To the Editor: In discussing my paper on Gihbs' paradox', J. J. McDonald2has correctlv noted that the treatment niven for entrow of isothermal mixing is restricted to samples with e q d numbers of molecules as well as equal initial pressures. McDonald's equations are more general and therefore constitute an imorovement on those in mv paper. Nevertheless, .. . the thrw main ronclusions oimy paper are nor affected.'l'he iirst i i that Gihhs' paradox arises frum thc unsound assumption that the entropy of each gas in an ideal mixtureis independent of the presence of the other gases, and the second is that e n t r o ~ i e of s ideal mixine cannot he deduced sulely from the laws;fclassicaldynan&sapplied to the ideal r m coudtion. The third is that indi\idual cntn)~ie.;cannot 6e assigned to the constituents of an ideal mixtire. The relevant point about all of the equations for mixing of similar and dissimilar molecules is that their derivations all depend on the equation for mixing of samples that have distinguishable molecules,
Although this equation is basic to the thermodynamics of ideal mixtures i t cannot, by correct means, he deduced from the classical laws. On the other hand, the equation
compression. Unless no = n~ the pressures of D and E will he unequal. Equalization of pressure is a spontaneous process and would therefore make a contribution to the entropy change A& when the samples are brought together. I t should he noted that ea. 1 for different molecules is applicable to samples of ga'with initial pressures either different or equal and regardless of whether the final volume is the same asthe original total volume. Hence it is correctly used in step c of Figure 1. In the calculation of the entropy of isothermal isobaric mixing of ideal gases, or of liquids, the best procedure is to treat all samples with indistinguishable molecules, D, E, . . . as a single sample with a comhined volume VD VE . . . and a comhined number of molecules nD + nE + . . .. Then, the various groups have different molecules so eq 1is applicable. For gases this procedure gives
+
A S ,,
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Journal of Chemical Education
Vf I Vfn + nBR In -
v~
v~
This procedure is also applicable if the final volume V f is greater than the total initial volume of the gases. A similar procedure used for isothermal isobaric mixing of ideal liquids leads to
, , S A can he deduced fr0.n the first two laws (and the ideal gas laws).%ut eq 2 applies only to the change in volume of a single gas sample from VAto Vfi and it is improper to use i t when the change in volume is accompanied by penetration into another gas. I t is unfortunate that this improper procedure leads to eq. 1since support is then gained for the false belief that the gases in a mixture can have individual entropies. The equations for entropy of mixing given in my original paper are restricted to equimolar samples because of step h of Figure 1 of the paper. This is the mixing of equivolume samples of the indistinguishahle molecules D, E, . . . with
= nAR
+
= -nAR in zA- ngR in xs
where x is the mole fraction of molecules in the final mixture. H. R. Kemp Royal Australian Naval College JBNISBay 2540. Australia
' Kemp, H. R. J. Chem. Educ. 1985, 62-47.
McDonald, J. J . J. Chem. Educ. 1985, 62, 47. Pitzer. Kenneth S.; Brewer, Leo "Thermodynamics"; McGrawHill: New York, 1961; p 81.
