ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR TRANSPORT PHENOMENA IN POLYATOMIC GASES BY Ca S. WANG CHANG G. E. UHLINBECK REPORT NO. CM-681 Project M604-6 U. S. NAVY DEPARTMENT, BUREAU OF ORDNANCE CONTRACT NOrd 7924, TASK UMIH-3F July 10, 1951

TABLE OF CONTENTS Section Page Introduction 1 II Transport Coefficients When the Exchange Between the Translational and the Interhal Energies is not Difficult the Enskog-Chapman Expansion 6 III Transport Coefficients When the Exchange Between the Translational and the Internal Energies is Difficult 22 IV Dispersion of Sound and the Connection of the Present Theory with the Existing More or Less Phenomenological Theories 31 APPENDIX Partial List of Symbols Used 44

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN TRANSPORT PEIEtIINOMIIA IN POLYATOMIC GASES I. INTRODUCTION For polyatomic gasest because of the existence of internal energy states one may expect that the transport coefficients will be given by expressions different from those for monoatomic gases. Furthermore9 one expects a new constant9 the relaxation time, which is a measure of the time required for the establishment of equilibrium between the translational and the internal energy states when the former is changed by some process. The effect of the relaxation time makes itself felt through various physical phenomena, most notably the dispersion and absorption of sound during its passage through a polyatomic gas. In fact, the effect of the relaxation time was first suggested by Lorentzl in 1881, in a paper entitled "Les Equations du Mouvement des Gaz, et la Propagation du Son suivant la Theorie cinetique des Gaz". Since then this effect has been seen experimentally, and many theoretical papers have been published on the theory of dispersion of sound in polyatomic 2 gases In an earlier report3 we have given general expressions for the transport coefficients when the passage of energy from the translational 1 H. As Lorentz9 Arch. neerl. 16, 19 1881. See, for instance, Lorentz Collected Papers~ Vplo VI. 2 For a comprehensive surmmary on this and related topics, see Wo T. Richards9 Rev. Mod. Phys. llp 36, 1939. 3 C6 S, Wang Chang and G& Eo Uhlenbeck. On the Transport Phenomena in Rarified Gases. CM-443, Feb. 20, 1948, UtM-3-F.

ENGINEERING RESEARCH INSTITUTE Page 2 UNIVERSITY OF MICHIGAN to the internal degrees of freedom is not difficult. We will report here a calculation from the opposite approximation, namely, when the probability of the transfer of energy from the translational to the internal degrees of freedom is very small. In this way, a more clear-cut definition of the relaxation time is obtained, which ties in very well with the existing more or less phenomenological theories on this matter. As beforep our starting point is the Boltzmann equation for the distribution function, f*o The state of a molecule is determined by its translational as well as internal states. Classically one could take, for instances a nonspherical. molecular models or some very queer kind of spherical models like the rough sphere or'the loaded sphere where rotational states can be excited, The orientation of each molecule is then specified by its Eulerian angles. The linear and angular velocities after collision can easily be expressed in terms of those before collision and some collision parameters. But in trying to work out the problem one meets with the difficulty which arises from the general nonexistence of the restituting collisionse except for very special models. In order to avoid this difficulty we choose to treat the problem semiquantum-mechanically. For the translational motion the classical description is retained, but for the internal motion we consider the molecules as being able to exist in different internal energy states, The distribution function, fly we define as the number of molecules per cubic centimeter at the times t, having coordinates lying between r % -.-I -4 -.1 -. — _ and r + dr and velocities between r and f + d; and internal energy states IEi where i stands for all the quantum numbers of the internal energy states, * A partial list of the symbols used in this report is given in the appendixo

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 3 When there are no outside forces acting on the intenal motion, the Boltzmann equation is where the right hand side is the change of f due to collisions: 00 age=, ' (2) The quantity I(go( is the differetial collision cross semiqutumechanical tion for thise colisionmp of partiles in tates arelyd cla wssich after cllrpision become states k and I; g and g' are the relative velocities before -after collision, respectively; and o( sy6 specify the orientation of g' with respect to g. One has This relation is really the reason why the semiquantuum-mecbanical treatment * In general, we write = -; / g j

