CORRESPONDENCE Fugacities in Gas Mixtures PIE: In the September issue of INDUSTRIAL AND ENGINHERING CHEMISTRY ( 4 ) a n article is presented by J. Joffe on “Fugacities in Gas Mixtures.” This paper is very similar in content t o part of the material in two papers by Gamson and Watson ( 1 , 2), which covered thermodynamics of solutions in ideal systems and high pressure vapor liquid equilibria. F:arly in their Rork, Gamson and Watson derived the most gcncral equation for the calculation of gas phase fugacities as:
The partial derivatives are then
and Equations 1 through 5 can be used t o calculate their respective functions and compared with experimental values. These calculations were made for partial molal enthalpies and volumes and fugacities. Unfortunately, the agreement left much t o be desired. Calculated fugacities of methane, ethane, and butane (Y,8) were in error by as much as 500% although the d a t a on argon-ethylene and hydrogen-nitrogen appeared satisfactory. Smaller errors were found in calculating partial molal volumes and enthalpies. I n attempting t o reconcile the differences between calculated and experimental fugacities, attention was focused upon the errors arising from inaccuracies in the definition of the pseudocritical constants arid the generalized enthalpy correction due t o lack of ideal behavior. Errors of more than 30% in the enthalpy are encountered when a generalized correlation is used. I t is evident t h a t such errorb in the region of the critical point, when fugacities are calculated by Equation 1, will lead t o serious errors. Accordingly, it was hoped that these differences would be reduced somewhat by the substitution of
prcdic.at ed upon several qualifying assumptions.
That it was possible to characterize the gross properties of ideal s y s t e m by applying the pseudocritical concept to the singlephase generalized charts and assuming t h a t they were singlevalued functions of pseudo reduced temperature and pressure. The ideal system was first defined by Gamson and Watson ( 1 ) as a group of chemically similar components which tend to form ideal liquid solutions a t low temperatures where the saturated vapor of each component behaves as a n ideal gas. The deviations from ideal behavior exhibited by ideal systems are designated as deviations due t o differences in molecular size. These deviations increase, in general, for liquid systems, as the temperature is incrmsed. I n the gaseous state at low pressure, deviations from ideal solutions decrease as the temperature is increased. The largest deviations for gases are observed in the region of the critical point, since the mass approaches a condensed liquid state where the effect of dissimilarity in size becomes effective. Ail expression for the pseudocritical temperature and pressure could be dcfined in terms of the molecular composition of the mixture. I t wa5 a190 indicated (%) that the properties of nonideal systems, which show deviations due t o differences in molecular type i n addition t o those of ideal systems, may be approximated a t high temperatures by the generalizations developed in references ( 1 ) and ( 2 ) . I n addition t o Equation 1 the following expressions were derived for partial molal quantities from the methods developed by Gamson and Watson for the gross properties of liquid and gas phase mixtures ( 1 , 2) :
Gas phase
On the basis of the above equations a procedure and equation for calculating fugacities in liquid phase mixtures were presented. I n this development, it was assumed that Kay’s pseudocritical constants could be used (5).
wherein tlhe
c(&)p,,
d In P,’ was evaluated from the
I
generalized compressibility chart instead of basing i t upon t h c best average form of the enthalpy correlation. Unfortunately, fugacities calculated in this manner were still not satisfactory. Although the use of Equations 6 and 7 for calculating pseudocritical constants is satisfactorv for estimating the total ~ , O- D erties of mixtures, i t was recognized t h a t small errors in the total property could be magnified manyfold upon differentiation. Accordingly, i t was necessary to include a cor(3) rection factor in Equations 8 and 9 t o account for the inexactness of the original definitions. The correction factors were developed as dimensionless multiplying factorb which are approximately equal to the ratio of the critical property of the component t o the pseudocritical propertv of the mixture raised to the 0.6 power. This exponent was bTr, __ evaluated on the basis of experimentally determined ani data on partial molal volumes i n both liquid and gaseous mixtures, fugacities of gaseous mixtures, and vapor liquid equilibria (6, Y, 8). With these niodificat)ions I
--
R, = H,*- 2‘0’
- T%[($$)p,l
e]
be’ dP ’ an, + (m)77r.
aT,’
1,iauid .~ phase
R,
= Hi*
- 1’‘‘ (EL+!&)-’ T,’Tc’(xni)
a
;”’>‘ _ I
bT,’
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2440
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 40, No. 12
It is intcrcsting to compare the partial derivatives obFOR GASEOVS RIIXTUREb TABLEI. CALCULATED AXD EXPERI\fEKTAL VALUES O F tained by Kay's and Joffc's Y1T Calculated from Equation pseudocrilicals. Table 11 Pressure, Mole Temp., Lb./ Experimental, 13 gives a comparison for the Refercnco Fractlon O K. Sq. I n . fi/uia (16) (17) (Joffe's) (11) niixture of methane butane of Ci 1.0C: = 0 344.3 3000 CI = 0 , 1 1 3 0,170 0,270 0.363 0.123 ('()7 ) CI = 1 . 0 Ca = 0 294.3 3000 C I = 0.063 ... 0.146 0.063 Table I a t a temperature of C I = 0,9685" 322.2 850 C7 = 0 022 .. .. .. . ,. 0.055 0.025 344.3 O I
