Multicomponent Systems and Gibbs Free Energy

Multicomponent Systems

• Mole Fraction: expressed as $x_i = \frac{n_i}{n}$, where $n = \sum_i n_i$
• Binary System: For a binary system $x_A + x_B = 1$

Gibbs Free Energy

• Ideal Gas Mixture: $G = \sum_i n_i \mu_i = \sum n_i \mu_i^0 + n_i RT \ln P_i = \sum n_i \mu_i^0 + n_i RT \ln (x_i P)$
• $\mu_i = \mu_i^0 + RT \ln (x_i P)$
• $G = \sum n_i \mu_i^0 + RT \sum n_i \ln (x_i P)$
• $G = \sum n_i \mu_i^0 + RT \sum n_i \ln x_i + RT \sum n_i \ln P$
• $G = \sum n_i \mu_i^0 + nRT \sum x_i \ln x_i + nRT \ln P$
• $\Delta G_{mix} = G - \sum n_i \mu_i^0 = nRT \sum x_i \ln x_i$ - Since $x_i < 1$, $\ln x_i < 0$, therefore $\Delta G_{mix} < 0$ - Mixing will always be spontaneous.
• $\Delta S_{mix} = - \left( \frac{\partial \Delta G_{mix}}{\partial T} \right)_P = -nR \sum x_i \ln x_i > 0$
• $\Delta H_{mix} = \Delta G_{mix} + T \Delta S_{mix} = 0$
• Ideal Solution
  • Molecules mix without interaction.
  • $\Delta H_{mix} = 0$
  • $\Delta S_{mix} = -nR \sum x_i \ln x_i$
  • $\Delta G_{mix} = nRT \sum x_i \ln x_i$
  • $\mu_i = \mu_i^* + RT \ln x_i$
    • Where $\mu_i^*$ stands for the chemical potential of the pure substance

Phase Diagrams

• Chemical Potential:
  • A plot of chemical potential vs composition at constant $T$ and $P$ can predict phase separation.
• Eutectic Point
  • The point at which the liquid phase has the same composition as the solid phase.
  • The melting point of the mixture is lower than that of the pure components.
  • $L \rightleftharpoons A(s) + B(s)$
• Eutectic Phase Diagram
  • Two components are miscible in the liquid phase but immiscible in the solid phase.
  • Solid $\alpha$ is mostly $A$ with some $B$
  • Solid $\beta$ is mostly $B$ with some $A$
  • At the eutectic point the liquid phase has the same composition as the solid phase.
• Lever Rule
  • Used to determine the relative amounts of each phase in a two-phase region.
  • $n_\alpha l_\alpha = n_\beta l_\beta$
  • $n_\alpha + n_\beta = n$
  • $n_\alpha = n \frac{l_\beta}{l_\alpha + l_\beta}$
  • $n_\beta = n \frac{l_\alpha}{l_\alpha + l_\beta}$
• Real Solutions
  • Molecules mix and interact.
  • If A-B interactions are stronger than A-A and B-B interactions, then the components will mix.
  • If A-A and B-B interactions are stronger than A-B interactions, then the components will not mix.
• Upper Critical Solution Temperature
  • The temperature above which the components are miscible in all proportions.
  • Above the curve, the components are miscible.
  • Below the curve, the components are immiscible.
• Compound Formation
  • If $A$ and $B$ form a stable compound $A_xB_y$, the phase diagram will look like two eutectic phase diagrams joined together