 4.23309 -2.03686

TABLE 12 BERLIN COEFFICIENTS FOR HELIUM Z = Al + B'P + C'P2 + D'P3 + E'P4 + F'P5 + G'P6 Pressure in Atmosphere Coefficient H Temperature Standard -1 (~K) Deviation A' B'xlO C'xlO D' xlO E'xl01 F'xl015 GxlO1 70 0.265x10-3 1.00061 1.80897 2.96340 -13.2222 30.1403 -3551756 16.2242 80 0.601xlo-3 1.00360 1.46855 3.96361 -20.0961 50.4692 -62.0908 29.4334 100 0.396x10-4 0.99954 1.43890 0.234588 -0.726644 0.216244 0.818254 -0.683837 120 0.316x104 1.00001 1.26602 -0.161639 0.240920 -0.512839 0.340978 - -

TABLE 13 BERLIN COEFFICIENTS FOR NEON Z = A' + B' P + C' P2 + Dt P3 + E' P4 + F' P5 + G' p6 Pressure in Atmosphere Standard Temperature Deviation A' B' x103 C'x105 D'x106 E' x109 F'xlO11 G xlO4 70.377x10-3 0.998479 -2.49453 -1.14477 0.276765 -0.697925 -0.139693 0.496004 80.156x10o- 0.998532 -1.41019 -0.448470 0.169866 -0.807773 0.158826 -0.113610 100.516x10-4 0.99573 -.386214 0.234405 0.024086 -0.095574 0.010881 ----- 120.817x10-4 0.999991 0.098132 0.307911 -0.012608 0.107566 -0.040149 0.050817

-126A comparison of compressibilities for neon computed from the Berlin expansion to experimental values is presented in Tables 18 through 21. D. Values of Compressibility for Even Values of Pressure The Berlin Expansion for each isotherm of neon and helium was used to compute values of the compressibility for even values of pressure for the full range of the experimental data. The results of these computations are presented in Tables 22 and 23.

TABLE 14 FIT OF THE 70~K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation 684.6179 2.38703 2.38709 -.00006 -.00239 575.2362 2.17780 2.17746.00034. )1573 477.0025 1.98494 1.98556 -.00062 -. 3098 408.e116 1. 84789 1.84796 -. 0007 -.00355 346.0660 1 71950 1.71911.00039.02281 301.4177 1 62591 1.62645.00046.02812 259.2827 1 53844 1.53832.~00012.00749 228.6351 1.47372 1.47379 -.00007 -.00482 199.1316 1.41104 1._41133 -.00029 -.02057 177.2556 1 36451 1.36484 -.00033 -. 0239 155.8527 1031895 1.31925 -.00030 -.02305 139.7372 1.28472 1.28492 -.00020 -.31557 r 123.7492 1., 25.; 7 8 1.25 09 O-.; O J 12 -.30-978 111.5619 1.22502 1.22504 -.C0 02 -.00178 99.3494 1.19934 1.19923.00011.00923 89.9440 1.17961 1.17945.00316.01386 80.4433 1.15988 1.15957.00031.02630 65.5593 1.129`3 1.12874.00029.02612 53.7108 1.1:481 1. 10451.00030.02710 44.1820 1.08549 1.08529. 0320.01833 36.4628 1 7' l 1 1. 6992. 9.806 30.1718 1.0 5755 1.05755 - 0300 - -.3 023 25.0185 1.04743 1.04753 -.00010 -.00960 20.7789 i.039 8 1.03937 -. 00329 -. 2775 17.2845 1.03240 1 327 -. 033 -.0291 The Average Per Cent Deviation For the Fit is 0.016.%

TABLE 15 FIT OF THE 80~ K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation - 693.3695 2.21331 2.21344 -.00013 -.00591 601.2871 2.05939 2,0.5883.0056. 02721 492.1884 1.87656 1.87726 -.00070 -0. 3721 433.8434 1.77490 1.77565 - 00075 -.04232 3621074 1.64925 _ 1.64843 00082.04971 323.3245 1.58022 1.57930.00092.05819 274.2133. 1.49211 -1.9157.00054.03609 247.1254 1.443C2 1.44299..000G3..O0194 212.1853 1.37949 1.37996 -.00047 -.03404 192.5948 1.34369 1.34438 -.00069 -.05141 166._9379 __ 1.29679 _ 1.29751 -.00072 -.05528 152.3615 1.27013 1.27075 -.00062 -.04894 133.0322 1.23480 1.23518. -.00038 -.03088 121.9451 1.21469 1.21477 -.COOC8 -.00618 107.0731 1.18756 1.18742. G00 14.01185 86*8495 1. 151 2 1. 15045.0. 0057.04.927 708718 1. 122 37 1. 112160.00077.06858 58.1139 1.09975 1.C9892.00083.07534 47.8305 1.C8162._.___.1.8096.OCc66.610o2 39.4923 1.C6718 1.0 666.0052.< 4890 32.6789 1L.(C5524 1.05518.CC006.00586 27.0986 1.04566 1.C4593 -..G0027 -.. 02616 22.5058 1.03777 1.03844 -.C0067 -.G6483 18.7183 131 42 1.C3235 -. C0093 -.09042 The Average Per Cent Deviation For the Fit is 0.041t

