El... E,, of negative total energy, as well as to continuous positive states. QUANTUM DYNAMICAL CORRECTION FOR THE EQUATION

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1 56 PHYSICS: H. MARGENA U University of Kansas, in partial fulfillent of the requireents for the degree of Doctor of Philosophy. The writer is indebted to Dr. D. McL. Purdy for suggestions and criticis. 1 Hecht, These PROCEEDINGS, 13, 1927 ( ); J. Gen. Physiol., Baltiore, 11, 1928 ( ). 2 Houstoun, R. A., Phil. Mag., London, 8, 1929 ( ); Houstoun, R. A., and Shearer, J. F., Ibid., 10, 1930 ( ). 3 Volkann, A. W., Physio!ogische Untersuchungen i Gebiete der Optik, Leipzig, QUANTUM DYNAMICAL CORRECTION FOR THE EQUATION OF STATE OF REAL GASES SLOANE PHYSICS By HENRY MARGENAU LABORATORY, YALE UNIVERSITY Read before the Acadey, Wednesday, Noveber 18, 1931 Recent progress in the understanding of interolecular forces has revived theoretical interest in the equation of state of real gases. Several investigations2 ake it appear very probable that the forces which cause the deviations fro the perfect gas law are due to the interactions of rapid electronic otions within the olecules; and calculations of the second virial coefficient B, based on these concepts, have produced reasonable agreeent with experiental findings. It was only in the case of very light gases, such as H2 and He, that satisfactory agreeent could not be obtained. Here the discrepancy proved to be such that for low teperatures the calculated values of B were uch saller (algebraically) than the observed data. A possible explanation of this difficulty was first pointed out by London,' who suggested that the existence of zero point energy associated with the vibratory otion of olecules in quantized collision states ight render the attractive Van der Waals' forces partially ineffective. However, a detailed quantitative investigation of this phenoenon and its effects upon the equation of state has not yet been presented. Such an investigation will be the object of this paper. We shall begin by assuing-in accord with the results of papers listed under 1)-that the forces between any 2 olecules have a potential energy e which if plotted against the distance of separation r of the two olecules, yields a curve siilar to figure 1. If now these olecules are regarded as ass points and their utual potential energy E(r) is substituted in the Schroedinger equation, this will lead to solutions corresponding to discrete vibrational energy states El... E,, of negative total energy, as well as to continuous positive states. It is not difficult to show that the relative spacing of the discrete states,

2 VOL. 18, 1932 PHYSICS: H. MARGENA U 57 and the position of the lowest possible state E1 depend on the depth of the iniu of e and on the asses of the olecules. In particular, for light gases like H2 and He the lowest energy state lies very high, indeed not uch below the 0-axis. Moreover, it is likely that for He there exists but this one vibrational state.3 This situation iplies that all pairs of olecules bound together by their -potential energy, i.e., having a negative total energy, cannot exist in that part of phase space which corresponds to the shaded region unless they be in one of the states E1....p while there is no liitation on the states of olecules which have a positive total energy. In order to obtain a correct equation of state it is necessary toodify the usual calculations by excludng the inaccessible region of phase space, and considering instead the set of discrete levels El...E^. FIGURE 1 The pressure of a gas ay be obtained fro the therodynaical relation p = kt log Z, (1) where Z is the coplete partition function ("Zustandssue" for the discrete states, phase integral for the continuous ones). If the gas cqnsists of N free ass points (e = 0) Z =h3n J... e -WkT d++n)2vd V (2) dv is written for dv,...dvn5, dv for dxl... dzn. (2) is easily evaluated and leads to the law for ideal gases pv = NkT, if substituted in (1). If the olecules, instead of being free, are considered to have a utual potential energy e(r) and the existence of quantized states is neglected, Z differs fro (2) only by the appearance of the total potential energy of the gas in the exponent of the integrand; that is,

