Engineering Research Institute The University of Michigan Ann Arbor, Michigan Data and Equations for-the Thermodynamic Properties of "Freon-C318" Perfluorocyclobutane by R. M. Kapoor 3B. G. Bray R. G. Renizus tI, L.,. Salive R, K, Bhada and J. J. Martin Project Supervisor Project 1777 E. I. du Pont de Nemours and Company Wilmington Delaware August, 1956

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page OBJECTIVE iii LIST OF TABLES iv NOMENCLATURE v SUMMARY vil I INTRODUCTION 1 II LITERATURE SURVEY 2 III EXPERIMENTAL WORK 3 IV. RESULTS 4 Vo PROCEDURE FOR CALCULATING TABLES OF THERMODYNAMIC PROPERTIES 11 VIo MARTIN-HOU EQUATION OF STATE WITH A5 TERM 21 VIIo BIBLIOGRAPHY 26, —----, ----— ii

The University of Michigan * Engineering Research Institute OBJECTIVE The purpose of this project is to develop the information necessary for calculating tables of thermodynamic properties of "Freon" refrigerantso The present report is devoted.to the work done on "Freon-C318", perfluorocyclobutane, iii.

The University of Michigan * Engineering Research Institute LIST OF TABLES Table Page I SUMMARY OF AVAILABLE DATA la II COMPARISON OF VAPOR PRESSURE EQUATION WITH THE EXPERIMENTAL VAPOR PRESSURE DATA 5 III COMPARISON OF SATURATED LIQUID DENSITY EQUATION WITH THE EXPERIMENTAL DATA 7 IV COMPARISON OF PVT EQUATION WITH EXPERIMENTAL DATA 9 V FUNDAMENTAL VIBRATION FREQUENCIES OF tFREON-C318" 12 VI COMPARISON OF HEAT CAPACITY AT ZERO PRESSURE WITH THE EQUATION 14 iv

The University of Michigan * Engineering Research Institute NOMENCLATURE A,B,C,D Constants of vapor pressure equation A2,B2,C2 A3,B3,C3 Constants in equation of state A4,A5,B5,C5 Equation of state a,b,cd Constants of heat capacity equation b Characteristic constant of equation of state Cp Heat capacity at constant pressure CV Heat capacity at constant.volume d1- Saturated liquid density e Base of natural logarithm flf2,f3 f4,f5 Temperature functions of the equation of state fl(Tc), s2(TC) f(T), f (Tc), Temperature functions of the equation of state f3(Tc), f4(Tc), at T=Tc f5(Tc) H Enthalpy AHv Latent heat of vaporization J Conversion factor k Constant of equation of state m Slope of critical isometric on pressure temperature diagram N,n Constants used in developing equation of state P Pressure R Universal gas constant S Entropy Sv Entropy of vaporization V

The University of Michigan * Engineering Research Institute T Absolute temperature, ~R (~F + 459.69) TB Boyle point temperature TI Absolute temperature for which the slope at Pr=0 of the isotherm on the compressibility chart equals the slope of line joining the critical point and (Z=1, Pr=O) V Specific volume Z Compressibility factor Subscri ts Bar indicates extensive property per unit mass V Constant volume P Constant pressure T Constant temperature C The value at critical point r Reduced property Superscripts g Property of gas, Eq. Vg 1 Property of liquid, Eq. V1 v Vaporization property, Eqo Hv * Star indicates zero pressure vi

The University of Michigan * Engineering Research Institute SUMMARY Experimental data on the critical temperature, vapor pressure, saturated liquid density and pressure-volume-temperature behavior of "Freon-C318", perfluorocyclobutane, have been determined. The ranges of these determinations are as follows: 1. Vapor pressure was measured from 419.94~R to 698.850R. The corresponding pressure range is from 2.7933 psia to 401.4 psiao 2. The saturated liquid density measurements were made from 473.04~R to 697,27~R. The equivalent liquid density range is from 101.66 lbs/ft3 to 57.9 lbs/ft3. 3. The pressure-volume-temperature measurements cover the following ranges: Pressure from 65.37 psia to 558.90 psia Density from 2.429 lbs/ft3 to 60.73 lbs/ft3 Temperature from 571.32~R to 875o49~R 4. The critical temperature observed by averaging the appearance and disappearance of the meniscus is 699o270R. The critical pressure estimated from the vapor pressure equation is 401.44 psia. The critical density obtained by the rectilinear diameter method is 38.70 lbs/ft3. 5. The specific heat of the gas at zero pressure was determined from the spectroscopic data reported by the Naval Research Laboratory. The range covered is from 360~R to 12600R. All the above data have been fitted by empirical equations. From these, derived equations have been developed to give the changes in thermodynamic properties such as enthalpy and entropy. vii

