Free Energy and Thermodynamics: Spontaneity, Entropy, and Gibbs Free Energy

Free Energy and Thermodynamics

Overview of Thermodynamic Laws

Thermodynamics is the study of energy transformations in chemical and physical processes. The laws of thermodynamics govern the direction and extent of these processes, determining whether a reaction is spontaneous and how energy is distributed.

• 0th Law of Thermodynamics: If system A is in thermal equilibrium with system B, and B with C, then A is in equilibrium with C.

• 1st Law of Thermodynamics: The energy of the universe is constant (conservation of energy). $\Delta E = q + w$ where $q$ is heat absorbed by the system and $w$ is work done on the system.

• 2nd Law of Thermodynamics: In a spontaneous process, the entropy of the universe increases. $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$ (if spontaneous)

• 3rd Law of Thermodynamics: The entropy of a pure crystalline substance at absolute zero is zero: $S(0\,K) = 0$.

Thermodynamics and Spontaneity

Spontaneous vs. Nonspontaneous Processes

Thermodynamics predicts whether a process will occur under given conditions. Spontaneous processes occur without ongoing outside intervention, while nonspontaneous processes require energy input. Spontaneity is determined by comparing the chemical potential energy before and after a reaction.

• If the system after reaction has less potential energy than before, the reaction is thermodynamically favorable.

• The direction of spontaneity can be determined by comparing the potential energy of the system at the start and end.

Kinetics Versus Thermodynamics

Spontaneity does not indicate the speed of a reaction. Thermodynamics determines if a reaction can occur, while kinetics determines how fast it occurs. For example, the conversion of diamond to graphite is spontaneous but occurs at a very slow rate.

Entropy and Spontaneous Processes

Definition and Role of Entropy

Entropy (S) is a thermodynamic function that increases with the number of energetically equivalent ways to arrange the components of a system (microstates). The second law states that for any spontaneous process, the entropy of the universe increases.

• Entropy is a state function, measured in J/(mol·K).

• Boltzmann equation: $S = k \ln W$ where $k$ is the Boltzmann constant ($1.38 \times 10^{-23}$ J/K) and $W$ is the number of microstates.

• Entropy increases with the number of microstates.

• Change in entropy: $\Delta S = S_{final} - S_{initial}$

Entropy and State Changes

When materials change state, the number of possible configurations (macrostates) changes. Gases have greater entropy than liquids, which have greater entropy than solids.

• Translational, vibrational, and rotational motions contribute to entropy.

• Entropy increases with freedom of motion: $S_{gas} > S_{liquid} > S_{solid}$

Microstates and Probability

Microstates are specific arrangements of energy among particles. The more microstates, the higher the entropy. Probability favors states with more microstates.

Entropy Change in State Change

Entropy increases when a substance changes from solid to liquid to gas. The more energetically equivalent configurations, the greater the entropy.

Calculating Entropy Changes

The change in entropy for a process at constant temperature is:

$\Delta S = \frac{q_{rev}}{T}$

where $q_{rev}$ is the heat exchanged in a reversible process and $T$ is the temperature in Kelvin.

Entropy of System, Surroundings, and Universe

Relationship Between System and Surroundings

For a process to be spontaneous, the total entropy change of the universe must be positive. If the system's entropy decreases, the surroundings' entropy must increase by a greater amount.

Heat Exchange and Entropy of Surroundings

• Exothermic process: increases entropy of surroundings ($\Delta S_{surr} > 0$)

• Endothermic process: decreases entropy of surroundings ($\Delta S_{surr} < 0$)

• Magnitude of $\Delta S_{surr}$ depends on temperature: $\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$

Gibbs Free Energy and Spontaneity

Definition and Criteria for Spontaneity

Gibbs free energy (G) is the maximum amount of work energy that can be released to the surroundings by a system at constant temperature and pressure. The change in free energy is given by:

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

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

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

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

Effect of Enthalpy, Entropy, and Temperature

The spontaneity of a process depends on the signs and magnitudes of $\Delta H$, $\Delta S$, and temperature ($T$):

