Entropy, Spontaneity, and Gibbs Free Energy: A Study Guide

The Concept of Entropy

Boltzmann’s View of Entropy

Entropy (S) is a thermodynamic property that quantifies the degree of disorder or randomness in a system. Boltzmann’s view connects entropy to the number of microscopic configurations (microstates, W) available to a system:

• Microstates: Specific arrangements of particles among energy levels.

• Boltzmann Equation: $S = k_B \ln W$

• Boltzmann constant (kB): Relates entropy to the number of microstates; $k_B = \frac{R}{N_A}$.

• As the number of accessible microstates increases (by increasing energy or volume), entropy increases.

Example: Expansion of a gas into a vacuum increases the number of accessible microstates, thus increasing entropy.

Example: Mixing two different gases increases the number of microstates and thus the entropy of the system.

Clausius’ View of Entropy

Clausius defined entropy change in terms of heat transfer in a reversible process:

• Mathematical Definition: $dS = \frac{\delta q_{rev}}{T}$

• For a finite process: $\Delta S = \int \frac{\delta q_{rev}}{T}$

• For a reversible isothermal process: $\Delta S = \frac{q_{rev}}{T}$

Evaluating Entropy and Entropy Changes

Entropy Changes in Physical Processes

Entropy increases in processes where disorder increases. Four common situations that increase entropy:

• Formation of liquids from solids

• Formation of gases from solids or liquids

• Increase in the number of gas molecules in a reaction

• Increase in temperature

Example: Melting, vaporization, and dissolving all increase entropy.

Qualitative Predictions of Entropy Change

Predicting whether a process increases or decreases entropy involves considering the number of particles, phase changes, and temperature changes.

Entropy Change for Phase Transitions

During a phase transition at constant temperature, the entropy change is given by:

• $\Delta_{tr}S = \frac{q_{tr}}{T_{tr}} = \frac{\Delta_{tr}H}{T_{tr}}$

Entropy Change for Heating or Cooling

When a substance is heated or cooled at constant pressure:

• $q = C_p \Delta T$

• $\Delta S \approx C_p \ln \left( \frac{T_f}{T_i} \right)$

Entropy Change for Ideal Gases

For an ideal gas undergoing isothermal expansion or compression:

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

• $\Delta S = -nR \ln \left( \frac{P_f}{P_i} \right)$

Summary Table: Enthalpy and Entropy Changes

Process	Enthalpy Change	Entropy Change
Phase transition at transition temperature	$\Delta_{tr}H^\circ$	$\Delta_{tr}S = \frac{\Delta_{tr}H^\circ}{T_{tr}}$
Heating/cooling at constant pressure	$\Delta H = C_p \Delta T$	$\Delta S = C_p \ln \left( \frac{T_f}{T_i} \right)$
Change in state for ideal gas	$\Delta H = C_p \Delta T$	$\Delta S = C_p \ln \left( \frac{T_f}{T_i} \right) - nR \ln \left( \frac{P_f}{P_i} \right)$

Calculating Standard Entropy of Reaction

The standard entropy change for a reaction is calculated using standard molar entropies:

• $\Delta_r S^\circ = \sum \nu S^\circ_{products} - \sum \nu S^\circ_{reactants}$

Entropy and Molecular Complexity

Standard molar entropy increases with molecular complexity (more atoms per molecule):

Criteria for Spontaneous Change: The Second Law of Thermodynamics

The Second Law of Thermodynamics

The second law states that the entropy of the universe increases for all spontaneous processes:

• $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$

Spontaneity criteria:

• $\Delta S_{univ} > 0$: spontaneous

• $\Delta S_{univ} < 0$: nonspontaneous

• $\Delta S_{univ} = 0$: reversible

Standard Gibbs Energy Change, ∆G

Gibbs Free Energy and Spontaneity

Gibbs free energy (G) is a thermodynamic potential that predicts the spontaneity of a process at constant temperature and pressure:

• $\Delta G = \Delta H_{sys} - T \Delta S_{sys}$

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

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

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

Applying the Criteria for Spontaneous Change

Case	ΔH	ΔS	ΔG	Result	Example
1	-	+	-	Spontaneous at all temp.	$2N_2O(g) \rightarrow 2N_2(g) + O_2(g)$
2	-	-	+/−	Spontaneous at low temp.	$H_2O(l) \rightarrow H_2O(s)$
3	+	+	+/−	Spontaneous at high temp.	$2NH_3(g) \rightarrow N_2(g) + 3H_2(g)$
4	+	-	+	Nonspontaneous at all temp.	$3O_2(g) \rightarrow 2O_3(g)$

Calculating ΔG for a Reaction

ΔG can be calculated from enthalpy and entropy changes:

• $\Delta_r G^\circ = \Delta_r H^\circ - T \Delta_r S^\circ$

Gibbs Energy Change and Equilibrium

Relationship Between ΔG and the Equilibrium Constant (K)

At equilibrium, the standard Gibbs energy change is related to the equilibrium constant:

• $\Delta_r G^\circ = -RT \ln K$

• If $Q < K$, the reaction proceeds forward (spontaneous).

• If $Q > K$, the reaction proceeds in reverse (nonspontaneous).

• If $Q = K$, the system is at equilibrium.

Predicting the Direction of Spontaneous Chemical Change

ΔrG	Spontaneous Reaction
< 0	Left to right (→)
> 0	Right to left (←)
= 0	Equilibrium (⇌)

ΔG° and K as Functions of Temperature

Temperature Dependence of Equilibrium Constant

The van’t Hoff equation relates the equilibrium constant to temperature:

• $\ln K = -\frac{\Delta H^\circ}{RT} + \frac{\Delta S^\circ}{R}$

• For two temperatures: $\ln \left( \frac{K_2}{K_1} \right) = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

Worked Example: Relating Equilibrium Constants and Temperature

Summary Table: Equilibrium Constants at Different Temperatures

T, K	1/T, K-1	K	ln K
800	12.5 × 10-4	9.1 × 102	6.81
850	11.8 × 10-4	1.7 × 102	5.14
900	11.1 × 10-4	4.2 × 101	3.74
950	10.5 × 10-4	1.0 × 101	2.30
1000	10.0 × 10-4	2.3 × 100	0.83
1050	9.52 × 10-4	5.0 × 10-1	-0.69
1100	9.09 × 10-4	1.1 × 10-1	-2.20
1170	8.5 × 10-4	1.2 × 10-1	-2.12

Additional info: This guide covers the core concepts of entropy, spontaneity, and Gibbs free energy, including their calculation, interpretation, and application to chemical and physical processes. Worked examples and tables are included to reinforce understanding and provide practical calculation strategies.