Question 24.3: As discussed in Lecture 23, in an ideal solution, μi = Gi =...
As discussed in Lecture 23, in an ideal solution,
\mu_{i}=\bar{G}_{i}=G_{i}(T, P)+R T \ln x_{i} (24.25)
Show that:
(i) \overline{ V }_{ i } is not a function of composition, that is, \overline{ V }_{ i }= V _{ i }( T , P ) (24.26)
(ii) \overline{ H }_{ i } is not a function of composition, that is, \overline{ H }_{ i }= H _{ i }( T , P ) (24.27)
(iii) \overline{ U }_{ i } is not a function of composition, that is, \overline{ U }_{ i }= U _{ i }( T , P ) (24.28)
In other words, show that the partial molar volume, the partial molar enthalpy, and the partial molar internal energy of component i in an ideal solution are equal to the molar values at the same T and P.
To prove (i), (ii), and (iii) above, we proceed as follows. In general, in a solution of n components, it follows that:
\left(\frac{\partial \overline{ G }_{ i }}{\partial P }\right)_{ T , x }=\overline{ V }_{ i } (24.29)
Recall that the subscript x in Eq. (24.29) indicates that all the mole fractions x _{ i } are kept constant. Recall also that x _{ i } denotes the mole fraction of component i in a condensed (liquid or solid) phase. On the other hand, y _{ i } denotes the mole fraction of component i in the gas phase.
According to the Problem Statement, in an ideal solution, \overline{ G }_{ i }= G _{ i }( T , P )+RTln x _{ i } . Using this expression for \overline{ G }_{ i } in Eq. (24.29) yields:
\left(\frac{\partial \overline{ G }_{ i }}{\partial P }\right)_{ T , x }=\left(\frac{\partial G _{ i }}{\partial P }\right)_{ T }+0= V _{ i } \Rightarrow \overline{ V }_{ i }= V _{ i }( T , P ) (24.30)
In addition, using the expression for \overline{ G }_{ i } given in the Problem Statement in the Gibbs-Helmholtz equation for a mixture yields:
\frac{\partial}{\partial T }\left(\frac{\overline{ G }_{ i }}{ T }\right)_{ P , x }=\frac{\partial}{\partial T }\left(\frac{ G _{ i }}{ T }\right)_{ P }=-\frac{ H _{ i }}{ T ^{2}} \Rightarrow \overline{ H }_{ i }= H _{ i }( T , P ) (24.31)
Having shown that, in an ideal solution, \overline{ V }_{ i }= V _{ i }( T , P ) \text { and } \overline{ H }_{ i }= H _{ i }( T , P ), it follows that:
\overline{ U }_{ i }=\overline{ H }_{ i }- P \overline{ V }_{ i }= H _{ i }- PV _{ i }= U _{ i } \Rightarrow \overline{ U }_{ i }= U _{ i }( T , P ) (24.32)