Thermodynamics

700 | Thermodynamicsof the pure component. Many liquid solutions encountered in practice, especiallydilute ones, satisfy this condition very closely and can be considered tobe ideal solutions with negligible error. As expected, the ideal solutionapproximation greatly simplifies the thermodynamic analysis of mixtures. Inan ideal solution, a molecule treats the molecules of all components in themixture the same way—no extra attraction or repulsion for the molecules ofother components. This is usually the case for mixtures of similar substancessuch as those of petroleum products. Very dissimilar substances such aswater and oil won’t even mix at all to form a solution.For an ideal-gas mixture at temperature T and total pressure P, the partialmolar volume of a component i is v i v i R uT/P. Substituting this relationinto Eq. 13–41 givesdm i R uTPdP R uTd ln P R u Td ln P i 1T constant, y i constant, ideal gas2(13–42)since, from Dalton’s law of additive pressures, P i y i P for an ideal gasmixture andd ln P i d ln 1y i P2 d 1ln y i ln P2 d ln P1y i constant2(13–43)for constant y i . Integrating Eq. 13–42 at constant temperature from the totalmixture pressure P to the component pressure P i of component i givesm i 1T, P i 2 m i 1T, P2 R u T ln P iP m i 1T, P2 R u T ln y i 1ideal gas2(13–44)For y i 1 (i.e., a pure substance of component i alone), the last term in theabove equation drops out and we end up with m i (T, P i ) m i (T, P), which isthe value for the pure substance i. Therefore, the term m i (T, P) is simply thechemical potential of the pure substance i when it exists alone at total mixturepressure and temperature, which is equivalent to the Gibbs functionsince the chemical potential and the Gibbs function are identical for puresubstances. The term m i (T, P) is independent of mixture composition andmole fractions, and its value can be determined from the property tables ofpure substances. Then Eq. 13–44 can be rewritten more explicitly asm i,mixture,ideal 1T, P i 2 m i,pure 1T, P2 R u T ln y i (13–45)Note that the chemical potential of a component of an ideal gas mixturedepends on the mole fraction of the components as well as the mixture temperatureand pressure, and is independent of the identity of the other constituentgases. This is not surprising since the molecules of an ideal gasbehave like they exist alone and are not influenced by the presence of othermolecules.Eq. 13–45 is developed for an ideal-gas mixture, but it is also applicable tomixtures or solutions that behave the same way—that is, mixtures or solutionsin which the effects of molecules of different components on each other arenegligible. The class of such mixtures is called ideal solutions (or ideal mixtures),as discussed before. The ideal-gas mixture described is just one cate-