Thermodynamics of Simple Mixtures: Gibbs Energy, Mixing, and Raoult’s Law

Thermodynamics of Simple Mixtures

Introduction to Mixtures and Gibbs Energy

Mixtures are systems containing more than one chemical component, and their thermodynamic properties are essential for understanding chemical processes. The Gibbs energy (G) is a central thermodynamic function used to predict the spontaneity of mixing and phase equilibria in mixtures.

• Gibbs Energy for a Binary Mixture: For a mixture of two components, A and B, the total Gibbs energy is given by: $G = n_A \mu_A + n_B \mu_B$ where $n_A$ and $n_B$ are the number of moles, and $\mu_A$, $\mu_B$ are the chemical potentials of A and B, respectively.

• Chemical Potential: The chemical potential $\mu_i$ is the partial molar Gibbs energy, representing the contribution of each component to the total Gibbs energy.

Mixing of Ideal Gases

Mixing two ideal gases is a spontaneous process, driven by an increase in entropy and a decrease in Gibbs energy. The process can be visualized as two separate containers of gases being combined into one.

• Spontaneity: Mixing is spontaneous if $\Delta_{mix}G < 0$.

• Gibbs Energy Change for Mixing: For ideal gases initially at the same pressure and temperature: $\Delta_{mix}G = nRT(x_A \ln x_A + x_B \ln x_B)$ where $x_A$ and $x_B$ are the mole fractions of A and B, respectively.

• Example: Mixing 1.0 mol A and 1.0 mol B at 298 K and 1 bar yields $\Delta_{mix}G = -3.4$ kJ.

Mathematical Derivation: Chemical Potential and Pressure Dependence

The chemical potential of an ideal gas depends on pressure at constant temperature. The relationship is derived as follows:

• $d\mu = V_m dp - S_m dT$ (general form)

• For isothermal conditions: $d\mu = V dp$

• For an ideal gas: $V_m = \frac{RT}{p}$, so $d\mu = RT \left(\frac{1}{p}\right) dp$

• Integrating from $p_1$ to $p_2$: $\mu_2 = \mu_1 + RT \ln \left(\frac{p_2}{p_1}\right)$

Mixing Gases with Different Initial Pressures

When two gases are initially at different pressures, the calculation of $\Delta_{mix}G$ requires determining the final and initial states, including partial pressures and mole fractions.

• Example: Mixing 3.0 mol H2 at 3p and 1.0 mol N2 at p.

• After mixing, total pressure is not simply the sum of initial pressures; it must be calculated based on the final volume and total moles.

• Partial pressures are determined by mole fractions: $x_{H_2} = \frac{3}{4}$, $p_{H_2} = \frac{3}{2}p$ $x_{N_2} = \frac{1}{4}$, $p_{N_2} = \frac{1}{2}p$

• Gibbs energy change: $\Delta G = (3.0)RT \ln\left(\frac{\frac{3}{2}p}{3p}\right) + (1.0)RT \ln\left(\frac{\frac{1}{2}p}{p}\right)$

Thermodynamic Functions of Mixing

Mixing affects several thermodynamic quantities, including enthalpy, entropy, and Gibbs energy.

• Enthalpy of Mixing ($\Delta_{mix}H$): For ideal gases and ideal solutions, $\Delta_{mix}H = 0$.

• Entropy of Mixing ($\Delta_{mix}S$): The increase in entropy drives the mixing process: $\Delta_{mix}S = -nR(x_A \ln x_A + x_B \ln x_B)$

• Gibbs Energy of Mixing ($\Delta_{mix}G$): For ideal mixtures: $\Delta_{mix}G = nRT(x_A \ln x_A + x_B \ln x_B)$

• Key Point: For ideal mixtures, the enthalpy change is zero, and the process is driven entirely by entropy.

Liquid Mixtures and Vapour Pressure

Raoult’s Law and Ideal Solutions

Raoult’s Law describes the vapor pressure of each component in an ideal liquid mixture. An ideal solution is one in which the intermolecular interactions between unlike molecules are similar to those between like molecules.

• Raoult’s Law: The partial vapor pressure of component A in a mixture is: $p_A = x_A p_A^*$ where $x_A$ is the mole fraction of A in the liquid, and $p_A^*$ is the vapor pressure of pure A.

• Total Vapor Pressure: For a binary mixture: $p_{total} = x_A p_A^* + x_B p_B^*$

• Ideal Solution Criteria: The best examples are isotopic mixtures or closely related substances (e.g., benzene/toluene).

Phase Equilibrium and Chemical Potential

At equilibrium, the chemical potential of a component is equal in both the liquid and vapor phases:

• $\mu_A^{liq} = \mu_A^{vap}$

• This condition determines the distribution of components between phases and is fundamental to understanding vapor-liquid equilibrium.

Summary Table: Thermodynamic Quantities for Ideal Mixtures

Quantity	Expression (Ideal Mixture)
Gibbs Energy of Mixing ($\Delta_{mix}G$)	$nRT \sum x_i \ln x_i$
Entropy of Mixing ($\Delta_{mix}S$)	$-nR \sum x_i \ln x_i$
Enthalpy of Mixing ($\Delta_{mix}H$)	0

Key Takeaways

• Mixing of ideal gases and liquids is driven by entropy, with no enthalpy change for ideal systems.

• Raoult’s Law applies to ideal solutions, predicting linear relationships between vapor pressure and composition.

• Phase equilibrium is established when the chemical potential of each component is equal in all phases present.