wheref(d) is a function of density alone and d, is the density of the crystal a t the triple point. Equation 2 converges for high densities, while Equation 1 is divergent for the liquid range. I n the following, it is assumed for purpose of discussion that Equation 2 holds for the liquid and the hypotheses that B is equal to the second virial coefficient of the gas and that J(d) is independent of temperature are checked against experimental data. As the explicit form of f(d) is unknown, precise absolute values of B cannot be calculated by Equation 2 from P V T data on liquid alone. I t is possible, however, to make a crude estimation of the second virial coefficient at the triple point. Let the isochore described by Equation 2 correspond to d = d L = density of the liquid at the triple point. As the temperature is decreased along the isochore to the triple point, tp,P on the left side will decrease to a negligible value ; hence,
The quantity (d, - dL) being small, the higher order terms of the expansion are negligible, and Equation 3 reduces to:
B,,
=
(Ad)-1
(4)
where Ad is the density change on melting. As shown in Figure 2, the B,, values calculated by Equation 4 seem to agree reasonably well with the known values of the second virial coefficient. Now consider another test of Equation 2. If subscripts o and i denote two temperatures along a high density isochore, so that all the higher order terms are negligible, elimination of f(d) gives: (Ptlpo) = (Ti/To)[1
+ Bi(ds - d ) l / [ l + Bo(ds - d)I
(5)
and B , at T , is expressible in terms of the reference data Bo a t Toand the Pi, Ti data along the isochore. Hence if Bo is equal to the known virial coefficient of the gas, at some specified
temperature, Equation 5 provides a means of calculating the temperature dependence of the second virial coefficient, from P V T data on the liquid. The result for parahydrogen is shown in Figure 3. The claim of Equation 2, that f(d) is independent of temperature, was checked for dense parahydrogen, based on known B, C values (Goodwin et al., 1964) and knowm P V T data (Goodwin et al., 1963). For instance, along the 0.040 mole per cc. isochore, between 24' and 46' K., the value off(d) is constant and equal to 0.113 + 0.0003 mole per cc. It seems that the speculative Equation 2 has a fairly sound basis, even though the hypothesis cannot be accepted without a good deal of checking and some theoretical interpretation of the symmetry phenomena. Conclusions
I t is important to elucidate the origin of symmetry, which suggests a 1- to- 1 correspondence between the low and high density distribution functions. The study of the symmetry phenomena could provide precise information on the distribution function of the liquid. Literature Cited
Beattie, J. S., Barriault, R. J., Brierley, J. A . , J . Chem. Phys. 19, 1222 (1951). Dwver. R. F.. Diller. D. E., Roder. H. M., Weber, L. A . , J . Chem. PAYS:43. SO1 11965). ' Goo;win, R . D.,'Diller, R. E., Roder, H. M., \Veber, L. A , , J.Res. 12htl. Bur. Std. 67A, 173 (1963). Goodwin, R. D., Diller, D. E., Roder, H. M., Weber, L. A , , J . Res. Katl. Bur. Std. 68A, 121 (1964). Gyorog, D. A , , Obert, E. F., A.I.Ch.E. J . 10, 621 (1964).
G. J. AUSLAENDER Institute Petrochim. Ploesti, Romania RECEIVED for review December 26, 1967 ACCEPTED July 10, 1968
COMPACT SONIC VELOCITY EQUATION FOR NONIDEAL GASES Following the idea presented in a previous communication where the sonic velocity, a, i s introduced as a thermodynamic variable replacing the specific volume, the equation a' = yz2RT i s derived, a not too obvious modification of the ideal gas relationship, a2 = yRT. The equation applies accurately for L values at least down to 0.80, thus covering many gaseous systems of practical importance. RECENT communication (Goring, 1967) presented the following equation relating the sonic velocity a ( p , T ) , the compressibility factor z ( p , T ) = pu/RT, and the specific heat ratio ~ ( pT, ) = C,/C,:
A
P z(p,T) =
(1)
R T l (ria') dp This may be regarded as a modified equation of state wherein the specific volume, u, has been replaced by the reciprocal of the integral c - y / u z dP.
The development starts with the basic equation
where
K
=
-
(g) ,
the isothermal compressibility.
T
The equation of state u = z R T / P may be differentiated to evaluate K as follows:
2(; =) -p'.+G(G)T= zRT RT
The present communication demon-
strates a similar equation, valid over a practical range of z , less rigorous than Equation 1 but considerably simpler. VOL. 7 NO. 4
NOVEMBER 1 9 6 8
669
or Table 1.
(3)
Combining Equations 2 and 3 gives (4)
The three-term virial expansion for z is
z = 1
+ B/L.+ C/o2
(5)
where coefficients B and C are functions of temperature alone. Lsing Equation 5 gives
T 32 - R7 (z)T
=
RT
ay
iU + $} + B
=
((.
-
1)
+-
RT
bz
2-1
- -(&r P
-
1)
(7)
A further approximation may be introduced for the bracketed 6 , then z2 = 1 28 a2. Now if quantity. If z = 1 6 1, the squared term may be omitted, giving z2 N 1 26 = 2 2 - 1 and
