Thermochemistry and Thermodynamics: Entropy, Free Energy, and Phase Changes

Thermochemistry and Thermodynamics

Entropy, Enthalpy, and Free Energy in Chemical Processes

This study guide covers key thermodynamic concepts relevant to General Chemistry, focusing on entropy (ΔS), enthalpy (ΔH), and Gibbs free energy (ΔG) changes in various chemical and physical processes. The problems and solutions provided illustrate the application of these concepts to gases, phase changes, and chemical reactions.

1. Entropy Change of a Gas with Temperature Change

When the temperature of a gas changes, the entropy change depends on whether the process occurs at constant volume or constant pressure.

• Constant Volume: The entropy change is given by: $\Delta S = n C_v \ln \left( \frac{T_2}{T_1} \right)$ where n is the number of moles, Cv is the molar heat capacity at constant volume, and T1, T2 are the initial and final temperatures (in Kelvin).

• Constant Pressure: The entropy change is: $\Delta S = n C_p \ln \left( \frac{T_2}{T_1} \right)$ where Cp is the molar heat capacity at constant pressure.

• Example: For 2.00 mol of CO2(g) heated from 25°C to 300°C, using ideal gas values for Cv and Cp, calculate ΔS for both constant volume and constant pressure conditions.

2. Work, Heat, and Internal Energy in Gas Expansion

For an ideal gas expanding isothermally (constant T), the changes in internal energy (ΔU), work (w), and heat (q) can be calculated as follows:

• Isothermal Expansion: For an ideal gas, ΔU = 0 (since internal energy depends only on temperature).

• Work Done by the Gas: $w = -P_{\text{ext}} \Delta V$ For reversible isothermal expansion: $w = -nRT \ln \left( \frac{V_f}{V_i} \right)$

• Heat Absorbed: Since ΔU = 0, $q = -w$.

• Example: 2.0 mol of an ideal gas expands from 1.0 L to 10.0 L in two steps against different external pressures. Calculate ΔU, q, w, and ΔH for each step.

3. Entropy Change in Gas Expansion

The entropy change for an isothermal expansion of an ideal gas is:

• $\Delta S = nR \ln \left( \frac{V_f}{V_i} \right)$

• For the surroundings: $\Delta S_{\text{sur}} = \frac{q_{\text{sur}}}{T}$

• Total entropy change: $\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{sur}}$

• Example: Calculate ΔS for the system, surroundings, and universe for the expansion described above.

4. Phase Changes and Enthalpy/Entropy Calculations

When a substance undergoes a phase change (e.g., vaporization), both enthalpy and entropy changes must be considered. The total change is the sum of heating, phase change, and further heating steps.

• Heating a Substance: $q = nC_p \Delta T$

• Phase Change: $q = n \Delta H_{\text{phase}}$

• Entropy Change: For heating: $\Delta S = nC_p \ln \left( \frac{T_2}{T_1} \right)$ For phase change: $\Delta S = \frac{n \Delta H_{\text{phase}}}{T_{\text{transition}}}$

• Example: Calculate ΔH, ΔS, and ΔG for vaporizing methanol at constant pressure, including heating from 25°C to 100°C, vaporization, and further heating.

5. Entropy and Free Energy in Gas Separation

Separating a mixture of gases into pure components increases entropy due to the increase in the number of possible microstates.

• Probability of Microstates: For N molecules, the probability of a particular arrangement is $\left( \frac{1}{2} \right)^N$.

• Entropy Change for Mixing/Separation: $\Delta S = -nR \sum x_i \ln x_i$ where $x_i$ is the mole fraction of component i.

• Gibbs Free Energy Change: $\Delta G = \Delta H - T \Delta S$

• Example: Calculate the entropy and free energy change for separating N2 and O2 in a box into two separate volumes.

6. Gibbs Free Energy of Formation and Temperature Dependence

The standard Gibbs free energy of formation (ΔGf°) for a reaction can be calculated from standard enthalpy and entropy values:

• $\Delta G_f^\circ = \Delta H_f^\circ - T \Delta S_f^\circ$

• To find the temperature at which a reaction becomes spontaneous, set $\Delta G = 0$ and solve for T:

• $T = \frac{\Delta H}{\Delta S}$

• Example: For the formation of CO2(g) from C(s) and O2(g), calculate ΔGf° at 25°C and the temperature above which the reaction becomes non-spontaneous.

7. Vapor Pressure and Free Energy of Vaporization

The vapor pressure of a liquid and the free energy change for vaporization can be calculated using the Clausius-Clapeyron equation and the relationship between ΔG and vapor pressure.

• Clausius-Clapeyron Equation: $\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• Gibbs Free Energy for Vaporization: $\Delta G = -RT \ln K$ where K is the equilibrium constant (for vaporization, K = Pvap)

• Example: Calculate the vapor pressure of methanol at 25°C and ΔGvap at 25°C, given ΔHvap and the normal boiling point.

Summary Table: Key Thermodynamic Equations

Process	Equation	Variables
Entropy change (heating at constant V)	$\Delta S = nC_v \ln \left( \frac{T_2}{T_1} \right)$	n = moles, Cv = heat capacity, T = temperature
Entropy change (heating at constant P)	$\Delta S = nC_p \ln \left( \frac{T_2}{T_1} \right)$	n = moles, Cp = heat capacity, T = temperature
Isothermal expansion (ideal gas)	$w = -nRT \ln \left( \frac{V_f}{V_i} \right)$	n = moles, R = gas constant, T = temperature, V = volume
Entropy change (mixing/separation)	$\Delta S = -nR \sum x_i \ln x_i$	xi = mole fraction
Gibbs free energy	$\Delta G = \Delta H - T \Delta S$	ΔH = enthalpy, ΔS = entropy, T = temperature
Clausius-Clapeyron equation	$\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$	P = vapor pressure, ΔHvap = enthalpy of vaporization, T = temperature

Additional info: These problems and solutions are representative of topics in Ch.8 (Thermochemistry: Chemical Energy) and Ch.16 (Thermodynamics: Entropy, Free Energy & Equilibrium) of a General Chemistry course. They integrate concepts of entropy, enthalpy, and free energy, and apply them to real chemical and physical processes, including gas expansions, phase changes, and chemical reactions.