Part II, Chapter 6

Chemical Reactions

Equilibrium constants, van't Hoff equation, Le Chatelier's principle, and electrochemistry

Historical Context

The connection between thermodynamics and chemical equilibrium was established through the combined work of Gibbs, van't Hoff, and Le Chatelier. Jacobus Henricus van't Hoff received the first Nobel Prize in Chemistry (1901) for his work on chemical dynamics and osmotic pressure. His equation relating the temperature dependence of equilibrium constants to reaction enthalpy remains one of the most useful results in physical chemistry.

Henri Le Chatelier (1884) formulated his qualitative principle predicting how equilibria respond to perturbations, while Walther Nernst connected electrochemistry to thermodynamics through the equation bearing his name (Nobel Prize 1920).

Key contributors: J. H. van't Hoff (1884, Nobel 1901), H. Le Chatelier (1884), W. Nernst (1889, Nobel 1920), Gibbs (1876).

Derivation 1: The Equilibrium Constant

Consider a general reaction $\sum_i \nu_i A_i = 0$ where $\nu_i > 0$for products and $\nu_i < 0$ for reactants. At constant T and P, the condition for equilibrium is $dG = 0$:

$$dG = \sum_i \mu_i \, dn_i = \left(\sum_i \nu_i \mu_i\right) d\xi = 0$$

where $\xi$ is the extent of reaction. Substituting $\mu_i = \mu_i^\circ + RT\ln a_i$:

$$\Delta G = \Delta G^\circ + RT \ln Q, \quad \text{where } Q = \prod_i a_i^{\nu_i}$$

At equilibrium ($\Delta G = 0$):

$$\Delta G^\circ = -RT \ln K \quad \Longleftrightarrow \quad K = \exp\left(-\frac{\Delta G^\circ}{RT}\right)$$

where $K = Q_{\text{eq}} = \prod_i a_i^{\nu_i}\big|_{\text{eq}}$

$\Delta G^\circ < 0$

K > 1, products favored

$\Delta G^\circ = 0$

K = 1, neither favored

$\Delta G^\circ > 0$

K < 1, reactants favored

Derivation 2: The van't Hoff Equation

Starting from $\Delta G^\circ = -RT\ln K$ and using$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$:

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

Differentiating with respect to temperature (assuming $\Delta H^\circ$ is approximately constant):

van't Hoff Equation

$$\frac{d \ln K}{dT} = \frac{\Delta H^\circ}{RT^2}$$

The integrated form between two temperatures:

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

A van't Hoff plot of $\ln K$ vs $1/T$ gives a straight line with slope $-\Delta H^\circ/R$. Endothermic reactions (positive slope on this plot) have K increasing with temperature; exothermic reactions have K decreasing.

Derivation 3: Le Chatelier's Principle

Le Chatelier's principle states: when a system at equilibrium is subjected to a perturbation, it shifts to partially counteract the change. This follows directly from thermodynamics.

Temperature Change

From the van't Hoff equation: $d\ln K/dT = \Delta H^\circ/(RT^2)$

Increasing T increases K for endothermic reactions ($\Delta H > 0$) and decreases K for exothermic reactions. The system shifts to absorb the added heat.

Pressure Change (Gas-Phase Reactions)

For ideal gases, $K_P = K_x \cdot P^{\Delta n}$ where $\Delta n = \sum \nu_i$ (gas moles).

Since $K_P$ is constant at fixed T, increasing P shifts the reaction toward the side with fewer gas moles (decreasing $\Delta n$).

Concentration Change

Adding a reactant increases Q beyond K, so $\Delta G = RT\ln(Q/K) > 0$ in reverse. The system shifts forward (consuming the added reactant) until Q returns to K.

Derivation 4: Gibbs-Helmholtz Equation

The Gibbs-Helmholtz equation relates the temperature dependence of G/T to the enthalpy, and serves as the foundation for the van't Hoff equation.

Starting from $G = H - TS$ and differentiating $G/T$:

$$\frac{\partial}{\partial T}\left(\frac{G}{T}\right)_P = \frac{1}{T}\left(\frac{\partial G}{\partial T}\right)_P - \frac{G}{T^2} = -\frac{S}{T} - \frac{G}{T^2} = -\frac{H}{T^2}$$

Gibbs-Helmholtz Equation

$$\left(\frac{\partial (G/T)}{\partial T}\right)_P = -\frac{H}{T^2}$$

Applied to standard reaction quantities, $\partial(\Delta G^\circ/T)/\partial T = -\Delta H^\circ/T^2$, which when combined with $\Delta G^\circ = -RT\ln K$ directly yields the van't Hoff equation.