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 4 equation of Boltmann for a quantity j(t, r.,~ Ei) is obtained by multiplying Eq (1) by -,, integrating over all velocities and summing over all internal energy states. A generally more useful equation is obtained when _ is expressed in terms of t, r E., and the molecular velocity C = f-'o, where Jo is the streaming velocity of the gas, _Dt as d and. There are of course the usual three conservation theorems: the number density, the linear momentum-,;and the total energy -translational plus internal The five sunmational invariants are

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page5 or i = 1,2/3 where m is the mass of the gas molecule. These quantities, when put into Eq (4), lead to the general hydrodynamical equations i or f S = o p is the density of the gas, and the other symbols are: Pij = ij component of the pressure tensor = mCiCj (qtr)i = ith component of the translational heat-flux vector m2 (qint)i = ith component of the internal heat-flux vector = EjCi IU = total energy = Cm + Ej = Utr + Uint2

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 6 For simplicity, we will limit ourselves in the following treatment to the case where there is only one mode of internal motion. The generalization to several modes is straight forward. IIo TRANSPORfT COEE'ICIENTS WHEN TE: EXCEANGE BETW THE TRANTSIATIONAL AND THE, TENAL ENERGIES IS NOT DIFFICULT --- — THE ESKOG-CAPMAN EXPANSION When the exchange of translational and internal energies is easy, the internal temperature will follow the translational temperature closely' To the zeroth approximation the state of the gas can be described by one temperature. To find the transport coefficients one expands f and hence all the average values obtainable from f by series expansions in a parameter:~9 (6) where Q is any of the average values Pij (qtrti-(qjnt)if Utr and Uint. Substituting these expansions into Eqs (1) and (5) eliminating the time derivatives of the quantities t and U by Eqs (5), and equating the coefficients of each power of 0 equal to zero, one finds that the lowaestorder equation is z/>/ X V2 k 2+9tX oxen A/ a I) = To J (7) da~j" b'~"cr~ c~je ~ 7

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 7 the solution of which is -e / e (8) 7s e The identification of n,yJ0, and T as the number density, the streaming velocity, and the temperature of the gas respectively, Smposes certain conditions on the f(n)'s. These conditions are satisfied by requiring that 4dyP"= y N= (9) 2/ (lc ~., f/ " - ~ for all n >O. The next-order equation is an integral equation for f/(1) Writing and after a calculation quite analogous to the case of the monoatomic gas i one finds: Cbhapman'and C.owling, The.athematical Theory of Nonuniform Gases, Chapter VIIo

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 8 /B ~ (10) 1 is the specific heat for the internal motion, and C1 is the total specific heat at constant volume. We have introduced dimensionless quantities. -z77 C- =,_ *;L/.- / I is an isotropic linear operator, defined by Eq (10) differs from the corresponding equation for a monoatomic gas in that there are two additional terms on the right-hand side. It will be seen that these terms are the transport of internal energy and the relaxationotime terms. The first two terms on the right-hand side of Eq (10) can be caomibined into ones but since they have distinct physical meanings we prefer to keep them separated, As in the case of moonoatomic gas, one can write as Ansatz for solution (1) _ /,,* _ X, s dac

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 9 where (Atr)k and (Aint)k are vectors, Bjk is a tensor of the second rank, and D is a scalar. Since I is an isotropic linear operator which changes an isotropic tensor, vector, and scalar5 into another isotropic tensor, vector, and scalar, respectively, it follows that and The integraLIEq (10' can now be separated into four componleat equations, (12);"aP= - +;B 5. P. Robertsoan, Proc. Camib Philo Soc0 56, II, 209, 1940.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 10 Eqs (12) are nonhomogeneous integral equations. The integrability conditions are tat the nonhomogeneous part must be orthogonal to the solutions of the homogeneous equation# namely, 1, mC, and + &Ei. It is easily verified that all these integrability conditions are satisfied for the component equations. Thus, our problem reduces to finding the four scalar functions Atr, Aint, B,3 and D, which satisfy Eqs (12). However, the requirementse Eqs (9), impose some auxiliary conditions on these functions. Aside fSrom those automatically satisfied the auxiliary conzditions are: (13) Before actually solving for these scalar functions, one can find formal epressions for the heat-flux vectors and the pressure tensor. They are (l4a)