TABLE 16 FIT OF THE 100~K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation 6 5.9706 1. 963 3 i.9633 3.C2. i88 596.0191 1.84474 1.84481 -.':CC7 -.C0388 503.0904 L ~ 71951 1 71942.:9 518 442.7845 1.6 667 1,63666.:ul i65 379,2859 1.54926 1.548262-5-.l'4'0 00:030 337.2588 1.48891 i 48899 -.CCtO8 -.054545 292.2335 1.42483 i.42482 -3 GC.CC008 261.9459 1.33129 1.3813- -. - -.i0044 228.9879. 3336 3 i 33 62__ - 2. 131 206.5452 1. 3 iC 1.3C-399 0.CC1. -.O-071 181.8324 1.26499 1.26494'.04. 0312. 164.8037 1.24;:4 1.24004 -.X 4 -. 00002 B 145.8905 1.2i241 1.21234. C'7.0 545 13'2.73 15 1.1930G5.19306 -. 1 -C68 118.CGC2 1.17'14:A 1.17146 -. 1 -.00101 107.6887 1.15632 1..15635 -.. 0 3 -.0 226 96.0702 1.13935. 13932.-03.00256 78,6122 1.11376 1.11376 -.CC -.0022 64. 5923 1.. -9325. 9327 C2.0178 53.2492..:7 7 1. 7672 ~. 2 -..166 44. 156.: 6 324 i. -632 7. -3. u260 36.4623 i. " 23 1.5228 -.CC5 -.05C"2 30.259 9 1..4. 428 4 8 —.. G 0345 25.15 2." 57 i. 3587. - 5 8. O' 00 4 2 20.9281 1.. 978 1. 2975.GOQ 3.O294 17.4326 1.,2477 1.2469. C C8.0C767 The Average Per Cent Deviation For the Fit is 0.0023%

TABLE 17 FIT OF THE 1200 K ISOTHERM OF HELIUM TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation 692.8584 1.81597 1.81597-.20.017 518.3546 1.62214 1.62211 300.(3175 396.7062 1.48243 1.48252- 000 7 4 439 309.0784 1.37927 1.37926.0 1.1.0079 244.2293 1.3C16 1.3I155 5 5.C421 195.1435 1.242'8 1.24205.0'03.C0244 157*3013 1.19579 1.19581 -; 2-.00169 127.6941 115939 1.15941 -.001 83 104.2411 1130431.13044 -.31-.1C4473 99.1653 1.12411 1.12415.30 ^4-.0366 85.4788.10716 1.1717..?7-.00076 61.3971.102061.12093239 -.00317 0 70.3492 1.08834 1.28834 f~J?,':2 2 67.0455 1."3422 1.08422. 25 2J37 58.0692 1.07302 1.)7302 3 55.3808 1.0697 1.0696 -34.3 6 7 45.8487 1.O5776 1.05773 * 3249 38.0308 1.4798 1.04793 ery 5 446 31.5946 1.O399 91.3985 ".-5'459 26.2807 1.*3319 1.3317 02 1 79 21.8844 1.02764 1.02764' 0'-.^Thl 7 18.2393 1.023011.-2305 rCn4 C-.'2354 15.213 01.01918 1.21923 -3 5 -.3496 12.6974 1Wi6Cn5 1.01606 Y-.;CO~1 -.Q&567 The Average Per Cent Deviation for the Fit is.0022%