The University of Michigan * Engineering Research Institute I INTRODUCTION This report presents the experimental data and empirical equations necessary for the calculation of a table of thermodynamic properties of t'Freon-C318" perfluorocyclobutane The thermodynamic properties of general interest are pressure, temperature, volume, enthalpy and entropy of saturated liquid and vapor, and the superheated vapor. The minimum experimental data necessary for this purpose ares 10 Critical temperature 2o Vapor pressure datao 3o Saturated liquid density datao 4o Heat capacity of the gas at zero pressure Critical pressure can be estimated from the vapor pressure equation once the critical temperature has been measuredo Critical density is obtained by the rectilinear diameter method which employs the liquid density as a function of temperatureo Knowing the critical constants and the vapor pressure data, the Martin-Hou equation of state (9) can be used to predict the PVT behavior of the gaso Thus, the PVT measurements are not absolutely essential; however, they serve as a good check for the calculated values from the equation of stateo Experimental data on vapor pressure, saturated liquid density, PVT behavior and the critical temperature for "FreonC318" were obtained at the University of Michigan and are reported hereo The spectroscopic data on "Freon-C318" reported by the Naval Research Laboratory (10) was used to calculate the spectroscopic specific heat data0 Empirical equations for vapor pressure, liquid density, PVT behavior and the specific heat of gas at zero pressure have been developed to represent these data with a high order of precisiono These equations are in a form ready for use in an electronic digital computer to calculate tables of thermodynamic properties The thermodynamic relationships involved in these calculations have been developed and an outline of the procedure for calculating the tables has been includedo A survey of the work done on this compound by other investigators has also been included as shown in Table Io 1

The University of Michigan * Engineering Research Institute o t- o to rd OCT -; ~0 c 0 (0 () o C e:O 0 0 mjo o 0 o Q0 1 O, 0- 4-) 6E C! Cd Cd < ) H 0 n 4 0 r tC FX tl0 I'- t 0o 8 o k- fi >H 0 0 0 F< 4ci+30C\ O 0 H M rI ^ 4^ 0> -P O 0 0 M 0 L t- 8 > O o O 0 Go-P (0 02 0 0 H H( 0 o O 40 )0 H 0 C0 o Hi V) Coc 0C- Hl 0 0 0 0 Or-4 LO Lf,_ 4 cnR p G o tog 4 U C) * 0 0 to H -P) ~4 )r t(C) m^V o4 co L0 ~ 4 O 0Q * 0 O L- nii'L U U 4) I C f*H 0 c t ) 0) ) O 0) r~? C ^H 1 0 0 Q4 4 Hq 0 04 p TO *H A — a

The University of Michigan * Engineering Research Institute IIo LITERATURE SURVEY The following experimental data and correlations have been reported by various investigators: lo CRITICAL TEMPERATURE: The only data reported are by Graham (6)o He found the critical temperature to be 2390o5Fo 2o VAPOR PRESSURES These data are available from two sources: (a) Graham (6) reported the results of his measurements from -41ol0~F to 212~Fo The data have been represented by an equation of the form: log P = A $ B + C log T + DT (b) Furakawa et al (5) measured the vapor pressure of "Freon-C318" from 77 to 274~Ko They have developed an equation of the form: logl0P = A + T + CT + DT2 T They claim that this equation represents the data within _ O05 mmo of Hgo in most caseso 30 SATURATED LIQUID DENSITY AND PVT BEHAVIOR: Graham (6) has reported empirical equations based on the preliminary work done on this compound at the du Pont laboratorieso 4. SPECTROSCOPIC DATA: (a) Edgell (2) measured the Raman and Infrared spectra of his compound in his study of the spatial configuration of cyclo compoundso His results were later used to check the calorimetric measurements of heat capacity made by Lacher (7) at the du Pont laboratories (b) The Naval Research Laboratory, in a report entitled "Spectroscopic Properties of Fluorocarbons and Fluorinated Hydrocarbons" (10) gives the results of measurements of the fundamental vibrational frequencies of perfluorocyclobutaneo The calculated heat capacity at zero pressure has also been reported at 5 values of temperature ranging from 2680 to 600~Ko 2