$\Delta H$	$\Delta S$	Low Temperature	High Temperature	Example
-	+	Spontaneous ($\Delta G < 0$)	Spontaneous ($\Delta G < 0$)	2 N2O(g) → 2 N2(g) + O2(g)
+	-	Nonspontaneous ($\Delta G > 0$)	Nonspontaneous ($\Delta G > 0$)	2 O3(g) → 3 O2(g)
-	-	Spontaneous ($\Delta G < 0$)	Nonspontaneous ($\Delta G > 0$)	H2O(l) → H2O(s)
+	+	Nonspontaneous ($\Delta G > 0$)	Spontaneous ($\Delta G < 0$)	H2O(l) → H2O(g)

Standard State and Absolute Entropy

The standard state is the state of a material at a defined set of conditions (1 atm for gases, 1 M for solutions, pure substance for solids/liquids at 1 atm and 25°C). The absolute entropy of a substance is always positive and is zero only for a perfect crystal at 0 K.

Standard Molar Entropies

Standard molar entropy ($S^\circ$) values are tabulated for substances at 298 K. Gases have higher entropy than liquids, which have higher entropy than solids. Entropy increases with molar mass and molecular complexity.

Calculating Standard Entropy Change ($\Delta S^\circ$)

The standard entropy change for a reaction is:

$\Delta S^\circ_{rxn} = \sum n_p S^\circ (\text{products}) - \sum n_r S^\circ (\text{reactants})$

where $n_p$ and $n_r$ are the stoichiometric coefficients.

Gibbs Free Energy Calculations

Standard Free Energy Change ($\Delta G^\circ$)

The standard free energy change for a reaction is:

$\Delta G^\circ_{rxn} = \sum n_p \Delta G^\circ_f (\text{products}) - \sum n_r \Delta G^\circ_f (\text{reactants})$

Alternatively, $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$ (at 25°C).

Temperature Dependence of Spontaneity

The sign of $\Delta G$ can change with temperature, especially when $\Delta H$ and $\Delta S$ have the same sign. The temperature at which a reaction changes from spontaneous to nonspontaneous can be found by setting $\Delta G = 0$:

$T = \frac{\Delta H}{\Delta S}$

Free Energy, Equilibrium, and the Reaction Quotient

Free Energy Under Nonstandard Conditions

For reactions not at standard state, the free energy change is:

$\Delta G = \Delta G^\circ + RT \ln Q$

where $Q$ is the reaction quotient, $R$ is the gas constant, and $T$ is temperature in Kelvin.

Relationship Between Free Energy and Equilibrium Constant

At equilibrium ($Q = K$), $\Delta G = 0$:

$\Delta G^\circ = -RT \ln K$

When $K > 1$, $\Delta G^\circ$ is negative (spontaneous forward). When $K < 1$, $\Delta G^\circ$ is positive (spontaneous reverse).

Temperature Dependence of the Equilibrium Constant

The relationship between $K$ and temperature is given by:

$\ln K = -\frac{\Delta H^\circ_{rxn}}{R} \frac{1}{T} + \frac{\Delta S^\circ_{rxn}}{R}$

This equation shows that a plot of $\ln K$ versus $1/T$ yields a straight line, with slope $-\Delta H^\circ_{rxn}/R$ and intercept $\Delta S^\circ_{rxn}/R$.

Summary Table: Effect of $\Delta H$, $\Delta S$, and $T$ on Spontaneity

$\Delta H$	$\Delta S$	Low Temperature	High Temperature	Example
-	+	Spontaneous ($\Delta G < 0$)	Spontaneous ($\Delta G < 0$)	2 N2O(g) → 2 N2(g) + O2(g)
+	-	Nonspontaneous ($\Delta G > 0$)	Nonspontaneous ($\Delta G > 0$)	2 O3(g) → 3 O2(g)
-	-	Spontaneous ($\Delta G < 0$)	Nonspontaneous ($\Delta G > 0$)	H2O(l) → H2O(s)
+	+	Nonspontaneous ($\Delta G > 0$)	Spontaneous ($\Delta G < 0$)	H2O(l) → H2O(g)

Additional info: This guide covers the core concepts of thermodynamics, entropy, and free energy as they relate to spontaneity and equilibrium in chemical systems, with examples and tables to reinforce key points.