TABLE 18 FIT OF THE 700 K ISOTHERM OF NEON TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation 302.5444 1.27578 1.27578 -.0000 -.00030 201.1379 1.01210 1.0C1197.00013.01235 149.0674.89524.89600 -00076 -.08438 118.1366.84686.84573 00113.1332 97.2274.83197.83179.00018.02200 81.6516.83404.83452 -.00048 -.05708 69,2561.84448.84501 -.00053 -.06294 58.9992.85879.85905 -.00026 -.03017 50.3235,87444.87438.00006.00684 42.9012.88991.88971.00020l.226 36.5347.90468.90424 00044.04903 H 31.0483.91780.91759.00021.02263 26.3429.92960.92953.00007.00761 22.3138.93999.94001 -.00002 -.00231 18.8712.94900.94910 -.00010 -.01013 15.9367.95672.95689-.00017 -.01791 13.4429.96338.96353 -.0015 -.01516 11.3265.96899.96915 -.00016 -.01620 9.5353.97382.97389 -.00007 -.00685 8.0208.97787.97787 -.00000 -.00048 6.7438.98149.98122.00027.02755 The Average Per Cent Deviation For the Fit is.0289%

TABLE 19 FIT OF THE 800 K ISOTHERM OF NEON TO THE BERLfN EXCPANSION p(atm) Z exp Z calc Z exp - Z calc % deviation 301.6140 1.25022 1.25022.00000.00003 216.3159 1.07020 1.07021 -.0000C1 -.00117 166.1444.98122,98116.00C06.00650 133.0937.93837.93843 -.00006-.00633 109.3310.92026.92019.00007.00722 91.0234.91471.91495 -.OC024 -.02587 76.4044.91667.91675 -.00008 -.00898 64.3973.92244.92227.00017.01836 54.3665.92977..92957.00020 902120 45.9233.93768.93749 O.00019 *02076 38.7807.94540.94531.00009.00908 a 32.7298.95262.95266 -.00004 -.00410 27.6038.95924.95932 -.00008 -.00797 23.2603.96505.96522 -.00017 -.01715 19.5856.97018.97035 -.00017 -.01789 16.4790.97460.97478 -.00018 -.01822 13.8561.97840.97855 -.00015 -.01570 11.6434.9816C.98176 -.00016 -.01607 9.7805.98446.98446 -.00000 -.00018 8.2123.98692.98674.00018.01838 6.8930,98903.98865.00038.03824 The Average Per Cent Deviation For the Fit is.0331%

TABLE 20 FIT OF THE 100~ K ISOTHERM OF NEON TO THE BERLIN EXPANSION P(atm) Z exp Z calc Z exp - Z calc % deviation 304.2405 1 24211 i.24211.00 OO 00000 232.6842. 113411 1.13411.COCO0.00010 184.1213 _ 1. 7144 1.C47145 -.Cccl -.00123 148.9C23 1.C 3457 1.03454.00003.00326 122.094 6 1. 129 - r i.012 91 -. )O l.00128 100.9979 1. C 47 1.C 0049 -.00 02 -.00198 84.0062.99363.99365 -.OCC2 -.00152 70.1102 99.992 99'C19.001.100069 58.6309.98879.98879.COCCO.00028 49.0917.98f859.98859.OOOs.0t 0022 1 41 1338.98911 _.989C77.000U4.00422 34.4771 0 98995.9899- 0-.00.50493 28.9013 99J91.99089.OCOG2.00243 24.2283 991 92.99190r.0 G0C2.o00179 20.30'85.99282 99288 -C.OC0 6 -.'0627 17.0217.99365.99379 -.00 14 -.01400 14.2673____.99451 99461 -.00010 -00966 11.9583. 99536.99533.CC2.C300311 10 021;.99 61.99596 *.00C 5.0487 8,3973.96 9966651.C0QC1 *.01014 The Average Per Cent Deviation For the Fit is.0036%

TABLE 21 FIT OF THE 1200 K ISOTHERM OF NEON TO THE BERLIN EXPANSION P(atm) Z exp Z caic Z exp - Z calc % deviation 304.7889 1.23859 1.23860 -.00001 -.00050 239.0719 1.15994 1.15988.0C006.00545 191.4889 1.10930 1.10947 -.00017 -.01554 155.5557 1.07600 1.07592.0008.00741 127.5737 1.05370 1.05356.0CC.14.01319 105.2952 1.03849 1.03846.00CC3.00272 87.2889 1.028 1028CC07 -.00007 -.00652 72.5757 1.0264 1.0276 -.00012 -.01192 60.4697 1.01547 1.01553 -.OCC06 -.00635 50.4512 1.01170 1.01173-003.00335 42.1355 1.00898 1.00894.C0C4.00420 4 35.2140 1.00694 1.00686.00008.00809 29.4441 1.0541 1.C00530.00011.01094 24.6269 1.C0417 1.00412.O00OC5.00473 20.6036 1.00323 1.00323.000C:.00027 17.2411 1.00248 1.0'0254 -.00006 -.00620 14.4303 1.C..1 9 5 1.00201 -.00CC6-.00643 12.0791 1.0.153 1.C0161 -.00008 -.00751 10.1135 1.136 1.00129,.0007.00738 The Average Per Cent Deviation For the Fit is.0068%