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN u ~T f ' =, ~-.~.,, ( a t4 I (14b) (14c) The bracket snymbol is defined as [9, B7= 47J di g6J27'(' 6K.), (15) which has the properties: because of Eq (3), and 2) f, J> o0. To thi order of approximstion one can write the sum of the two heat-flux vectors into one simple term where

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 12 The first term in the expression ~for the pressure tensor is the ordinary viscosity term with.z ( /, 7YX Y The second term is new. It is proportional to the scalar -; the proportionality factor can be interpreted as the dilatation vriscosity coefficient6. To see more clearly the meaning of this term and its relation with the relaxation time we now calculte the first-order correction to the translational and rotational energies: /L / _i3 _ /.e The corresponding temperature corrections are6 L. Tisza, Phys. Rev. 61, 531, 1942.

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN Pag 13 (,) hsen e to-n is a rn T The second equation is only an approximation. The dependence of,B on T has been neglected. This is all right when Tt is smalli which is consistent with our approximation calculation. The difference between the temperature corrections is the quantity of interest. It is the difference between the tempef3rature of the translational and the internal states of motion, due to the dilation. One gets 7pL Zti Z'-S -- F- +' d = - - 'g where [D,D is positive. The quantity 1 [D, D has the dimension of n time. Other conditions remaining constants the smaller the value of n [D, D], the smaller the teaqperature difference, or the deviatian of the internal energy frcm the equilibrium value, Thus it can be interpreted

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 14 as a relaxation time. We vill, haoever, define the relaxation time Z* by (16) The requirement (9c) states that /Ci f+ E. - The above equation together with the equation giving Ttr- int s namely leads to,Ci =- - ' ce /~ nl ~ Thus._ t/dr dr( '2-J d'@ We have altered the definition of Zfrom that given in our former report, see ref. (2); so that will have the same meaning as the relaxation time to be defined in Sec. III.

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN Or, LA(r a/6(9 6 j7 -d a) s 9d; (17) where both beingp sittive. ' is to be interpreted as the dilationviscosity coefficient. It is due to the finite time required for the internal motion to reach equilibrium with the translational motion. To this order of approximation, the generalization to the case where there are re more than ane internal modes of motion is clear. One gets for each additional mode of motion a new heat-conduction coefficient. The dilatation-viscosity coefficient is then a sum of the dilatation viscosity oofrcientg of the individual modes of motion. fir7- ~e - A~w/pr, where i is the relaxation time for the,th mode of motion and X is the corresponding specific heato The evaluation of the transport coefficients which reduces now to the evaluation of the square bracklet follows the same line as given by Chapman4. All the formal proofs go through as given there. The present case is slightly more camplicated because of the additional variable &i but oae still can make pawer series expansions in terms of 42 and 6i.

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN We will give, as an example, the evaluation of Z= [ D, 31 to the lowest order in this expansion. D satisfies the integral equation and the auxiliary conditions S/e Pr, = _ ~ We develop D 'into a double series in the conmplete set.s of orthogoneal functions uk(2)v (E ) These series will be chosen to be of the forms with uo = v = 1 and weight factors, 62 e and e respectively. This choice determines the u's and the v's uniquely. For instance, wit/u6 = ev * ' < = a f? *

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICEHGAN Page The first auxiliary condition requires that d * 0, and the second puts a restriction on d01 and dlo, or, Up to the order k + = 1, -/ 7 fZo CO,/ From the laws of conservation of the number density and the total energyp one finds that By multiplying by u1 and vl respectively, integrating over all velocities and summing over all the internal states i, we obtain: so, tat= ze, W, _ 8 80 thSt

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN.age 18 One hba to find do1 next. The integral equation is, to this order of appraximation, /t/47(4q' t' =/ _r or Similarly, iR {g' roy + diz0, /) =4/ /, These two equations are identical on account of the relations between the square brackets. This shows also that the restriction on do, and d~o is consistent with the theory. With the help of this relations we find // re~~~~~L ~ 01

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 1 where - 00. A (0) being defined aa tJ_ 4/ ' - and J) (,) - d v