The University of Michigan * Engineering Research Institute 5. SPECIFIC HEAT OF GAS: Specific heat measurements by a calorimetric method have been reported by the following: (a) Masi (8) measured the heat capacity of gaseous perfluorocyclobutane with a constant flow calorimeter at three pressures at each of the temperatures 10, 50 and 90~Co He claims an accuracy of 0ol percent in these measurementso The specific heat of the gas at zero pressure, obtained by extrapolation and believed reliable to Oo15 percent has also been reportedo (b) Furukawa et al (5) obtained the heat capacity data in an adiabatic calorimeter from 17~ to 270~Ko They claim an accuracy of about 0o2 percent except in the transition regions of 141o3~ and 1740o6K, the anomalous region of 970K (attributed to glass transformation) and below 50~Ko (c) Lacher (7) determined the heat capacity of the gas at the du Pont laboratories in a preliminary investigation on this compoundo IIIo EXPERIMENTAL WORK The experimental work in this study involved the measurements of vapor pressure, saturated liquid density, critical temperature and pressure-volume-temperature behavior of the gaso The sample of "Freon-C318", perfluorocyclobutane, was supplied by Eo Io du Pont de Nemours and Company0 It was a fractionated product which contained no impurities detectable by infrared analysiso The moisture content of the sample was O00005 weight percent and an air content in the vapor phase of 0oOll volume percent0 The equipment used and the experimental technique employed were essentially the same as described in the "Freon23", (trifluoromethane,) report (3) The vapor pressure measurements were made by the static methodo A sample of the perfluorocyclobutane was charged to the system and the vapor pressure exerted by it was measured at intervals of temperatureo The saturated liquid density was measured by the method of Benning and McHarness (1)o In this system, the temperature was found at which the saturated liquid had a known density The critical temperature measurement was based on the phenomena of disappearance and reappearance of the meniscus 3

The University of Michigan * Engineering Research Institute between the vapor and liquid phases The average of the two temperatures at which the disappearance and reappearance of the meniscus occur was considered to be practically the critical temperature The pressure-volume-temperature measurements were carried out as a series of approximately constant volume and hence, constant density runso The pressures at various temperatures for each run were recorded, making sure that the system was leakproof during the runo For a detailed description of the equipment and the experimental procedure, a reference to the "Freon-253 report is suggestedo IVo RESULTS The following equations have been developed to represent the experimental data obtained in this investigations lo VAPOR PRESSURE The general form of the equation chosen to represent the vapor pressure data is: log0 P = A + T l C logl0T + DT (1) The constants of the equation were obtained by fitting the experimental data by the method of least squareso The constants are A = 46o8587746 B = -4,270 76331 C = -14o573528 D C 0 o00473182 The pressure is in lbs/sqoino and temperature in ~R (~F+459o69) The experimental data and its comparison with the calculated values are shown in Table II on the following pageo 2o LIQUID DENSITY EQUATION The following form of empirical equation was developed in the "Freon-23" report (3) to represent the density of the saturated liquido 1 m 1/5 T 2/3 T 4/1 (2) d a0 # a1(l- ) + a2 (-) 2/ a (l4) a4(l ) (2) Tc Tcc At the critical temperature, the equation reduces to the form of d1 = ao; so that a0 is the critical densityo This value was obtained by the rectilinear diameter method, 4

The University of Michigan * Engineering Research Institute TABLE II COMPARISON OF VAPOR PRESSURE EQUATION WITH THE EXPERIMENTAL VAP OR PRESSURE DATA VAPOR PRESSURE EQUATION: logl0P = 46o8587746 - 49270o76331 14o573528 log1T + 0Q00473182 T Percent T~R Pe lb/sq.in. P.a, - lb/sqin. Deviation+ — expP _alco bo_ 419 94 2o7933 2,7976 -0o15 424 69 3 2635 3 2504 +0.4 428.80 3.7081 3.6873 +0.56 4370.99 4 8521 4.8433 +0.18 466 13 10.205 10.296 -0 89 479o14 14 024 14,025 -0o07 480053 14.393 14,549 1-o08 482,17 14,969 15.105 -0 91 498,19 21o391 21.529 -0o64 520.67 34,300 33.894 +1,18 531.55 41.88 41o55 +0.79 535155 42o14 41o55 +1o40 549.36 57.19 56,86 +0o58 549.70 57.83 57o18 +1o12 571.88 81o74 81o87 0.o16 549002 113o09 113o72 -o056 600o59 124o59 124o72 -0o26 619.36 159,46 160,54 -0o68 637o85 201o26 202.67 -0o70 653.18 242o06 243.35 -0o53 669.62 291.46 293,11 -0.57 677o09 317.14 318.17 -0o32 687.98 358.33 357.15 +0o33 696o69 393o03 390o95 +0o53 698.85 402,83 395o55 +1.81 +Percent Deviation Exp. - Cal x 100 Exp o ----------------------— 5 ----------— I

The University of Michigan * Engineering Research Institute The other constants of this equation were obtained by fitting the equations to four evenly distributed values of temperature and liquid density. The constants thus obtained are: a0 = 38.70 a1 = 70o85831830 a2 = 23o60975930 a3 = 15.98918211 a4 = 8.924385653 1 ^3 The liquid density, d is in lbs/ft and the temperature is in OR, The comparison of calculated values with the experimental data is shown in Table III. 3. EQUATION OF STATE The PVT behavior is represented by the Martin-Hou equation of state (9). This equation has been modified by the addition of an A5 term, this addition being necessitated in the higher density range (about 1o5 critical density), A full discussion of this is given later. The modified form of the equation is: kT kT kT Tc Tc Tc RT A_+BT+Ce AA3+B3T+c3e A4 A5+B5T+C5e p + _..........+.+ + (V-b) (V-b)2 (V-b)3 (V-b)4 (V-b)5 (3) The information numbers used in evaluating the constants of the equation are: Tc = 699o27~R V =.0258397932 ft3/lb Pc = 401,44 psia P 3.24 M = 4.68 Tf = 0.81 Tc TB = 1575~R -----------------— g ------------------

