Gibbs Free Energy (ΔG) - Chemistry

1. Fundamental Concepts

Definition: Gibbs Free Energy (\$$ \Delta G \ $$) is a thermodynamic potential that measures the maximum reversible work that may be performed by a thermodynamic system at constant temperature and pressure.
Equation: \$$ \Delta G = \Delta H - T \Delta S \ $$, where \$$ \Delta H \ $$ is the change in enthalpy, \$$ T \ $$ is the temperature in Kelvin, and \$$ \Delta S \ $$ is the change in entropy.
Spontaneity Criteria:
- \$$ \Delta G < 0 \ $$: The process is spontaneous.
- \$$ \Delta G = 0 \ $$: The system is at equilibrium.
- \$$ \Delta G > 0 \ $$: The process is non-spontaneous.

2. Key Concepts

Enthalpy (\$$ \Delta H \ $$): The heat content of a system at constant pressure.

Entropy (\$$ \Delta S \ $$): A measure of the disorder or randomness in a system.

Temperature (\$$ T \ $$): Measured in Kelvin (K).

Application: Used to predict the spontaneity of chemical reactions and phase changes.

3. Examples

Example 1 (Basic)

Problem: Calculate \$$ \Delta G \ $$ for a reaction with \$$ \Delta H = -100 \, \text{kJ} \ $$ and \$$ \Delta S = -50 \, \text{J/K} \ $$ at \$$ 298 \, \text{K} \ $$.

Step-by-Step Solution:

Convert \$$ \Delta S \ $$ to kJ/K: \$$ \Delta S = -50 \, \text{J/K} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = -0.050 \, \text{kJ/K} \ $$
Use the Gibbs Free Energy equation: \$$ \Delta G = \Delta H - T \Delta S \ $$
Substitute the values: \$$ \Delta G = -100 \, \text{kJ} - (298 \, \text{K}) \times (-0.050 \, \text{kJ/K}) \ $$
Simplify: \$$ \Delta G = -100 \, \text{kJ} + 14.9 \, \text{kJ} = -85.1 \, \text{kJ} \ $$

Validation: Since \$$ \Delta G = -85.1 \, \text{kJ} < 0 \ $$, the reaction is spontaneous.

Example 2 (Intermediate)

Problem: Determine if the reaction is spontaneous at \$$ 298 \, \text{K} \ $$ given \$$ \Delta H = 50 \, \text{kJ} \ $$ and \$$ \Delta S = 100 \, \text{J/K} \ $$.

Step-by-Step Solution:

Convert \$$ \Delta S \ $$ to kJ/K: \$$ \Delta S = 100 \, \text{J/K} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 0.100 \, \text{kJ/K} \ $$
Use the Gibbs Free Energy equation: \$$ \Delta G = \Delta H - T \Delta S \ $$
Substitute the values: \$$ \Delta G = 50 \, \text{kJ} - (298 \, \text{K}) \times (0.100 \, \text{kJ/K}) \ $$
Simplify: \$$ \Delta G = 50 \, \text{kJ} - 29.8 \, \text{kJ} = 20.2 \, \text{kJ} \ $$

Validation: Since \$$ \Delta G = 20.2 \, \text{kJ} > 0 \ $$, the reaction is non-spontaneous.

4. Problem-Solving Techniques

Unit Conversion: Always ensure that units are consistent (e.g., convert J to kJ, K to °C if necessary).
Sign Convention: Pay attention to the signs of \$$ \Delta H \ $$ and \$$ \Delta S \ $$. Positive \$$ \Delta H \ $$ indicates endothermic, negative \$$ \Delta H \ $$ indicates exothermic. Positive \$$ \Delta S \ $$ indicates increased disorder, negative \$$ \Delta S \ $$ indicates decreased disorder.
Temperature Consideration: Temperature in Kelvin (K) is always positive and must be used in the Gibbs Free Energy equation.