# 16.2: Raoult's Law and Ideal Solutions


An ideal solution is a homogeneous liquid solution that is at equilibrium with an ideal-gas solution in which the vapor pressure of each component satisfies Raoult’s law$${}^{1}$$. Since the gas is ideal, the partial pressure of $$A$$ is $$P_A=x_AP$$. Raoult’s law asserts a relationship among the gas- and solution-phase mole fractions of $$A$$, the vapor pressure of the pure liquid, and the pressure of the system:

$P_A=x_AP=y_AP^{\textrm{⦁}}_A$

(Raoult’s law)

For a binary mixture of $$A$$ and $$B$$ that satisfies Raoult’s law, we have also that $$P_B=x_BP=y_BP^{\textrm{⦁}}_B$$, and the total pressure becomes $$P=P_A+P_B=y_AP^{\textrm{⦁}}_A+y_BP^{\textrm{⦁}}_B$$. The lines sketched in Figure 4 show how $$P_A$$, $$P_B$$, and $$P$$ vary with the solution-phase composition when the solution is ideal.

When the standard state for $$A$$ in solution is taken to be pure liquid $$A$$ at its equilibrium vapor pressure, substitution of Raoult’s law into the results in Section 16.1 gives the activity of component $$A$$ in an ideal solution as

${ \ln \left[{\tilde{a}}_A\left(P,y_A,y_B\right)\right]\ }={ \ln \left[\frac{x_AP}{P^{\textrm{⦁}}_A}\right]\ }={ \ln \left[\frac{y_AP^{\textrm{⦁}}_A}{P^{\textrm{⦁}}_A}\right]\ }={ \ln y_A\ }$ and

${\tilde{a}}_A\left(P,y_A,y_B\right)=y_A$

(ideal solution, Raoult’s law)

In general, the activity and chemical potential of a component depend on pressure. If the solution is ideal, we see that the system pressure is fixed by $$P=y_AP^{\textrm{⦁}}_A+y_BP^{\textrm{⦁}}_B$$, and the pure-component vapor pressures depend only on temperature. Since for the binary solution, $$y_B=1-y_A$$, we can write the chemical potential of component $$A$$ as

${\mu }_A\left(P,y_A,y_B\right)={\mu }_A\left(y_A\right)={\widetilde{\mu }}^o_A\left(\ell ,P^{\textrm{⦁}}_A\right)+RT{ \ln y_A\ }$ (ideal solution)

We can also use relationships we develop earlier to find another representation for $${\widetilde{\mu }}^o_A\left(\ell ,P^{\textrm{⦁}}_A\right)$$. The chemical potential of $$A$$ in the liquid phase is the same as in the gas. Using the chemical potential for $$A$$ in the gas phase that we find in Section 16.1, we have

${\mu }_A\left(P,y_A,y_B\right)={\mu }_A\left(g,P,x_A,x_B\right)={\Delta }_fG^o\left(A,{HIG}^o\right)+RT{ \ln \left[\frac{x_AP}{P^o}\right]\ }={\Delta }_fG^o\left(A,{HIG}^o\right)+RT{ \ln \left[\frac{P^{\textrm{⦁}}_A}{P^o}\right]\ }+RT{ \ln y_A\ }$ and hence, ${\widetilde{\mu }}^o_A\left(\ell ,P^{\textrm{⦁}}_A\right)={\Delta }_fG^o\left(A,{HIG}^o\right)+RT{ \ln \left[\frac{P^{\textrm{⦁}}_A}{P^o}\right]\ }$

In Section 15.4, we find, for an ideal gas,

${\Delta }_fG^o\left(A,{HIG}^o\right)+RT{ \ln \left[\frac{P^{\textrm{⦁}}_A}{P^o}\right]\ }={\Delta }_fG^o\left(A,\ell \right)+\int^{P^{\textrm{⦁}}_A}_{P^o}{{\overline{V}}^{\textrm{⦁}}_A\left(\ell \right)}dP$

so that the chemical potential of the pure liquid at its vapor pressure is also given by ${\widetilde{\mu }}^o_A\left(\ell ,P^{\textrm{⦁}}_A\right)={\Delta }_fG^o\left(A,\ell \right)+\int^{P^{\textrm{⦁}}_A}_{P^o}{{\overline{V}}^{\textrm{⦁}}_A\left(\ell \right)}dP$

The integral is the difference between the Gibbs free energy of the pure liquid at its vapor pressure and that of the pure liquid at $$P^o=1\ \mathrm{bar}$$. Note that we can obtain the same result much more simply by integrating $${\left(dG^{\textrm{⦁}}_A\right)}_T={\overline{V}}^{\textrm{⦁}}_AdP$$ between the same two states. In Section 15.3, we see that the value of the integral is usually negligible. To a good approximation, we have

${\widetilde{\mu }}^o_A\left(\ell ,P^{\textrm{⦁}}_A\right)\approx {\Delta }_fG^o\left(A,\ell \right)$

(ideal solution)

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