The University of Michigan * Engineering Research Institute TABLE III COMPARISON OF SATURATED LIQUID DENSITY EQUATION WITH THE EXPERIMENTAL DATA SATURATED LIQUID DENSITY EQUATION: 1 2 T 3 T 3 d 38.70+ 70.85831830 (1 Tc-) + 23.60975930 (1- T ) 4 T T3 + 15.98981211 (1- T-) - 8o924385653 (1- c Tc Percent T~R exp lb/tf t Deviation+ exp. lb/ft3 cale. 698.67 45.237 45.659 -0 93 *697.27 49.276 49.276 0.00 688o 91 57o386 57o732 -0.60 *65 920 70.246 70.246 0.00 650.07 72o962 72.842 +0o16 *583o97 86o484 86.484 0000 532.15 94 132 94o263 -0o14 *473,04 101 661 101o661 0o00 * Equation fitted to these points (dl) eXpo (dl)cale. +Percent Deviation - (dl)exp 7 -

The University of Michigan * Engineering Research Institute k = 5o0 n = 1.7 N = 17 M The constants of the equation obtained by the method discussed later are: A2 -1.782832574 B = +0o8288016876 x 10"3 02 = -29,98281801 A3 = +2.220141064 x 10i2 B3 = -0.7000454923 x 10'6 C3 = +0.6970502981 A4 = -2.49243233 x 10"4 A5 = +1,027671206 x 10~6 B5 = +0.2444029514 x 10'9 C5 = -3.742878007 x 10-5 R Oo0536456979 (psia) (ft3)/(lb) (R) k= 50 b = 0 005655630365 It will be noticed that the value of k has been changed from 50475, as suggested in the paper by Martin and Hou (9), to 5.0. This change was made to improve the curvature of the isometrics, _ands i keeping with the suggestions in _their -oao-er.

The University of Michigan * Engineering Research Institute TABLE IV COMPARISON OF CALCULATED PVT DATA WITH EXPERIMENTAL DATA FOR "FREON-C318" Density lb/sq.in Percent b/cuft T~R exp lb/sqin Pcalc Deviation 2.148 572,63 58.4 58,59 -0,325 641.52 69o2 67o76 +0.21 711o18 77.2 76o70 +0o65 785o07 86.5 85o95 +0.64 862.42 96o7 95 48 +1o26 2.429 571o32 65.37 64o97 +0.612 645.07 76,35 76.27 +0.105 710.19 87.95 85.83 +2.41 781.52 97 47 96o 04 +1.47 851 83 107 o 70 105 94 +1.634 7,900 651.44 199,5 199.45 -0o03 698o42 227.7 227.42 +0.123 768o55 266o7 266.91 -0o0787 813o03 291o6 291o00 +0o0206 849.84 312.0 310.55 +0,465 15.03 680,89 313040 312o02 +0.440 720o65 365o40 365o43 -0.0082 761o67 41900 418o05 +0.o227 809.01 480.50 476.47 +0o839 820o86 495o50 490.79 +0,951 875o49 558.90 555.99 +0.521 22,31 694.68 37600 374,98 +0.271 728o25 454o5 451.20 +0,726 769.11 544o0 540o45 +0.653 820.64 651.3 648.93 +0.364 864.33 742o0 740o34 +0.224 33.30 699o52 40309 402o14 +0.436 738.79 555.4 553o58 +0,328 776.94 695.4 698o10 -0,388 825.55 874o4 879,52 -0.586 852.59 973.4 979o42 -0.618 44.45 707.28 453.0 446.09 +1.525 74 999 681,0 687.30 -0.925 782.89 873.4 876 97 -0.409 816 25 106904 1071.95 -0.238 857.52 131304 1316.00 -0.198 9

The University of Michigan * Engineering Research Institute TABLE IV (continued) Density Percent lb/cu~fto T~R P ab/sqin, P l~b/cuft TO~R exp, q ocalc, Deviation 50.84 698o80 402o0 402.13 -0o032 704o49 444*0 440o60 +0o766 752.14 781.4 774.76 +0.850 796.81 1111.4 1103o54 +0o707 838o86 1430.4 1422 o97 +0.519 862o04 1612.4 1602e22 +0.631 55o77 695.95 405.9 398o01 +1.944 721 19 616. 4 61008 +1 025 763 76 987. 4 98561 +0,181 801.60 13384 1334o 09 +0.322 861,36 1895.4 1904o62 -0.486 60.73 688,25 373,4 386.89 -30613 716o71 668.4 705,33 -5.525 754o51 1080,4 1143o88 -5.8756 792.09 1502.9 1593,66 831.08 1946o9 2071,39 -6.4216 10