-135TABLE 22 VALUES OF COMRESSIBILITY FOR HELIUM FOR EVEN VALUES OF PRESSURE TEMPERATURE (~ K) v P(atm) cc/gm 70 80 100 120 10 v 146,22155 167.05729 207.85583 249.10711 Z 1,01899 1,01866 1.01395 1.01265 20 v 74.46615 84.81940 105.40945 126.10496 Z 1,03788 1,03440 1.02841 1.02526 30 v 50.56913 57.43824 71.26328 85.10180 Z 1,05722 1.05072 1,04290 1.03785 40 v 38.63447 43.76747 54,19194 64,59847 Z 1.07694 1,06752 1.05743 1,05040 50 v 31.48312 35.57814 43,95036 52.29512 Z 1.09700 1.08472 1.07198 1,06293 60 v 26,72221 30.12752 37.12354 44.09180 1,.11733 1,10225 1.08657 1,07543 70 v 23.32633 26.24037 32.24788 38,23138' 1.13789 1.12004 1.10117 1,08791 80 v 20.78287 23,32922 28.59160 33.83529 Z 1.15865 1.13803 1.11579 1,10036 90 v 18,80710 21.06784 25.74815 30.41545 Z 1.17956 1.15619 1.13043 1.11278 100 v 17.22827 19.26060 23.47362 27.67899 Z 1.20060 1.17445 1.14508 1,12519 1lo v 15.93775 17.78311 21.61278 25.43955 Z 1,22173 1.19279 1.15973 1.13756 120 v 14.86316 16,55251 20.06214 23.57287 z 1.24294 1. 1118 1.. 1 714 39 1,14992 130 v 13.95446 15.51152 18.75008 21.99295 z 1.26419 1.22960 1.18905 1.16225 140 v 13.17589 14.61926 17.62542 20.63833 Z 1.28548 1.24801 1.20371 1.17456 150 v 12.50130 13.84580 16.65064 19.46397 Z 1.30678 1.26641 1.21836 1.18685 160 v 11.91107 13.16874 15.79760 18.43606 Z 1.32809 1.28478 1.23301 1.19912 170 v 11.39021 12.57099 15.04479 17.52876 Z 1.34939 1.30312 1.24764 1.21136 180 v 10.92711 12.03926 14.37548 16.72197 Z 1.37068 1.32141 1.26226 1.22358 190 v 10.51258 11.56309 13.77644 15.99981 Z 1.39194 1.33965 1.27687 1.23578 200 v 10.13931 11.13415 13.23713 15.34960 Z 1.41317 1.35785 1.29145 1.24796 210 v 9.80136 10.74568 12.74899 14.76104 z 1.43438 1.37600 1.30602 1.26012 220 v 9.49391 10.39218 12.30501 14.22574 Z 1.45554 1.39410 1.32057 1.27225 230 v 9.21296 10.06910 11.89943 13.73674 Z 1.47667 1.41216 1.33509 1.28436 240 v 8.95520 9.77267 11.52743 13.28825 Z 1.49776 1.43018 1.34958 1.29644 250 v 8.71783 9.49971 11.18497 12.87540 Z 1.51882 1.44816 1.36405 1.30850 260 v 8.49851 9.24755 10.86863 12.49409 Z 1.53983 1.46611 1.37849 1.32054

