Mixing and Excess Quantities

When considering mixtures of two or more components, the thermodynamics of the mixtures is often cast in the form of mixing and excess quantities. Here we outline the treatment of such mixtures. The mixing volume, enthalpy, entropy, and Gibbs free energy are defined as

$$\Delta V_{mix} = V - V^* = \sum N_i (v_i - v_i^*)$$ (1.66)

$$\Delta H_{mix} = H - H^* = \sum N_i (h_i - h_i^*)$$ (1.67)

$$\Delta S_{mix} = S - S^* = \sum N_i (s_i - s_i^*)$$ (1.68)

$$\Delta G_{mix} = G - G^* = \sum N_i (\mu_i - \mu_i^*) = \Delta H_{mix} - T \Delta S_{mix}$$ (1.69)

The starred quantities refer to the separate pure components. Mixing quantities have the same kinds of thermodynamic relationships as the total quantities. For example,

$$\left(\frac{\partial G_{mix}}{\partial p}\right)_{T,{\vec N}} = \frac{\partial}{\partial p} \sum N_i (\mu_i - \mu_i^*)$$ (1.70)

$$= \sum N_i (v_i - v_i^*)$$ (1.71)

$$= \Delta V_{mix}$$ (1.72)

Similarly, we find

$$\left(\frac{\partial G_{mix}}{\partial T}\right)_{p,{\vec N}} = -\Delta S_{mix}$$ (1.73)

An ideal solution is defined as one with $\Delta H_{mix}^{id} = \Delta V_{mix}^{id} = 0$ and

$$\Delta S_{mix}^{id} = - k_B \sum N_i \ln x_i$$ (1.74)

Therefore, for an ideal solution

$$\Delta G_{mix}^{id} = k_B T \sum N_i \ln x_i$$ (1.75)

The excess quantities are given by

$$V^{E} = V - V^{id}$$ (1.76)

$$H^{E} = H - H^{id}$$ (1.77)

$$S^{E} = S - S^{id}$$ (1.78)

$$G^{E} = G - G^{id} = H^E - TS^E$$ (1.79)

We take the pure liquid as the reference system. Then

$$G = \sum N_i \mu_i = \sum N_i (\mu_i^* + k_BT \ln \gamma_i x_i)$$ (1.80)

and

$$G^{id} = \sum N_i (\mu_i^* + k_BT \ln x_i)$$ (1.81)

Then

$$G^{E} = k_B T \sum N_i \ln \gamma_i$$ (1.82)

Also, by subtracting and adding $G^*$ from/to Eq. 1.79, we have

$$G^E = \Delta G_{mix} - \Delta G_{mix}^{id}$$ (1.83)