The University of Michigan * Engineering Research Institute symmetry species of this assignment is Blu and it was shifted from 613 to 972~ This gave a better comparison of the calculated heat capacity with the experimental data reported by Masi (8). The fundamental vibration frequencies generally used to calculate the heat capacity are given in Table V. Results of calculations of the heat capacity of the gas at zero pressure were also given in the above mentioned report at temperatures ranging from 2680K to 600~K. These results were modified for the change in the fundamental assignment indicated. Additiona alalculations of heat capacity were made at various values of temperature within the above range to get a good spread of the data All the results thus obtained are shown in Table VIo HEAT CAPACITY EQUATION: The calculated spectroscopic heat capacities were represented by an equation of the form: C* a + bT + CT2 + dT (4) The constants of this equation are: a 6o49044393 b = 7o399783877 x 10-2 c = -3o297575755 x 105 d = 4,306508915 x 10-9 Here T is in ~R and Cp is Btu/(lb mole)((OR) A comparison of the calculated values with the spectroscopic results is shown in Table VI o At three values of temperature given in Table VI MasiVs (8) data obtained by calorimetric measurements were also available, These are shown within parentheses and the deviations have been calculatedo V. PROCEDURE FOR CALCULATING TABLES OF THERMODYNAMIC PROPERTIES The calculations of tables of thermodynamic properties, in general, are based on exact thermodynamic relations, The approach to the problem varies with the type of experimental data availableo The purpose of this section is to indicate how the empirical equations presented earlier can be utilized to obtain tables of thermodynamic properties, The discussion is presented under two headings: 1, Properties of the saturated liquid and vapor. 2. Properties of superheated vapor. 11

The University of Michigan * Engineering Research Institute TABLE V FUNDAMENTAL VIBRATION FREQUENCIES OF PERFLUOROCYCLOBUTANE ACTIVE VIBRATIONS *1 SYMMETRY APPROX;MATE *2 WAVE NUMBER SPECIES CHARACTER INFRARED GAS)RN (GAS) Alg CF Stretching 1431.3 cm1 Alg Ring Stretching 699.3 Aig CF2 Deformation 358,2 A2u CF Stretching 1239 A2u CF2 Rocking 338 BLg CF Stretching 1220 Blg CF2 Wagging 258 B2g CF Stretching 1008 Bgg CF2 Deformation 659,5 B2g In-plane ring bending 192 Eg CF Stretching 1285 Eg CF2 Twisting 439 Eg CF2 Twisting 273 Eu CF Stretching 1340 Eu Ring Stretching 962 Eu CF2 Deformation 569 Eu CF2 Wagging (285) *1 Fundamentals not observed directly are enclosed in parentheses. *2 Only very rough meaning can be attached to the terms used in this column 12

The University of Michigan * Engineering Research Institute TABLE V (continued) INACTIVE VIBRATIO NS SYMMETRY APPR OXIMATE CALCULATED SPECIES C HARACTER VALUE Alu CF2 Twisting (173) A2g CF2 Wagging (745) Blu CF Stretching (1385) Blu CF2 Rocking (972)*3 Blu Out of plane ring bending (86) Bgu CF2 Twisting (250) *3 This assignment has been modified. The original assignment reported is (613) 13

The University of Michigan * Engineering Research Institute TABLE VI COMPARISON OF HEAT CAPACITY AT ZERO PRESSURE WITH THE EQUATION HEAT CAPACITY EQUATION C = 6.490443930 + 7.399783877 x 102 T - 3.297575755 x 105T2 + 4.306508915 x 109T3 Btu/(lb mole) (R) SPECTROSCOPIC HEAT CAPACITY+l1 HEAT CAPACITY EQUATION C*. Percent T~R p CP Deviation +2 360,00 28o947 29.057 -0029 432 00 32 651 32 651 0.00 509.69 36o215(36,19) 36 210 +0,01 (06) 536o40 37o369 37 360 +0.03 563o40 38o491 38,484 +0.02 581o69 39o224(39,23) 39,224 0.00(0,02) 653o69 41.974(42.01) 41.974 0.00(0,09) 720o00 44.282 44 282 0o00 900o00 49.493 49,518 -0o05 1080.00 53o387 53,370 +0.03 1260o00 56o272 55.990 +0,50 lMasi's calorimetric values are given in parentheses (C'*)spec. - (C)Calco +2Percent Deviation = -_C.. G. x 100 Spec4 14

