Thermodynamics: Entropy, Spontaneity, and the Laws of Thermodynamics

Thermodynamics: Entropy, Spontaneity, and the Laws of Thermodynamics

Entropy and the Second Law of Thermodynamics

Entropy (S) is a fundamental thermodynamic property that quantifies the degree of disorder or randomness in a system. It is a measure of how dispersed the energy of a system is among the different possible ways that a system can contain energy. The second law of thermodynamics states that in any spontaneous process, the total entropy of the system and its surroundings always increases.

• Definition: Entropy is a measure of the number of possible microstates (ways to arrange energy) in a system. More microstates mean higher entropy and less order.

• Example: When heat flows from a hot cup of coffee to your hand, the entropy of the universe increases because energy becomes more dispersed.

• Second Law: In any spontaneous process, $\Delta S_{universe} > 0$.

• Applications:

  • Power plants cannot convert all heat from fuel into work; some energy must be lost to the surroundings, limiting efficiency.

  • Refrigerators transfer more heat to the surroundings than they remove from the system.

The First and Second Laws of Thermodynamics

• First Law (Law of Energy Conservation): Energy cannot be created or destroyed, only transformed. This is an equality: $\Delta E_{universe} = 0$.

• Second Law: The direction of spontaneous change is toward increased entropy. This is an inequality: $\Delta S_{universe} > 0$.

• Entropy of Activation: In transition state theory, the entropy of activation refers to the change in entropy as reactants form the activated complex.

Evaluating Statements about Spontaneity (Example 1)

• Spontaneous reactions do not always release heat (can be endothermic or exothermic).

• Spontaneity is not related to the rate of reaction; it is a thermodynamic property, not a kinetic one.

• Entropy of a system may increase or decrease during a spontaneous change, but the entropy of the universe always increases.

• Energy of a system does not always increase in a spontaneous change; free energy decreases.

Standard Entropy and Phase Changes

Standard entropy values ($S^\circ$) depend on temperature and phase. Entropy increases with temperature and when a substance changes from solid to liquid to gas.

Phase	Entropy ($S^\circ$)
Solid	Lowest
Liquid	Intermediate
Gas	Highest

Example: The standard entropy of bromine increases sharply at the temperatures where it melts (fusion) and boils (vaporization).

The Third Law of Thermodynamics

The third law states that the entropy of a perfect crystal at absolute zero (0 K) is zero. At this temperature, there is only one possible way to arrange the particles (one microstate).

• Mathematical Relationship: Entropy is related to the number of microstates ($W$) by Boltzmann's equation: $S = k \ln W$, where $k$ is Boltzmann's constant.

Comparing Entropy in Different Systems (Example 2)

• More moles of a substance at the same conditions have higher entropy.

• Gaseous states have higher entropy than liquids or solids of the same substance at the same temperature and pressure.

• For solids, higher temperature means higher entropy.

Entropy Change for a Process

When a process occurs very close to equilibrium (reversible), the entropy change is given by:

• $\Delta S = \frac{q_{rev}}{T}$

• For non-equilibrium (irreversible) processes: $\Delta S > \frac{q}{T}$

Criteria for Spontaneity and Gibbs Free Energy

The spontaneity of a process can be predicted using entropy and enthalpy changes:

• $\Delta S > \frac{\Delta H}{T}$

• $T\Delta S > \Delta H$

• Gibbs free energy is defined as $G = H - TS$

• A process is spontaneous if $\Delta G < 0$

Entropy and the Universe

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

• For spontaneous processes: $\Delta S_{universe} > 0$

• Relating to Gibbs free energy: $\Delta G = \Delta H - T\Delta S$

• If $\Delta S_{universe} > 0$, then $\Delta G < 0$

Identifying Spontaneous Processes (Example)

• Sugar dissolving in hot tea: Spontaneous

• Rust turning shiny: Nonspontaneous (reverse of natural rusting)

• Stopped pendulum starts swinging: Nonspontaneous

• Water decomposing into H2 and O2: Nonspontaneous

Thermodynamic Analysis of a Hypothetical Reaction

For the reaction: A(g) + 2B(g) → AB2(g)

• Change in Entropy ($\Delta S$): The number of gas molecules decreases (from 3 to 1), so $\Delta S$ is negative.

• Change in Enthalpy ($\Delta H$): At equilibrium, $\Delta G = 0$, so $\Delta H = T\Delta S$. Since $\Delta S$ is negative, $\Delta H$ is also negative.

• Change in Free Energy ($\Delta G$): Before equilibrium, $\Delta G$ is negative (spontaneous). At equilibrium, $\Delta G = 0$.

Temperature Dependence of Spontaneity (Example: N2O4 Decomposition)

For the reaction: N2O4(g) → 2NO2(g)

• Breaking bonds requires energy, so $\Delta H$ is positive (endothermic).

• Increasing number of molecules increases entropy, so $\Delta S$ is positive.

• Spontaneity: $\Delta G = \Delta H - T\Delta S$. At low temperatures, $\Delta G > 0$ (nonspontaneous); at higher temperatures, $\Delta G < 0$ (spontaneous).

• Concentration and Rate: The concentration of NO2 increases with temperature, and the rate of reaction increases with temperature in both directions.

Summary Table: Entropy and Spontaneity Criteria

Condition	Spontaneity
$\Delta G < 0$	Spontaneous
$\Delta G = 0$	Equilibrium
$\Delta G > 0$	Nonspontaneous

Key Equations:

• $\Delta S = \frac{q_{rev}}{T}$

• $\Delta G = \Delta H - T\Delta S$

• $S = k \ln W$

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