-136TABLE 22 VALUES OF COMPRESSIBILITY FOR IIELIUM FOR EVEN VALUES OF PRESSURE (CONT'D) TEMPERATURE (~ K) v P(atm) cc/gm 70 80 100 120 270 v 8.29522 9.01391 10.57550 12.14079 Z- 1.56081 1.48403 1.39290 1.33255 280 v 8.10625 8.79681 10.30309 11.81252 Z 1.58174 1.50193 1.40728 1.34454 290 v 7.93013 8.59459 10.04923 11.50667 z 1.60264 1.51981 1.42163 1.35651 300 v 7.76557 8.40576 9.81209 11.22101 Z 1.62350 1.53767 1.43594 1.36844 310 v 7.61145 8.22906 9.59002 10.95356 Z 1.64432 1.55553 1.45023 1.38036 320 v 7.46680 8.06335 9.38162 10.70263 Z 1.66511 1.57337 1.46448 1.39224 330 v 7.33076 7.90765 9.18563 10.46671 z 1.68585 1.59121 1.47869 1.40410 340 v 7.20255 7.76108 9.00097 10.24447 Z 1.70656 1.60904 1.49288 1.41593 350 v 7.08152 7.62285 8.82666 10.03474 Z 1.72724 1.62686 1.50702 1.42774 360 v 6.96705 7.49226 8.66184 9.83647 z 1.74787 1.64468 1.52114 1.43951 370 v 6.85861 7.36869 8.50574 9.64872 Z 1.76846 1.66248 1.53521 1.45126 380 v 6.75572 7.25157 8.35766 9.47068 Z 1.78901 1.68028 1.54926 1.46298 390 v 6.65794 7.14038 8.21700 9.30158 Z 1.80951 1.69805 1.56327 1.47467 400 v 6.56488 7.03466 8.08320 9.14076 Z 1.82997 1.71581 1.57724 1.48633 410 v 6.47619 6.93399 7.95575 8.98761 z 1.85038 1.73353 1.59118 1.49797 420 v 6.39153 6.83798 7.83422 8.84157 Z 1.87073 1.75123 1.60509 1.50957 430 v 6.31063 6.74628 7.71817 8.70216 Z 1.89103 1.76888. 118)97 1.52114 440 v 6.23320 6.65857 7.60725 8.56892 Z 1.91126 1.78648 1.63281 1.53268 450 v 6.15901 6.57455 7.50112 8.44144 Z 1.93144 1.80403 1.64662 1.54420 460 v 6.08783 6.49396 7.39946 8.31935 Z 1.95154 1.82151 1.66040 1.55568 470 v 6.01946 6.41654 7.30198 8.~O229 Z 1.97157 1.83892 1.67415 1.56714 480 v 5.95372 6.34207 7.20844 8.08996 Z 1.99153 1.85625 1.68786 1.57856 490 v 5.89043 6.27035 7.11858 7.98207 Z 2.01141 1.87350 1.70155 1.58996 500 v 5.82943 6.20119 7.03220 7.87835 Z 2.03120 1.89065 1.71521 1.60132 510 v 5.77060 6.13443 6.94909 7.77856 Z 2.05092 1.90770 1.72883 1.61266 520 v 5.71379 6.06990 6.86905 7.68247 z 2.07054 1.92464 1.74243 1.62397 530 v 5.65889 6.00749 6.79192 7.58988 Z 2.09009 1.94148 1.75600 1.63525

-137TABLE 22 VALUES OF COMPRESSIBILITY FOR HELIUM FOR EVEN VALUES OF PRESSURE (CONT'D) TEMPERATURE (V K) P(atm) cc/gm 70 80 100 120 540 v 5.60580 5.94706 6.71754 7.50060 Z 2.10954 1.95822 1.76953 1.64651 550 v 5.55442 5.88853 6.64575 7.41445 Z 2.12891 1.97485 1.78304 1.65774 560 v 5.50466 5.83179 6.57641 7.33126 Z 2.14821 1.99138 1.79652 1.66894 570 v 5.45647 5.77678 6.50940 7.25088 Z 2.16742 2.00783 1.80997 1.68012 580 v 5.40977 5.72344 6.44459 7.17318 Z 2.18657 2.02419 1.82338 1.69127 590 v 5.36450 5.67174 6.38187 7.09801 Z 2.20566 2.04048 1.83677 1.70240 600 v 5.32063 5.62164 6.32112 7.02527 Z 2.22470 2.05674 1.85012 1.71352 610 v 5.27811 5.57313 6.26226 6.95483 Z 2.24370 2.07298 1.86344 1.72461 620 v 5.23693 5.52622 6.20518 6.88659 Z 2.26269 2.08922 1.87673 1.73568 630 v 5.19706 5.48093 6.14979 6.82045 Z 2.28168 2.10552 1.88997 1.74674 640 v 5.15850 5.43729 6.09601 6.75632 Z 2.30070 2.12191 1.90318 1.75778 650 v 5.12124 5.39536 6.04375 6.69411 Z 2.31977 2.13845 1.91635 1.76881 660 v 5.08529 5.35520 5.99294 6.63375 Z 2.33893 2.15519 1.92947 1.77982 670 v 5.0o5068 5.31691 5.94351 6.57515 Z 2.35821 2.17220 1.94255 1.79083 680 v 5.01743 5.28057 5.89538 6.51826 z 2.37765 2.18955 1.95558 1.80183