The University of Michigan * Engineering Research Institute The general procedure is as follows: lo Saturation properties: The properties to be evaluated ares (a) Volume of saturated vapor, (b) Volume of saturated liquido (c) Enthalpy and entropy of vaporization. (d) Enthalpy of saturated vapor and liquid. (e) Entropy of saturated vapor and liquid. Before any calculations of these properties are undertaken, it is necessary to decide on the temperature range available and the intervals of the temperature desired in the tables. The former is limited by the triple point at one end and the critical point at the othero The temperature interval in the intermediate range is mostly a matter of choice depending on the ease of the user, and the extensiveness of the tables desired. Having fixed this, the properties are evaluated as follows: (i) Saturation pressure is obtained from the vapor pressure equation (1) by substituting the values of temperature fixed above. (ii) The volume of saturated vapor is obtained by substituting the values of pressure and temperature in the equation of state (3)o (iii) Saturated liquid volume is simply the reciprocal of saturated liquid density which has been represented by an equation as a function of temperature (Eqno 2)o (iv) [a. The enthalpy of vaporization is calculated from the Clapeyron equations: d: HV cdT% I l) (5) T(Vg;Vl)J where Vg = volume of vapor (ft3/lb) V1 =volume of liquid (ft3/lb) J conversion factor The vapor pressure equation (1) is differentiated to give dP _ [2 302585 (A+ +DT)+ClnT]+ B L35s \T2 0.* la) 15

The University of Michigan * Engineering Research Institute [b] The entropy of vaporization is obtained from the following equation: Sv dP (g V1)J (5a) dT - (v) [a] The calculations of enthalpy and entropy are based on some datum termperature at which both these properties are zero. This is an arbitrary choice and can be any value whatsoevero [b] The equations used to calculate the enthalpy of vapor are next developedo o VEnthal of Vao The fundamental relation which expresses the effect of temperature and pressure on enthalpy is: dH = CpdTp + [V - T(dy/dT)p]dPT (6) But VdPT = d(PV)T -PdT (7) and (dV/dT)pdPT (dP/dT) dVT (8) Substituting (7) and (8) in equation (6), we get dH = CpdT + d(PV)T PdVT + T()v dV (9) Integrating between T at 0 pressure, the datum conditions, and any given T and P or V9 and substituting the equation of state and its derivative with respect to T in the above equation we have: Tk / oT V V VA2 + Ce T(1 + k OH r T= CFdT V(Px)' (V - b)2 0 T-T \ Ag + C3e Te( + 4 "~T~dT A5 C 1 + Ik...5..J.dV-T (10) (V - b) 16

The University of Michigan * Engineering Research Institute where subscript, o, indicates datum condition. Integrating: T Tc T T A2+C2e (1 + k~ H -H = f CdTP + P) + 2 4?b - V=0 (V b) sm ii w T k L T (V-b) A3+C3e c( + k ) 2(V- b)2 3(V - b)3 kT / \ V A +C e c + kT+ e ( (11) 4(V - b) V= oo Now AH = H H (H2 H0) (H1 H.) and (12) C*= C*' + R (13) P V Substituting equation (11) into (12), combining with (13) and simplifying we get T A +C e GKI + kw-J AH = GdT ~ PV + -^ T (V - b) A3+C3e + 4 - T c A4 2(V - b)2 3(V - b)3 + T V2T2 (14) +- (14) 4(V - b)4 V T1 Using the references properties as P1, VI, T1 and H1, equation (14) will give Hi-Hr Dropping the subscript 2 in order to i ref~ generalize the properties for any temperature and pressure, we have 17

The University of Michigan * Engineering Research Institute A +C e c + bT2..3 dT4 +...... H = (a-R)T + bT + T3 + PV + -- 2 3 4 V - b T 33e A3C3..+ +,. -. — 2(V - b)2 3(V - b)3 -ki T \ A5C5e C + T +......-.......... + X (15) 4(V - b)4.==, whefre X = Href 3 ref + b Tre r ef Tre~k:ref ref k A T C ( T A2 3+ Tc A4 + P efref +................ 4 (vref b- b Tref (1 Tref -k- Tre A3 +C3 e A^. 3 l k +..... —2 —b.2...... (16) (ef - b) b T T-ref 4 (_ b) Equation (15) is the final form of equation used for calculating the vapor enthalpy. For the given set of constants for the heat capacity equation and the equation of state, values of enthalpy at the temperatures fixed can be thus evaluated by simple substitution of corresponding volume and pressure valueso (v) [c] The liquid enthalpy is the difference between the enthalpy of vapor and latent heat of vaporizations and can be directly calculated. 18

The University of Michigan * Engineering Research Institute Entropy of Vapor (vi) [a] The equations used to calculate the entropy of vapor are developed as follows: The fundamental equation for entropy is: dS = C + dV (17) Integrating the above equation between T at 0 pressure and any given T and P or V, we have r V rB2 T 3 3V~ B C k e Tc q0 Ij C* dT t- 2 2 - V T _ + b (V b)3 (V -b)5 d or S - C' dT+Rln(V-b) 2 - 3 * *.... (19) Also AS = (S - SO) - (S - (20)V Substituting (19) into (20), C72- T k B S SAMS C *+ V-bRln )-(V-b) _ B32 5 ~ TG (V ) (V-(Vb) 2(b)2 4(V-b ( TI Subst~~1 -ituti -....ioT k T Cg c V-2B ~TO~~1