ENGINEERING RESEARCH INSTITUT Page 20 UNIVERSITY OF MICHIGAN. 20 where the b's are he ollwing square brackets: ]fch of the squrs brackets is a eight-rold integ' axl four-fold sum. Five of the ZintegrEls ca be carried aot -rithort ang Specific assmUption as to the intermoleoClar frcres, The expressiaas for A, htr' k~int,,X and ' are: / _ _ /. x fie = (1 a) * i/ A1-_rv Jat (Iga

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN Z7j (,l) -I' 4,do a /~o (19b) where the b s are the following expressions: ~~~~,Z r~~ -- -X fhi / LK/ jvOr '7, t' f_/_ / s/ 2-e'- 1o 2i

ENGINEERING RESEARCH INSTITUTE Page Page 22 UNIVERSITY OF MICHIGAN (20) (21) III. 'TANSP0~iT COEFaFICIENTS WHEN TEE EXCANGE B TEH TRANSLATIONAL AND l TE IlGERNAL ENERGI IS DI-FFICULT In this case we proceed slightly differently. Because of the slowness of energy exchange, there will, in general, be two different temperatures, the translational temperature Ttr and the internal temperaturew Tint, where Tint may or may not be equal to Ttr. To allow for the description of this fact# we employ a partial perturbation calculation. We assume that the inelastic collision cross section is much smaller than the elastic collision cross sections and write where Iel and Iinel are the elastic and inelastic collision cross sections, respectively and ' i a parameter to indicate the order of agnitude. We will take i' of the same order as 0, and after the sorting out of the

ENGINEERING RESEARCH INSTIUTE P 2 UNIVERSITY 0F MICHIGAN successive orders we will put as usual 9 = O = 1. The expansion of fp and cansequently of all the average valuesj remains as given in Eqs (6). Making substitutions as beforey the lowest-order equation (coefficient of 9-4) is Ale //;J4/f &tgbz.9/tXJ fa) Zf<4w-/62fy/') so. (22) Eq (22) differs from Eq (7) only in that the collision cross section I ij is replaced by Iel. (Here, for short, we have omitted the indices indicating the internal energy states.) It is this replacement which allows for the possible existence of different temperatures for the translational and internal motions respectively. The solution of Eq (22) is ~CA (0) where F(Ei) is arbitrary since the dependence of f on the internal energy state is not determined by Eq (22). However, it seems most natural to choose for F(El) also the local Mawell-Boltzmann distribution bIt with a temperature Tint which may or may not be the same as Ttr. Thuss we take L j;(t e (23) The conditions to be imposed on f(n) for the identification of n,fo, Ttrp and Tint as the number density, the streaming velocity, the translatioal and the internal temperatures, respectively, are satisfied if

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN (24) for all n >O. These are to be compared with Eqs (9). The first-order equation (coefficient of 9~ and @'~) is fe ~r~ t -L- -4 >}22 (25) x(/'7'4o lef 6'7 /() /o' C Contrary to the case considered in Sec. II, the hydrodynamical equations are not enough to eliminate the time derivatives. There.a.t six quantities, n, Jo, Ttr, and Tint, defining the state of the gas, b't there are only five bydrodynamical equations. However, there is one additional equation of special physical significance which one can derive from f(O), This equationa describes how the internal temperature (or the translational temperature) varies with time. It can be looked upon as an equation supplemen1 2 tary to the hydrodynamical equations. Inserting Ei (or 9 M ) for J in Eq (4), keeping terms up to the zeroth order in 9 and 9', one finds:

| ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 25 where we have again put f(1) = f(O) (1) The second term on the left hand side of the above equation is zero because (qint)i(O) = the subscript et meaning that the total differential collision cross section is to be replaced by the elastic differential collision cross section. On account of the first property of the bracket symbol, and since the collision is to be limited to elastic ones only, it follows that _ (26) The equation for i(l)1 which takes the place of Eq (10), is: (27) fY 3.-: A)/ - i