3 58 PHYSICS: H. MARGENA U = M Nf..J'k [2(v' + *-- DN) + E(ri-n) +... f(rn -rn)1d,jv (3) Here the integration extends over all velocity coponents fro - co to + c, and over all space. By (ri - rj), the arguent of e, we designate the relative distance between olecule i and olecule j. A calculation of (3) ay be found in textbooks on Statistical Mechanics.4 The result is approxiately /(27rkT 3N/2 Z = VN(1 + t)n (4) with 2irN c e(r) = (e kt - l)r2dr. This expression, inserted in (1), V yields the usual equation of state: pv = NkT 1 - -V (e kt - l)r2dr}. (5) It is evident that this equation ignores the effects of quantized collisions. A better approxiation to the actual state of affairs ay be obtained as follows: We consider the gas as a ixture of n pairs of olecules in states of -utual quantization and neglect their interaction with siilar pairs and with all other olecules. N is the nuber of olecules in continuous energy states. The total partition function now consists of 3 parts: (1) the su of state resulting fro the vibratory otion of the pairs (e-kt (2) the phase integral due to the classical otion of F(47r-kT)3/2V1 translation of these pairs, which reduces to [ h3 and (3) Zp, an integral like (3) but extended only over the accessible part of phase space. B,.i (4irk~T)3/21f (Ve ) T (6) Substitution in (1) shows that the first part of (6) contributes nothing nkt to p, the second part only the partial pressure,v X which ay be neglected since, by ordinary statistical reasoning, n ay be shown to be very uch saller than N at all teperatures of interest. The proble is then to evaluate Zp. The reasoning which leads fro (3) to (4) involves the assuption that only "binary encounters" between the olecules can occur. This sae restriction will here be iposed. The integrand of Zp ay be transfored by the identity

4 VOL. 18, 1932 PHYSICS: H. MARGENA U 59 2;a; e 1= Ht(Ca- 1) +,-ll-i(e~$1) + E-21-2(ei- 1) + * + 1. Hr indicates the product of r ters (ea_- 1) with different indices i and selected fro the total nuber of such ters, while 2 denotes the su over all possible products () in nuber of r ters. If now we write e, for e(r, - rj), etc., allowing i to take on values fro 1 to N(N - 1) t 2 and W for the total kinetic energy of the N olecules, ZP = () Xf f e {kt(e -l) + E-l 1(e kr- **+le kt - 1) + 1 }dvdv. (7) The liits of integration p are defined as follows: If one olecule X is so far fro all others that e(rx- r,,) practically vanishes, where j, ay be any other olecule of the assebly, then the velocity coponents of X are to be integrated fro- co to + 00; however, if e(r<- r,,) is different fro zero the velocity coponents of X and j, run fro values corresponding to a relative kinetic energy e to = co. We illustrate the integration of (7) by considering a typical ter: Ir r _ w /e(r - ri) \ / (rk- ri) y \ Tte kt te kt - 1iV. Er /-e There are altogether r factors of the for (ekt- 1). The assuption of binary encounters iplies that if e(r, - rj) 6 0, all other e's involving either r, or rj ust vanish. This in turn reduces to zero every ter of Mr in which any index occurs ore than once and liits the nuber of N! ters Of Mr to (N - 2r)! r! 2'- These reaining ters contain the coordinates of 2r different olecules, so that the space integration over the other N- 2r olecules siply gives 7N -, while the corresponding velocity integration over these olecules yields gn-, where g= (-wt)/2.the reainder of the integration over our typical ter decoposes into successive independent integrations over the velocity and configuration spaces of pairs of olecules. Thus Ir= (N - 2r)!r!2r I g 7717

5 60 0 PHYSICS: H. MARGENA U where rw= J...J ei 2kT (D1' + [ Le kt - 1 dv1dv2d Vld V2. (8) The evaluation of (7) by substitution of the various Ir thus derived proceeds by a well-known analysis, for the details of which we wish to refer to footnote (4). If, for convenience, we define N = 2g2V27'7 (9) the result ay be written in a for siilar to (4): zp (2kT)3/2 VN (1 + )N. (10) It is now necessary to calculate r, defined by (9) and (8). The actual work involved in this process is soewhat lengthy and will not be written down in full. It is convenient to put 77 = 1-02, where ni is the result of an integration over all phase space, while 772 is over the excluded doain of phase. Changing the variables in the volue integration to relative coordinates,,7 is at once obtained: e(r) 771 = g2vje kt - 1) dv. (11) Using the abbreviations 2e a2 = 2kT' p2 =-r = p2i- (y2-yl)2 - (Z2 - Z1)2,M =jp2 (Z2 Z1) and writing dv for an eleent of volue, as before, but x, y, z for the coponents of velocity, the coplete for of 772 is seen to be 72 = J'J'dVidV2(e kt - af+p e 2 - ) -a2z12 e -co r I2 a'y22 I' e -ckxi2 + -a2x22 dy2 Xj e e dx2. e 22dyl e- (12) p is to be considered as a function of the distance of separation, r, - r2, of the two olecules. X2,Is2 and p2 can never be greater than 2I axi, so that the product of Ca2 and any of these three quantities has the axiu value eax We shall develop (12) as a series of increasing powers of this kte paraeter and thereby insure its rapid convergence for ost cases of

