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

Gibbs Energy and Thermodynamics

Drivers of Physical and Chemical Change

Chemical and physical changes are driven by fundamental energetic and probabilistic factors. Thermodynamics helps us understand why certain processes occur spontaneously, while others do not.

Spontaneity in Chemical Systems

Spontaneity refers to the natural tendency of a process to occur without external intervention. Thermodynamics predicts whether a reaction is spontaneous, while kinetics describes how fast it occurs. A process can be spontaneous but extremely slow.

• Spontaneous process: Occurs naturally under given conditions.

• Nonspontaneous process: Requires continuous input of energy to proceed.

Thermodynamics vs. Kinetics

Thermodynamics determines the direction and extent of a reaction (spontaneity), while kinetics concerns the reaction rate (speed). Both are essential for understanding chemical change.

Spontaneity and Enthalpy

Many spontaneous reactions are exothermic (release heat, ΔH < 0), but not all. Some endothermic reactions (ΔH > 0) can also be spontaneous, indicating that enthalpy alone does not determine spontaneity.

• Exothermic reactions: Often spontaneous, but not always.

• Endothermic reactions: Can be spontaneous if other factors favor the process.

Entropy and the Second Law of Thermodynamics

Definition of Entropy (S)

Entropy is a measure of the disorder or randomness of a system. It quantifies the number of ways energy can be distributed among the particles of a system (microstates).

• Higher entropy: More possible arrangements (microstates) of energy.

• Lower entropy: Fewer possible arrangements.

Energetically Equivalent States and Microstates

A macrostate (overall state) can be achieved through different microstates (specific arrangements of particles and energy). The macrostate with the highest number of microstates has the greatest entropy.

Entropy and Phase Changes

Entropy increases as matter changes from solid to liquid to gas, reflecting greater molecular motion and more ways to distribute energy.

• Solid → Liquid → Gas: Entropy increases (ΔS > 0).

The Second Law of Thermodynamics

The second law states that for any spontaneous process, the entropy of the universe increases:

$\Delta S_{univ} > 0$

• Spontaneous process: Increases the entropy of the universe.

• Nonspontaneous process: Decreases or does not change the entropy of the universe.

Calculating Entropy Change

Entropy change for a process is a state function, depending only on the initial and final states:

$\Delta S_{rxn} = S_{final} - S_{initial}$

Examples of Entropy Change

• Water vapor condensing: Entropy decreases (ΔS < 0).

• Sugar dissolving in tea: Entropy increases (ΔS > 0).

• 2NH3(g) → N2(g) + 3H2(g): Entropy increases (ΔS > 0).

• Ag+(aq) + Cl−(aq) → AgCl(s): Entropy decreases (ΔS < 0).

Entropy Change in the Universe

For a process where the system's entropy decreases, the surroundings must increase in entropy by a greater amount for the process to be spontaneous.

Temperature Dependence of Entropy Change in Surroundings

The magnitude of entropy change in the surroundings is inversely proportional to temperature:

$\Delta S_{surr} = -\frac{\Delta H_{rxn}}{T}$

• At higher temperatures, the same heat transfer causes a smaller change in entropy of the surroundings.

Standard Entropy and the Third Law of Thermodynamics

Standard Entropy Change (ΔS°rxn)

The standard entropy change for a reaction is the difference in entropy between products and reactants under standard conditions (1 bar, 298 K):

$\Delta S^{\circ}_{rxn} = \sum S^{\circ}_{products} - \sum S^{\circ}_{reactants}$

The Third Law of Thermodynamics

The entropy of a perfect crystal at absolute zero (0 K) is zero:

$S = 0$ at $0\ \mathrm{K}$ for a perfect crystal.

Factors Affecting Entropy

• Phase: Gases > Liquids > Solids

• Molar mass: Higher molar mass, higher entropy (for similar types of substances).

• Molecular complexity: More complex molecules have higher entropy.

• Allotropes: Different forms of the same element (e.g., diamond vs. graphite) have different entropies.

• Dissolution: Dissolving a solid increases entropy.