ENGINEERING RESEARCH INSTITUTE Page 2 UNIVERSITY OF MICHIGAN e ~e re Ei/k ia2 aid Ent = i The Ansatz fcr solution is (28) The =in difference between Eq (27) and Eq (10) is that nl the operator, Is tIe elastic differntial collision cross section tams tbe place of the tctal collision cross section. The expressions for the heat-flux vectors asd the presstre tenor are: The ~ssre tensor difes fri Eq. (I4) in tat there is no term due to te e4lat.!a time, Folloving the sazse argent as used in the derivatlo: of Eq (25) one finds, to this order of approxtiitc the equation supplementary to the bydrodynamiiial equaticois: a-;~~~~i~~uc ~ ~,d z.Pr~ra "LT~/~i''~ ~~;~'/9~~

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 27 IA the above equaticn, 1iel can be replaced by II becawse the elastic collisioas give no aoatribution to the integral. skinkg use of the conservatiin of the total energyl one writes for the equatim supplementary to the kh-drn eiCal equAtims I 9' (30) rcam Eq (30) p me derives an expression for the elaxation time. When Ttr and Tit do not differ too muah, as is in the case of the passage oC saotd waves tVbroud a gast to a first apprmlmction in (Tt' Tjt) 1tr e fi2 t wbere evezywiere except in the faotor T - Tint, Tint is to be Trplaoed by Ttr, The equation for is then: X22 t4 ~j 72. J if~C'rC;Z Cyr (/ f/QxZ

ENGINEERING RESEARCH INSTITUTE Pa UNIVERSITY OF MICHIGAN Page 28 We define: / = -c ~ / and the eqution for Tt becomes: ad tGhe eb/ aa(t- ftor J b c 7s) V 749 4&JA. I)(32) whoee 'is alvay,.~: posit"4ve. From l2q (32) it is seen that Eq (31) is a sensible definit:.on:for '. Apar~' frorl the heat-conduction term=, Eq (32) says that the total tine rate of change of the internal energyr is equal to the deviatsion of the instantaneous internal energr from its equilibrium value divided by the relaxation time. One can easily verify that Eq (31) is the same as Uie first approximatioan value for '2i obtained in the previous sectionp E.q (16). In Eqs (29), both (qtr)j &nd (qint) are the Um of to terms, proportional to the radient of the tLranslational and of the

I ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 29 internal temperature, respectively. The first-order approximation for the solution of the integral equations for (Atr)j and (Aint)j gives, haoever, the value zero for the bracket expression l(Atr)o, (Aint)j]. Thus rg' = -C = 0.; W) fd (4) 4C~~~~~~~~~(3) The constants Xtr, it andA can be deduced from the expressions we obtained before,Eqs (18) and (19), by remembering that in the operator I, the collision cross section is the elastic collision cross section, so that AE1 = ACC - O0 We find ~x~~i~~ E X;/ (34a) 4 =y z> ee (34b) /7,-~7~j~T~ A e-~jL cj faR yfFe e ew,~~~~~~~3E%

ENGINEERING RESEARCH INSTITUTE Page 30 UNIVERSITY OF MICHIGAN Page / = aJ~~ A "Y ESS e,&~2'/ ' z.- JX (34c) K ( The relaxation tine, s, given by Eq (31) can also be written as =_ J- X — g ---d _Z-/zHi, J 52Z e1x ') A (34da) In vq2 (34d), the CI.ferential cross section appearing under the integral s:'. should reall; be Iel; we have replaced it by Ij because the presence of the tfactor (A i)2 makes the contribution due to the added term zero. It is to be noted that the same integral appears in both Eq (34a) and Eq (34b), so that as for monoatomic gases. The gaeneral-zation t Caess awhere there are more than one internal mode of motion is again very simple. There will be n + I temperat-ures, where n is the number of internal modes of motion. Besides the five hytrodyzaanical equations, there will be n saupplementary equations i; ~. nrf thle tQsi tine rate of change of these internal tempemtures. There vwill alslo be z- relaxat:on times.

