[[Thermodynamic properties of mixtures]] ### Ideal Gas Mixtures: The volume, [[Enthalpy (H) is the heat energy of a system|enthalpy]], and [[Internal Energy (U) of a system is its total energy content|internal energy]] do not change during ideal mixing, while the entropy increases and the Gibbs free energy decreases. $\begin{align} V = \sum x_iV_i \qquad ΔV_{mix} = 0 \qquad S = \sum x_iS_i - R \sum x_i ln(x_i) \qquad ΔS_{mix} = -R \sum x_iln(x_i) \\ H = \sum x_iH_i \qquad ΔH_{mix} = 0 \qquad G = \sum x_iG_i + RT \sum x_i ln(x_i) \qquad ΔG_{mix} = RT \sum x_iln(x_i) \\ U = \sum x_iH_i \qquad ΔU_{mix} = 0 \qquad S_{gen} = ΔS{mix} - \frac{ΔH_{mix}}{T_{bath}} \qquad W_{lost} = T_0S_{gen} = work to separate \end{align}$ The final temperature is calculated by setting the sum of ΔU for each gas to 0, using ideal gas heat capacities. Pressure is calculated by substituting the ideal gas law into the equation for volume. The partial molar properties of the components are calculated from a reference state using the ideal gas equations. Monatomic Ideal Gas: C_V = 3/2, Diatomic Ideal Gas: C_V = 5/2 ### Real Mixtures: The properties of ideal mixtures are modified to fit real mixtures by adding residual properties, which can be calculated from equations of state. The same equations are used, but special mixing rules are used to calculate the constants $a$, $b$, and $da/dT$. $b = \sum_i x_ib_i \qquad a = \sum_i \sum_j x_ix_ji $