CORRESPONDENCE
On the Benedict-Webb-Rubin Equation of State Sir: I n connection with the recent claim of Barner and Adler (7) that the B-W-R equation is capable of representing with one set of constants, both vapor and liquid states in the critical region, we should like to quote several other studies, so that those interested in the problem will be in a better position to decide for themselves. Zudkevitch and Kaufmann (8) reported that for argon, a t temperatures below the critical, “two C0)s are needed” to express vapor pressures and densities, and Ramalho and Frizelle (6) said for ammonia, separate sets of eight constants are necessary for each phase. Other authors also confirmed the need of separate constants for the gas and the liquid, respectively. We suppose the assertion that two sets of constants are needed, probably means that with two sets the data fitting is better than with one set. Therefore, the claim of Barner and Adler expresses probably their opinion that even with a single set of constants, the B-W-R equation is still reasonably precise for both phases. As their calculated equilibrium vapor pressure data for n-butane agree well with the experimental values, they conclude that: “Inasmuch as the equation must evaluate both liquid and vapor densities and fugacities in the course of calculating the equilibrium pressure, it may be concluded that the equation describes both the saturated liquid and vapor phases well in this region.” Now, a final agreement of a stepwise calculation with experimental data is not a proof, in itself, that a similar agreement should exist for each intermediate step. Ree (7) found that vapor pressures obtained by using the van der Waals equation agree closely with the experimental data of several gases, even though the equation cannot describe the densities adequately. O u r own experience with the B-W-R equation does not confirm the finding of Barner and Adler for n-butane; insertion in this equation, with constants from Hirschfelder et al. (4) of the orthobaric densities of the liquid (5), gives calculated pressures 10 to 20% higher than the actual vapor pressure. According to Barner and Adler, the prediction of enthalpy is a particularly sensitive test for an equation of state, because its expression involves a temperature differentiation of the density equation. We agree that their calculated enthalpies fit well the experimental data for
n-butane, but again we cannot disregard the opinion of others. Harrison and Douglin (3) have found that “a closed equation of state (they use this term for the B-W-R and the Redlich-Kwong equation) that had the ability to represent the P V T data over a given range of variables with an average deviation less than 1%)would often, because of large percentage errors in ( d P / d T ) , produce numbers for the enthalpy that deviated an order of magnitude greater over the same range of variables; some individual points were off by as much as 27$&” Now we consider that the prediction of the specific heat a t constant volume is a far more sensitive test for an equation of state than the prediction of enthalpy, because the estimation of c, involves (d2P/dT2)li values, but we doubt the ability of the B-W-R equation to predict such data with one set of constants. The B-W-R isochores are expressed by P = a bT c/T2where a, b, c are functions of volume, hence (d2P/dT2)v= G c/TB,and this formula cannot represent the slightly sigmoid behavior of real isochores, nor the change of the sign of (#P/dT2),at, or close to, the critical point. The equation of state of Goodwin (2) recently proposed for para-hydrogen, seems to be unique in its ability to represent accurately the inflections of the isochores. In view of the assertions quoted above, we consider it unlikely that the B-W-R equation is capable to describe with one set of constants the pertinent thermodynamic functions of the vapor and liquid phases. I n defense of the B-W-R equation, we should like to mention that although the reports on its ability to describe the liquid are contradictory, it is unanimously agreed that it predicts remarkably well the state behavior up to densities not surpassing the critical.
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LITERATURE CITED (1) Barner, H. E.,Adler S.B., IND.END.C H E M (7), . ~ ~60 (1967). (2) Goodwin, R.D., J. of Res. Null. Bur. Std. (Phys. and Chem.) 71A (3), 203 (1967). (3) Hamson, R. H.,Douglin,D. R.,J. Chsm. Eng. D Q ~11U (3),383 (1966). (4)Hirschfelder, J. O., Curtiss, C. F. Bird, R. D., “Molecular Theory of Gases and Liquids,” Wiley, New York, 1964. (5) Landoldt-Bocrnstein, “Zahlenwerte und Funktioncn,” Vol. 2, IIa, p. 195, Springer Verlag, Berlin, 1960. (6) Ramalho, R.S., Frizelle, W. E., J.Chem. Eng. Datu 10 (3),366 (1965). (7) Ree, F. H.,J . Chsm. Phys. 36 (12)3373 (1962). (8) Zudkevitch,D.,Kaufmann, T.G., A.I.Ch.E.J. 12 (3), 577 (1966).
Institute Petrochim, Ploesti Romania VOL 6 0
G. J. AUSLAENDER
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