Multi-Component Systems

Cheatsheet Content

### Key Equations, Relationships, and Variables #### Colligative Properties - **Chemical Potential of Solvent in Ideal Solution:** $$\mu_A(l) = \mu_A^*(l) + RT \ln x_A$$ - **Boiling Point Elevation:** $$\Delta T_b = K_b m_B$$ where $K_b = \frac{RT_b^2 M_A}{\Delta H_{vap}}$ - **Freezing Point Depression:** $$\Delta T_f = K_f m_B$$ where $K_f = \frac{RT_f^2 M_A}{\Delta H_{fus}}$ - **Osmotic Pressure (van't Hoff Equation):** $$\Pi = [B]RT$$ #### Real Gases and Solutions - **Chemical Potential of Real Gas:** $$\mu = \mu^\circ + RT \ln \left(\frac{f}{P^\circ}\right)$$ where $f = \gamma P$ - **Fugacity Coefficient from Compressibility Factor:** $$\ln \gamma = \int_0^P \frac{(Z-1)}{P} dP$$ - **Raoult's Law (Ideal Solvent):** $$P_A = x_A P_A^*$$ - **Henry's Law (Ideal Dilute Solute):** $$P_B = x_B K_B$$ - **Chemical Potential of Component in Real Solution:** $$\mu_A(l) = \mu_A^*(l) + RT \ln a_A$$ where $a_A = \gamma_A x_A$ #### Thermodynamics of Mixing (Ideal) - **Gibbs Energy of Mixing:** $$\Delta G_{mix} = nRT(x_A \ln x_A + x_B \ln x_B)$$ - **Entropy of Mixing:** $$\Delta S_{mix} = -nR(x_A \ln x_A + x_B \ln x_B)$$ - **Enthalpy of Mixing (Ideal):** $$\Delta H_{mix} = 0$$ #### Variables - $\mu$: Chemical potential - $x$: Mole fraction - $R$: Gas constant ($8.314 \text{ J K}^{-1} \text{mol}^{-1}$) - $T$: Temperature (K) - $P$: Pressure - $f$: Fugacity - $\gamma$: Fugacity coefficient / Activity coefficient - $K_b$: Ebullioscopic constant - $K_f$: Cryoscopic constant - $m_B$: Molality of solute - $M_A$: Molar mass of solvent - $\Delta H_{vap}$: Enthalpy of vaporization - $\Delta H_{fus}$: Enthalpy of fusion - $\Pi$: Osmotic pressure - $[B]$: Molar concentration of solute - $Z$: Compressibility factor - $a$: Activity - $n$: Total moles - $\Delta G_{mix}$: Gibbs energy of mixing - $\Delta S_{mix}$: Entropy of mixing - $\Delta H_{mix}$: Enthalpy of mixing - $\Delta U_{mix}$: Internal energy of mixing - $\Delta V_{mix}$: Volume of mixing