-138TABLE 23 VALUES OF COMPRESSIBILITY FOR NEON FOR EVEN VALUES OF PRESSURE TEMPEATURE (~ K) v P(atm) cc/gm 70 80 100 120 10 v 27.68238 32.01054 40.49397 48.85133 Z.97266.98414.99597 1.00127 20 v 13.46336 15.77149 20.18591 24.47035 Z.94611.96977.99296 1.00310 30 v 8.72996 10.36678 13.42621 16.35167 Z.92022.95616.99067 1.00544 40 v 6.37647 7.67547 10.05452 12.29826 Z.89619.94391.98918 1.00827 50 v 4.98064 6.07249 8.03863 9.87083 Z.87501.93347.98857 1.01157 60 v 4.06737 5.01544 6.70101 8.25636 Z.85748.92518.98889 1.01535 70 v 3.43225 4.27139 5.75119 7.10647 Z,84418.91925.99017 1.01959 80 v 2.97231 3.72347 5.04387 6.24698 z,83549.91581.99245 1.02432 90 v 2.62976 3.30647 4.49826 5.58113 Z.83160.91490.99573 1.02953 100 v 2.36937 2.98104 4.06582 5.05087 z.83251.91650 1.00001 1.03524 110 v 2.16829 2.72197 3.71565 4.61923 z.83804.92054 1.00527 1.04144 120 v 2.01090 2.51235 3.42709 4.26156 z.84787.92689 1.01149 1.04815 130 v 1.88609 2.34037 3.18582 3.96078 z,86152.93539 1.01864 1.05535 140 v 1.78573 2.19754 2.98160 3.70465 z.87842.94587 1.02667 1.06304 150 v 1.70365 2.07762 2.80688 3.48420 z.89790.95813 1.03555 1.07119 160 v 1.63518 1.97590 2.65602 3.29267 Z.91927.97196 1.04522 1.07980 170 v 1.57669 1.88876 2.52466 3.12487 z.94178.98717 1.05562 1.08882 180 v 1.52539 1.81344 2.40943 2.97677 z.96474 1.00355 1.06670 1.09823 190 v 1.47920 1.74774 2.30767 2.84517 Z.98750 1.02093 1.07841 1.10799 200 v 1.43659 1.68995 2.21724 2.72750 z 1.00953 1.03913 1.09068 1.11807 210 v 1.39654 1.63871 2.13641 2.62170 z 1.03046 1.05800 1.10347 1.12843 220 v 1.35850 1.59294 2.06378 2.52606 z 1.05012 1.07743 1.11671 1.13904 230 v 1.32234 1.55179 1.99820 2.43921 z 1.06863 1.09730 1.13038 1.14988 240 v 1.28833 1.51458 1.93873 2.36000 z 1.08642 1.11756 1,14441 1.16091 250 v 1.25717 1.48079 1.88456 2.28752 z 1.10431 1.13815 1.15879 1.17214 260 v 1.22994 1.45000 1.83505 2.22098 z 1.12361 1.15906 1.17348 1.18357 270 v 1.20811 1.42189 1.78966 2.15979 Z 1.14611 1.18031 1.18847 1.19523 280 v 1.19357 1.39622 1.74794 2.10347 Z 1.17426 1.20193 1.20376 1.20717 290 v 1.18861 1.37281 1.70952 2.05164 Z 1.21114 1.22397 1.21935 1.21947 300 v 1.19594 1.35150 1.67410 2.00404 z 1.26063 1.24653 1.23526 1.23226.......

APPENDIX A LIST OF PHYSICAL CONSTANTS Local gravitational acceleration g = 32.1618 ft/sec2 Standard gravitational acceleration g = 32.1740 ft/sec2 Universal Gas Constant R = 82.0575 liter-atm/gm mole ~K Planck's Constant h = 6.6252x1034 joule-sec Boltzman's Constant k = 1.3804x10-23 joule/~K Avogadro's number N = 6.0249x1023 molecules/gm mole Molecular weight of Helium 4.003 gm/gmmole Molecular weight of Neon 20.183 gm/gmmole Density of Mercury at 0 ~C Phg = 13.5951 gm/cm3 Standard atmosphere 14.6959 lbf/in2 Absolute Zero of Temperature - 273.15~C -139

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