ENGINEERING RESEARCH INSTITUTE Page 31 UNIVERSITY OF MICHIGAN Page IV. DISPERSION OF SOUND AND THE CONNECTION OF THE PRESET THEORY WITH THE EXISTING MORE OR LESS PHENOMENOLOGICAL THEORIES A. Dispersion of Sound The one-dimensional hydrodynamical equations for a small disturbance have the same form for the case considered in Sec. II as for the case of a monoatomic gas. The only differences are: 1) the heat-conduction coefficient is now replaced by the sum of the heat-conduction coefficients for the translational and the internal motion, and 2) the viscosity coef3; ficient is replaced by /,a+ s-, where aX is the so-called dilatationviscosity coefficient. Up to the first approximation in the Chapman-Enskog development, the velocity of sound remains the same as the velocity of sound at zero frequency, Vo. The absorption coefficient is given by: where ) is the frequency of sound and Cp is the specific heat at constant pressure. We do not get the characteristic dispersion curve for a polyatomic gas for this case because we have started with one temperature, and our expansion extends only to values of frequencies such that p2a is small. In this treatment, the effects of heat conduction and viscosity are assumed to begin in the same frequency range as the effect of the energy transfer between translational and internal degrees of freedom, so that these effects cannot be separated from each other, The corresponding hydrodynamical equations for the case considered in Sec. III are:

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 32 77gI~~~~~~~ / / ~(35b) (35c) whereyXo is the streaming velocity in the x-directions and p is the static pressure. To this set we add the supplementary equation: /09.7ffi~ ~ ~ ~ ~~~7; = a(36) Eqs (35) and (36) are the same as the equations of Herzfeld and Rice7 except that Eq (35c) is the sum of the third and the fourth equations of Herzfeld and Rice1 and that their relaxation time is 3k/2R times our relaxation time. From Eqs (35) and (36) one can again derive expressicnst for the velocity and the absorption coefficient of sound. Howevers to see more clearly the effect of the relaxation time in polyatomic gases,. we drop the viscosity and the heat conduction terms which amounts to assuming Ke. F. Herzfeld and E. 0. Rice3 Phys. Rev. 31, 691, 1928.

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 33 that all absorption and dispersion effects are due to the slow transfer of enerrgy from the translational to the internal degrees of freedom. For the passage of sound of frequency? and wave number 1i one obtains from Eqs (35) and (36), by dropping/ and: _tC ~ —/- - C-= 5- -. From the aove es: ation it follows tht for extrely hih ad low freque na cies: There is no absorption = ~ - F= ~ the velocity of so8md at zer frequency: b)Z/ >~ /: There is no absorption the velocity of sound when the internal C- -/ state is not excited. Between these two asymptotic values the velocity of sound increases monotonically with the. frequency. B. Application to the Rougha,Sphere Model For a rough, but perfectly elastics spherical model9 Pidduck8 bas calculated the viscosity and the two heat-conduction coefficints. F. B. Pidduck, Proc. Roy. Soc. A, 101, 101, 1922.

23ecause of the way the temperature was defined, no relaxtici time was found. From our calculation$ howeverp one obtains a relaxation time. By -the worc. "rough" is meant that "when two molecules collide, the two sphereS 7. ea.ch other without slipping; first each sphere is strained by the other aend then the strain energy is reconverted into kinetic eneri7 of translati:ion 2nd rotation, no enery being lost;; the efac! We ' i'.elc-ltIe velocity1 of the spheres at their point of impact:;7 reversed by the impact". The expressions for/s and the two X's as g-ven by PiciducL andL also derivable from our expressions, Eqs (18), (19a), s:nd (1.9bc ar.: __ _( /#< ) (37a). =...t. _(r9(+.) -- ~r 4(37) /46 7 2fj~l c /0//f /OJ / q(37c),"- r b, bc-i;,; the uotcnt o2 inertia, E; measures the ease with which the energy i.. ';:. % the K:io!2 hi;::.*U?7 44g the internml states and which depends on

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 35 the mass distribution of the sphere. K = 0 corresponds to complete concentration of the mass at the center of the spheret so that for K = 0 the inelastic collision cross section is zero. For a rough sphere, the linear and angular velocities of the molecules after collision are restricted by the conservation of the angular momentum and the reversal of the relative velocity of the points of the spheres which come into contact, in addition to the other five conservation theorems. The former we Vill take into account by a special form of the collision cross sections Let w be the angular velocityp and k be the unit vector in the direction of the line from the center of the second molecule to that of the first. We write: 'Zf (W 4-, -)- ar ~ -)'7/ XZ (o) I/ x c s where the first Sc-function is a dimensionless expression stating the conservation of the angular momentum while the second S-functior expresses the reversal of the relative velocity of the pointsin n contact due to collision. Inserting this expression for I in the follwing equations X % A t t - - - -