The University of Michigan * Engineering Research Institute Using the reference properties at Pi, V1, Ti and Si, equation (21) will give S - Srf * Dropping the subscript 2 for generalising the propert es for any temperature and pressure, we have: C 2 d 3B2 B3 S = (a-R)lnT + bT + 2 T2 + T5 + Rln(V-b) - V- - 2 b B5 C2 C3 C5 - 4(4V-b)4 (V-b) 2(V-b)2 4 T- ek B5,_____ + 03.. + 4(Veb)~j (22) + Y where Y = Se - a-R)lnT + bT + T red 3 ref ref ref (r 3 ref B2 B3 B5 + Rln(Vrefb) -- B -- V -b 2(V -b)2 4(V -b)4 -ref -ref -ref +k ~ce T C b 4 (23) Ci vref-b 2(Vref-b)2 4(V b) Equation (22). is the final form for the entropy of the vapor and can be programmed for use on any digital computer. For a given set of constants of the heat capacity equation and the equation of state, the values of the vapor entropy can thus be run off. (vi) L[b The liquid entropy is the difference between the the vapor entropy and the entropy of vaporization. The procedure for obtaining these properties has been outlined above so that the liquid entropy can be obtained directly. Superheatred Vap or The properties to be evaluated as functions of P and T or V and T for the preparation of superheat tables are: (a). Volume or Pressure (b}) Enthalpy (c).a Entropy In this case also, it is necessary to fix the range of temperature and the pressure or volume, whichever is desiredO This is an arbitrary choice and depends mostly on the range which would be encountered in practice 20

The University of Michigan * Engineering Research Institute Having fixed these values, the following equations are to be solved for obtaining the various properties. (a. The equation of state No. 3 is to be used, It will be solved by an explicit or implicit method depending on whether vdlume is fixed or is desired, (b) For the known PVT conditions, the values of enthalpy and entropy are obtained from equations (15) and (22) respectively. VI. MARTIN-HOU EQUATION OF STATE WITH A5 TERM Martin and Hou examined the pressure-volume-temperature characteristics of pure gases in detail and developed an equation to fit the PVT characteristics of different gases. This equation requires only the critical properties and a low pressure point on the vapor pressure curve, The equation was used to check the PVT behavior of widely different gases and the agreement with the experimental data was excellent. It has later been used to correlate the PVT data for various "Freon" refrigerants with a high degree of accuracy. The various hypothesis used in the development of this equation have been fully discussed in their paper entitled "Development of an Equation of State for Gases" (9). The conditions are summarized below: 1. PV = RT as P -- 2. (dP/dV)T = 0 at critical 3. (d2P/dV2) = 0 at critical 4. (d3P/dV3) 0 at critical 5. (d4P/dV4)T = at critical 6. dZ/dP ) TP = -(1-Zc) at T' = 0.8T 7 ^ (dZ/dPPr)TrP r= = 0 at the Biyle point temperature 8. (d2P/dT2) V= 0 at V=Vc 9. (dP/dT)v = M -MP/T at V_=Vc 21

The University of Michigan * Engineering Research Institute The general form of the equation developed was: T T 0kii- akA2+B2T+C2e c A +B3+C3e c A RT 2 3 A4 P fV-b+ +.. + T_ + (V-b)2 (V-b) (Vb)4 B5T + (3a) (V-b)5 This equation was shown to predict the data up to densities of 1.3 times the critical within the experimental accuracy. Above these densities, it was found that the equation predicted too much curvature of the isometrics. This situation was improved by the addition of one more curvature constant, C5o The condition utilized to fix this constant was that at some very high density (approximately twice the critical density), the isometric is straights Thus, the modified equation was: T T A2+B +C e C A +B +e Tc RT 2 2 2 3' 3 3 A4 p = (V, )...................... + +. V-b) (V-b)2 (V-b)3 (V-b)4 T B5T + C5eT c (+ 5 (3b) (V-b)5 With the addition of this constant, the procedure for the evaluation of the constants was also modified, The changes in the various formulae were described in the "Freon-12", dichlorodifluoromethane, report (4) The addition of the C5 term did improve the curvature of the isometrics as was expected. However, the calculated values for "Freons-12, -23 and-C318t indicated that the equation predicted a higher slope of the isometrics at about lo51 times the critical density. This led to further investigation of the characteristics of the isometrics at the higher densities. From a study of the general trend of the isometrics at higher densities, it was concluded that the isometrics, in addition to being straight, have a definite slope which could be correlated with the slope of the critical isometric. This condition was finally taken as an additional hypothesis, It was thus possible to add one more constant to the equationo Since only the high density isometrics were subject for improvement, it was decided to add this constant preferably in the fifth or the fourth term, In the fifth term, with B5 and Cg already fixed, the only constant left was A5. Thus, two new equations with B4 and A5 terms 22

