Gibbs Free Energy and Spontaneity of Chemical Reactions

Gibbs Free Energy

Definition and Importance

Gibbs Free Energy (G) is a thermodynamic quantity that measures the amount of energy in a system available to do work at constant temperature and pressure. It is crucial for predicting whether a chemical or physical process will occur spontaneously.

• Spontaneity of a reaction is determined by the sign of the change in Gibbs Free Energy ($\Delta G$).

• If $\Delta G < 0$, the process is spontaneous.

• If $\Delta G > 0$, the process is nonspontaneous.

• If $\Delta G = 0$, the system is at equilibrium.

Equation:

$\Delta G = \Delta H - T\Delta S$

• $\Delta H$ = change in enthalpy

• $\Delta S$ = change in entropy

• $T$ = temperature in Kelvin

Spontaneity and Equilibrium

The sign of $\Delta G$ and the equilibrium constant ($K$) are related:

• If $K > 1$, $\Delta G$ is negative, and the reaction is spontaneous in the forward direction.

• If $K < 1$, $\Delta G$ is positive, and the reaction is nonspontaneous in the forward direction.

• If $K = 1$, $\Delta G = 0$, and the system is at equilibrium.

Example: Interpreting $\Delta G$

• If $\Delta G$ is small and positive, the forward reaction is nonspontaneous and system is far from equilibrium.

• The reverse reaction is spontaneous and system is far from equilibrium.

• The nearest reaction to equilibrium is the one with $\Delta G$ closest to zero.

Predicting Spontaneity

Relationship with Enthalpy and Entropy

Spontaneity depends on both Enthalpy ($\Delta H$) and Entropy ($\Delta S$):

• If $\Delta H$ is negative and $\Delta S$ is positive, $\Delta G$ is always negative (spontaneous at all temperatures).

• If $\Delta H$ is positive and $\Delta S$ is negative, $\Delta G$ is always positive (never spontaneous).

• If both $\Delta H$ and $\Delta S$ are negative, spontaneity depends on temperature.

• If both $\Delta H$ and $\Delta S$ are positive, spontaneity also depends on temperature.

Example: Temperature Dependence

• For the reaction $\text{PCl}_3 (g) + \text{Cl}_2 (g) \rightarrow \text{PCl}_5 (g)$ at 25°C, $\Delta H^\circ = -88$ kJ:

• This is an exothermic reaction ($\Delta H < 0$).

• If the temperature is increased, the ratio of $[\text{PCl}_5]/([\text{PCl}_3][\text{Cl}_2])$ decreases.

• $\Delta G^\circ$ for this reaction has to be negative at all temperatures.

Practice Problems

Application of Concepts

• Example 1: For the reaction $2\text{NO}_2 (g) \rightarrow 2\text{NO} (g) + \text{O}_2 (g)$, $K_{eq} = 4.50 \times 10^1$. Does the reaction increase the entropy of the universe?

• Answer: Yes, because the number of gas molecules increases, leading to an increase in entropy ($\Delta S > 0$).

• Example 2: Predict the signs of $\Delta S$, $\Delta H$, and $\Delta G$ for the conversion of a solid into a gas.

• Answer: $\Delta S > 0$ (entropy increases), $\Delta H > 0$ (energy required), $\Delta G$ depends on temperature.

• Example 3: Combustion of butane: $\text{C}_4\text{H}_{10} (g) + \frac{13}{2} \text{O}_2 (g) \rightarrow 4\text{CO}_2 (g) + 5\text{H}_2\text{O} (g)$

• Answer: $\Delta S > 0$, $\Delta H < 0$, $\Delta G < 0$ (spontaneous reaction).

Interpreting $\Delta G$ Values

• If you calculate $\Delta G$ for a chemical reaction and get a positive value, the most accurate way to interpret this result is:

• At equilibrium, the mixture will contain more reactant than product.

• If a mixture of reactants and products is created and left to equilibrium, the equilibrium mixture will contain more reactant than product.

Summary Table: Spontaneity Based on $\Delta H$ and $\Delta S$

$\Delta H$	$\Delta S$	Spontaneity ($\Delta G$)	Temperature Dependence
Negative	Positive	Always spontaneous ($\Delta G < 0$)	All temperatures
Positive	Negative	Never spontaneous ($\Delta G > 0$)	All temperatures
Negative	Negative	Spontaneous at low T	Low temperatures
Positive	Positive	Spontaneous at high T	High temperatures

Additional info: The notes expand on the relationship between Gibbs Free Energy, equilibrium, and spontaneity, providing context for interpreting $\Delta G$ values and their implications for chemical reactions.