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page replacing the sunnmmations by integrations over the angular velocities Zt8Ps and making the proper change of normalization of f one finds: /?7 //J ~, _ 7 3 -,( ( ' X' a -4 (/ s ( /,W — / All the integrals involved are simple; the evaluation yields: / =- X{;2@t// (38) For this models the relaxation time can also be calculated by simple kinetic methods. One calculates the time rate of c ge dt, the time rate of change of translational energyp which is m/2n times the time rate of change of the average value of C2. The latter is equal to the total number of collisions per second multiplied by the change of C2 by each collision, or 2 2 12 2dtr (C1 + C2 - C12 C2 )-av2 In this way dt is found to be proportional to Utr Uint. Identifying the proportionality constant as 1/T, as is consistent with our definition,

UNIVERSITY OF MICHIGAN Page we obtained an expression which agrees with Eq (38). An estimate of the numerical value of 2 shows that o = 'o for K = 0 as is to be expected, and that V decreases monotonically to -10'10 second for K = 2/3, correspon.Cndg.o t.ue complete concentration of the mass on the surface of the sphere. Co The Eucken Constant By a simple mean free-path argument Eucken9 reached the conclusion that: (39) when there is no correlation between the velocity of the molecule and the amount of the internal energy carried in the heat-transport process. In Eq (39) 65 is the Eucken constants and is the ratio Cp/Cv a = 2.5, 1.9, and 1.75, respectively, for r= 5/3, monoatomic gas, 7-= 7/5, diatomic gas with no vibrational statesp and 1= 4/3, rigid triatomic gas like the rough spheres. Without a knowledge of the collision cross section we cannot calculate/a and X. But we can make two estimates of the value of the Ueq is the energy if the internal motion is in equilibrium with the translational motion. It is equal to Utr in this example. 9 A. 2uckens Thys. Zeits. 14, 324, 1913.Euecen.s'forma is i= (K tr tr +.Krot.. Crot + Kvib Crib). EKtr = 2.5, rot = land Kvib = 1 to 15K, depending on the molecular structure. EKvib = 1 if there is no correlation between the velocity of the molecule and the amount of vibrational energy carried.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN.a.g 3. Eucken constant from the results obtained so far. For a rough sphere, it follows from Eqs (37) that /o /~t + 7J-/V Z /t/ff ~/o /Y /0 /xz -A F' icr/. #/,a,&,4 1.71 for K= 2 3 1.85 for 7 = 0 From the expressions we obtained in Sec. IIIt a rough estimate of c can also be reached. As has been remarked before tr// = 15k/4. If we now assume that the elastic cross section is independeit of the states i, j of the molecules, and write X /Rgsf rivA~y/ ef/ re MS 9 then: Jw '

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGANge c is temperature dependent through -. For very laow temperatures, p = 0 the value for monoatomic gases. For very high temperatures, the internal motion is completely excited, p is independent of T. Assuming that the elastic collisions can be described by the elastic sphere model so that one obtains: -- -(90-3 (40) 1.98 for rigid diatomic molecule 1.85 for rigid triatomic molecule, also the rough sphere D. Derivation from Eq (34d) of the landau-Teller Expression for the Relaxation Time v for Vibrational Degrees of Freedom1o For the case of the transfer of translational into vibrational energy (which is the most interesting and most often occurring case), landau and Teller have given a derivation of the relaxtion time, based on the following simplifying assumptions: 10 L. landau and E. Teller, Phys. Zeit. d. SowJetunion 10, 34, 1936.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 40 1) The vibrations are harmonic. 2) The interaction between the molecules are, aside from the radial part, proportional to the normal coordinates of the oscillators. 3) First-order perturbation calculation is sufficient. 4) The translation motion can be separated off completely from the vibrational motionp so that the differential cross section has the form: where TUk is the transition probability from the states is j to the states ij k, 1. From assumptions 1)p 2), and 3) it follows that 77=; s>ei7,/,t, e 70 T0L being the transition probability for a molecule from the state 0 to the state 1p while T10 is that for the reverse process. We will now show how with these assumptions the Landau-Teller derivation fits in with our conls ideration. ge (ir,y / ~% J~,~ ~~~ ~~ A ~" a~~lrL K% - I~,; l