The University of Michigan * Engineering Research Institute respectively were studied. It was found that the A5 term did a better job' in the correlation of the data, and so it was finally included in the equation. Thus, the final form of the modified Martin-Hou Equation is: T T A2+B2T+C2e c A3+B3T+C3e Tc p =RT 2 2 3 3 + (V-b) (V-b)2 (V-b)3 (V-b)4 T A5+B5T+C5e c +. —-.-.. (3) (V-b) The addition of the A term necessitated a revision of the procedure for the evaluation of constantso The various steps involved are given belowo PROCEDURE FOR EVALUATING THE CONSTANTS OF THE MARTIN-HOU EQUATION OF STATE WITH C5 AND A5 TERMS The procedure for evaluating the constants of the equation with the B; term alone has been described in the paper by Martin and Hou (9). This procedure employed the nine conditions summarized earlier for the evaluation of the constants. The same formulae hold good for evaluating the temperature functions and the constants A2, B2, C2 and A4 even with the two additional conditions for the C5 and A5 terms. These formulae are reproduced below. fl(Tc) = RTc (24) f2(Tc) = 9Pc(Vb)2 - 3.8RTc(Vc-b) (25) f3(Tc) = 5o4RTc(Vc-b)2 l17Pc (Vcb)3 (26) f4(T) = lSPc (Vc"b)4 - 34RTc(VcGb) (27) f5(T)c = Oo8RTc(Vc -b)4 - 3Pc (V b)5 (28) B Pc c b = V 1 - BZ where Z = - (29) lS2c C C 2 RTc 2(T,)+bRT+ (RT') (1.zc (TB-TCT)+f (Tc)+bRT (Tc-T') (TBT c)'(ke kT e Te (TTc-TT ) )(ek TB/Tc e-k) (30) 2:3

The University of Michigan * Engineering Research Institute / -k TB/Tc =k -f2(T) -bRTB-C2 TB e B2 = ---- (31) (TB )Tc A2 = f(Tc) - B2Tc C02ek (32) A4 = f4(Tc) The derivation of expressions for the two remaining curvature term.constants, C3 and C5 has been given in the "Freon-12" report (4)o The conditions utilized were: lo d2P/dT2 = 0 at critical density 2, d2P/dT2 0 at n times critical density Thus, we have: d2p C2 C3 C5 d2p _ -— 2 + + -— 5 - 0 at V and V /n (33) dT2 (V-b)2 (V-b)3 (b)5 0 a'-c c Therefore, eliminating C5 and solving for C3 in terms of C2p we have: C2 c-b) -C b) 3] 3 [ b) (Vc-b) and 5 - C(V-cb)3 C3(Vo-b)2 (35) The remaining constants to be evaluated are: A39 B39 A5' B5 The conditions used for solving the constants are (i) At V = (V n (1ii) A t V - n d ( N = y times m *d () ^f R B2 B3 B5 00.,. I,, + _ + O V-b (V b) I 2 (Vi-b)3 (Vcb) ( —— + (Vc-b)3 (V T- e)) m 24

The University of Michigan * Engineering Research Institute R B B B5 and v +v2 3 and ()v " V V 2 V 3 V 5 VdM TSc" 1 - C -- -c - n — b — b -b — b n n n n k C2 C3 C5 k c + +e T0V 3 V 5 T rn n e n N ym where y is an arbitrary constanto Eliminating B5 from the above two equations, we have: (V -b) -N — b) - R V +( -b) (Vb)3 R (V)b) +(- _ (V - b (b ban A5 = fs (T') BgT - Ce The value of n is approximately 26 I-' 435

The University of Michigan * Engineering Research Institute VII. BIBLIOGRAPHY 1, Benning, A.F., and McHarness, R.C., Jackson Laboratory Report 30-89, No. 11, SN-15954, 2. Edgell, W,F,, J. Amer. Chem. Soc., 69, 660(1947)o 3. Physical and Thermodynamic Properties of "Freon 23", May,1955, Engineering Research Institute, Ann Arbor. 4. Data and Equations for Some Thermodynamic Properties of "Freon-12", May,1955, Engineering Research Institute, Ann Arbor. 5. Furukawa, G.T,, McCosky, R.Eo and Reilly, M.L,, U.SBureau of Standards, Journal of Research 11-16, 52, January,1954* 6* Graham, D.P,, Jackson Laboratory Report, JLR-69-13, No.13, SN-193353 7. Lacher, Jackson Laboratory Report, JLR-71-3, Noo3, SN-19427, 8. Masi, J.F,, AmerChemSocoJ., 75, 5082-84, October 20, 1953. 9. Martin, JJ. and Hou, YoCo, JoAmeroInstoChemoEngro, Vol.1, No. 2 (1955)o 10. Naval Research Laboratory, NRL Report 3567, 26