Relative Entropies: Table of Standard Molar Entropies

Substance	S° (J K−1 mol−1)
H2(g)	130.7
Ar(g)	154.8
H2O(l)	70.0
H2O(g)	188.8
C(s, diamond)	2.4
C(s, graphite)	5.7
NO(g)	210.8
NO2(g)	240.1
N2O4(g)	304.4

Gibbs Free Energy and Spontaneity

Definition of Gibbs Free Energy (G)

Gibbs free energy is the chemical potential energy available to do work at constant temperature and pressure. It combines enthalpy, entropy, and temperature into a single value:

$G = H - TS$

• Spontaneous process: $\Delta G < 0$

• Nonspontaneous process: $\Delta G > 0$

• Equilibrium: $\Delta G = 0$

Gibbs Free Energy Change for a Reaction

The change in Gibbs free energy for a reaction is given by:

$\Delta G_{rxn} = \Delta H_{rxn} - T\Delta S_{rxn}$

• If $\Delta G_{rxn} < 0$, the reaction is spontaneous.

• If $\Delta G_{rxn} > 0$, the reaction is nonspontaneous.

Temperature Dependence of Spontaneity

The sign of $\Delta G$ can change with temperature, especially when $\Delta H$ and $\Delta S$ have the same sign. For example, the sublimation of dry ice (CO2(s)) is spontaneous at high temperature but not at low temperature.

ΔH	ΔS	Low Temperature	High Temperature	Example
+	+	Nonspontaneous (ΔG > 0)	Spontaneous (ΔG < 0)	2H2O(s) → 2H2O(g)
−	−	Spontaneous (ΔG < 0)	Nonspontaneous (ΔG > 0)	H2O(g) → H2O(l)
+	−	Nonspontaneous (ΔG > 0)	Nonspontaneous (ΔG > 0)	H2O(l) → H2O(s)
−	+	Spontaneous (ΔG < 0)	Spontaneous (ΔG < 0)	2H2(g) + O2(g) → 2H2O(g)

Standard Gibbs Energy of Formation (ΔG°f)

The standard Gibbs energy of formation is the change in Gibbs energy when 1 mol of a compound forms from its elements in their standard states. For pure elements in their standard states, ΔG°f = 0.

Hess's Law for Gibbs Energy

If a reaction can be expressed as the sum of a series of steps, the overall ΔG°rxn is the sum of the ΔG°rxn values for the individual steps.

Gibbs Energy at Non-Standard Conditions

The Gibbs energy change under non-standard conditions is given by:

$\Delta G_{rxn} = \Delta G^{\circ}_{rxn} + RT \ln Q$

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

Gibbs Energy and Equilibrium

At equilibrium, ΔGrxn = 0. The relationship between standard Gibbs energy and the equilibrium constant K is:

$\Delta G^{\circ}_{rxn} = -RT \ln K$

Temperature Dependence of K: The van't Hoff Equation

The van't Hoff equation relates the equilibrium constant at two temperatures:

$\ln \left( \frac{K_2}{K_1} \right) = -\frac{\Delta H^{\circ}_{rxn}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

Summary Table: Key Equations

Equation	Description
$\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$	Total entropy change in the universe
$\Delta S_{surr} = -\frac{\Delta H_{rxn}}{T}$	Entropy change of surroundings
$\Delta G_{rxn} = \Delta H_{rxn} - T\Delta S_{rxn}$	Gibbs free energy change
$\Delta G_{rxn} = \Delta G^{\circ}_{rxn} + RT \ln Q$	Gibbs energy at non-standard conditions
$\Delta G^{\circ}_{rxn} = -RT \ln K$	Relationship to equilibrium constant
$\ln \left( \frac{K_2}{K_1} \right) = -\frac{\Delta H^{\circ}_{rxn}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$	van't Hoff equation

Example: Calculate the standard entropy change for the reaction: 4NH3(g) + 5O2(g) → 4NO(g) + 6H2O(g) using standard molar entropies from the table above.

Additional info: These notes cover the core concepts of thermodynamics relevant to spontaneity, entropy, and Gibbs free energy, as outlined in a typical general chemistry curriculum.