ENGINEERING RESEARCH INSTITUTE Page 41 UNIVERSITY OF MICHIGAN, i; /'2i,,A... It can easily be seen that the collision frequency, F, is: thus -2 y~q'~-/ *_' But To1 T=oe' /k hence, atlring use of the smunation formulas: 'eb I eW/ 2; z e ~~ F 77-W

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN and one obtains: A / ( ti- (16741) where FTLO is the number of collisions per second in which molecules in internal states 1 are deexcited into the ground states 0, and FTl01 is the number of collisions per second in which molecules in internal states 0 are excited into the states 1. Eq (41) is also the expression obtained by Landau and Teller. In fact, one can show that from the above assumption dUint one finds that.int is proportional to Ueq - Uint with the proportionalit dt constant given by Eq (41). Let nj be the number of molecules per cc in the state J. The equation of change can then be written as By using the relationships among the Tl Ims and the summation formulas, one obtains =Fg- k3AnLi 7 o, At 7?4_ As _AS =~, - / I ej 8;iJ

ENGINEERING RESEARCH INSTITUTE Page 43 UNIVERSITY OF MICHIGAN where Uint = tAj nj. For equilibrium the average energy is &/g= k~ (Jat ) ' so that if V is given by Eq (41). Thus our definition of the relaxation time is the same as that of Landau and Teller.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 44 APPIMX Par-tial List of Symbols Used C malecular Velocity t6 a;a=sionoess molecular velocity %C 3zpecifi'c hea'o at constant pressure JCv cok coolstic E. energr of -the internal states the aubscript stlands for all quantum numtbers of the intermal enerE7 states & aucQ n conjSteant f distribut-ioni function rt' (, r,i) f(n) nth orter term of f in the present development F collision frequency g.la"'-,.: -elocity o::. '.& c.oiding,ie!.ec-uci3 before collision; g ';~I~9 * '2C 'U2' ": ".iL 'rn ~.n...... 2b2CJA~ v~lC g erelativ'e Lvelocity of the coll.i,: ng z:loecules after collisi.on I moment of inertia Il differential collision cross section for the collision of particles j in states i, j# which after collision become states ko 1. Iel elastic differential collision cross section Iinel inelastic differential collision cross section k unit vector in the direction of the line from the center of the second molecule to that of the first K = I4/md2, dimensionless quantity measuring the ease with which the energy goes from the translational to the internal states 3iJ = mCiqjp i, j component of the pressure tensor qtr = mC2-C/2g the translational heat-flux vector

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 45 qi =t E= iC, the internal heat-flux vector T temperature of the gas Ttr translational temperature Tint internal temperature Tkl transition probability from the states ij J to the states k# 1 ij Tik transition probability for a molecule from the state i to the state k Utr = mC2/2, translational energy Uint = Eio internal energy U = Utr + Uint, total energy Ueq internal energy if the internal motion is in equilibrium with the translational motion V velocity of sound Vo velocity of sound at zero frequency 13 specific heat for the internal degree of freedom Bg specific heat for the Ltbh intrtnal degree cof freedom ei energy of the internal state i measured in units k1 or kint 8gint average internal energy measured in units kT or kTint 9 Chapmun:Enskog expansion parameter of order of magaitude parameter for the inelastic collision cross section Zx.dilatation-viscosity coefficient 'L dilatation-viscosity coefficient corresponding to the lth internal degree of freedom k het-conduction coefficient Xtr tr tanslational heat-conduction coefficient Xint internal heat-conduction coefficient

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN f/f 9ic8viscosity coefficient frequency of the harmonic oscillator in the vibrational motion _J velocity of a molecule streeaming velocity diameter of molecule relaxation time relaxation time belonging tb the /th internal motion any function of the coordinates, velocities~ and internal states of a molecule# in particulars the summational invariants at collisions z" defined byf(l ) = f(O) (1) ae) angular velocity of a molecule