6 VOL. 18, 1932 PHYSICS: H. MARGENA U physical interest. Only the evaluation of the last two integrals of (11) will be discussed as an exaple. Denoting their product by J and changing variables, Now put J fox e "' duf ec dv, where v = ax\ fee dv = G(x), so that 61 Expand J = if2e [G(u + v) -G(u - v)]du. (13) a2]c G(u + v) -G(u-v') =2 {G'(u) +GjG(3) (u) + G +... The derivatives of G are easily coputed and inserted in (13). The integration is then to be perfored only over ters like ule-2"'; it vanishes when n is odd and otherwise yields 1.3 (n 1) The result for J is J = 2;( This ay now be substituted in (11), and it ust be reebered that J, through v = ax, is a function of the y's and z's. One continues by defining J(x)e dx = H(y) and developing the integral over Y2 in powers of a/a in the anner just illustrated. After the integration over the velocity coordinates of (12) has been copleted, the space integration over one of the d V's ay be carried out and there results (_ 3/2 ( )5/2 g02vf{ )T '12 ~~~~kt/kt/ (2e) (k) + ** }(etkt-1)dv. The liits of r, the arguent of e, are clearly the point C in figure 1, at which e becoes positive, and + x. Hence if we define a function (p(x) to be 0 when x < 0, and equal to x" x" X" O ' for x > 0,

7 62 PHYSICS: H. MARGENA U 772 = 4lrg2VJ (P )(e kt- 1) r2dr, so that, recalling (9) and (11), = 27rNf@[{-1_o( )](ekt - 1)r2dr. (14) We have entioned that in ost cases Zp (Eq. (10)), is the only quantity which needs to be retained in (6) if it is desired to derive an equation of state. Substituting (10) in (1), with t given by (14), the final for of our result becoes: pv=nkt - 7f [ - {o(k)]e 1)r2dr}. (15) It is needless to state that, like all siilar classical results, (15) is valid only for sall pressures, since in the last substitution r has been taken to be < 1, (log (1 + r) = r), and all through the analysis only binary encounters have been considered. (5) ay be regarded as a first approxiation to (15) uch in the sense in which the perfect gas law approxiates to (5). But the correction ter in (15) is seen to becoe appreciable only at low teperatures, as a nuerical application shows.5 Experiental observations on the second virial coefficient of He definitely call for this correction. However, perfect nuerical agreeent between experient and theory has not yet been achieved.5 Possibly this can be brought about only by a ore basic fusion of the concepts of quantu dynaics and statistics in a anner that will not require the unrelated use of the Schroedinger equation for the purpose of obtaining e, and of Gibbs' statistics in the rest of the proble. This is a very fundaental and at present rather difficult issue; it involves aong others the question as to the justification for treating the distance between two olecules, in the wave echanical perturbation proble for deterining e, as a paraeter and not as an independent coordinate. As long as these points reain obscure, the ethod eployed in this paper is probably the only available one. 1 Eisenschitz and London, Z. f. Phys., 60, 491 (1930). London, Ibid., 63, 245 (1930). Slater and Kirkwood, Phys. Rev., 37, 682 (1931). Margenau, Ibid., 37, 1425 (1931). 2 Margenau, Phys. Rev., 36, 1782 (1930). Wohl, Z. f. Phys. Che. B, 14, 36 (1931). 3 Margenau, Phys. Rev., 37, 1014 (1931). The accuracy of the results there stated ay now be iproved by using a better approxiation for A2E; but they are qualitatively correct. 4 Cf. for instance, R. H. Fowler, p. 168 et seq. 5 Margenau, Phys. Rev., 38, 1785 (1931).