Derivation 5: The Nernst Equation

Electrochemistry connects thermodynamics to electrical work. For an electrochemical cell, the maximum electrical work equals the Gibbs energy change:

$$\Delta G = -nFE$$

where n is the number of electrons transferred, F = 96485 C/mol is the Faraday constant, and E is the cell potential. Combining with $\Delta G = \Delta G^\circ + RT\ln Q$:

Nernst Equation

$$E = E^\circ - \frac{RT}{nF}\ln Q$$

At 298 K: $E = E^\circ - \frac{0.02569}{n}\ln Q = E^\circ - \frac{0.05916}{n}\log_{10} Q$

At equilibrium, E = 0, so:

$$E^\circ = \frac{RT}{nF}\ln K \quad \Longleftrightarrow \quad \ln K = \frac{nFE^\circ}{RT}$$

This provides a direct experimental route to equilibrium constants: measure the standard cell potential, and calculate K. For the Cu-Zn cell with $E^\circ = 1.10$ V and n = 2,$K \approx 10^{37}$, indicating the reaction goes essentially to completion.

Applications

Haber-Bosch Process

$\text{N}_2 + 3\text{H}_2 \rightleftharpoons 2\text{NH}_3$: exothermic with $\Delta n = -2$. Le Chatelier predicts high P and low T favor products, but low T is too slow. Industrial compromise: 400-500 C, 150-300 atm, with iron catalyst.

Batteries & Fuel Cells

The Nernst equation governs battery voltage at all states of charge. Lithium-ion batteries, hydrogen fuel cells, and flow batteries all rely on electrochemical thermodynamics for design and optimization.

Biochemistry

Metabolic reactions are governed by $\Delta G$ under cellular conditions. ATP hydrolysis ($\Delta G^\circ = -30.5$ kJ/mol) drives otherwise unfavorable reactions by coupling.

Corrosion Science

Pourbaix diagrams (E-pH diagrams) combine the Nernst equation with acid-base equilibria to predict which metal phases are stable under given electrochemical conditions.

Simulation: Equilibrium, van't Hoff, and Nernst

This simulation visualizes the Gibbs energy minimum at equilibrium, generates a van't Hoff plot for the N2O4/NO2 system, shows Le Chatelier's principle for temperature effects, and computes the Nernst equation for a Cu-Zn galvanic cell.

Chemical Equilibrium and Electrochemistry

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Standard States and Thermodynamic Data

The standard Gibbs energy of reaction $\Delta G^\circ$ is computed from standard formation data:

$$\Delta G^\circ = \sum_{\text{products}} \nu_i \Delta G_f^\circ - \sum_{\text{reactants}} \nu_j \Delta G_f^\circ$$

Similarly for enthalpy and entropy:

$$\Delta H^\circ = \sum \nu_i \Delta H_f^\circ, \qquad \Delta S^\circ = \sum \nu_i S_m^\circ$$

Gases

Pure gas at P = 1 bar, ideal gas behavior

Liquids/Solids

Pure substance at P = 1 bar

Solutes

Unit activity (1 M or 1 mol/kg), ideal dilute behavior

The temperature dependence of $\Delta H^\circ$ is given by Kirchhoff's equation:

$$\Delta H^\circ(T_2) = \Delta H^\circ(T_1) + \int_{T_1}^{T_2} \Delta C_P^\circ \, dT$$

Coupled Reactions in Biological Systems

Thermodynamically unfavorable reactions ($\Delta G > 0$) can be driven forward by coupling them to favorable reactions. This is a central principle of biochemistry.

Example: ATP Hydrolysis Coupling

Unfavorable reaction: Glutamate + NH3 → Glutamine + H2O ($\Delta G^\circ = +14.2$ kJ/mol)

ATP hydrolysis: ATP + H2O → ADP + Pi ($\Delta G^\circ = -30.5$ kJ/mol)

Coupled reaction: Glutamate + NH3 + ATP → Glutamine + ADP + Pi ($\Delta G^\circ = -16.3$ kJ/mol)

Since Gibbs energies are additive for coupled reactions:

$$\Delta G_{\text{coupled}} = \Delta G_1 + \Delta G_2$$

and $K_{\text{coupled}} = K_1 \times K_2$

Biological systems exploit this principle extensively. The cell maintains [ATP]/[ADP] ratios far from equilibrium, making the actual $\Delta G$ for ATP hydrolysis even more negative (about -50 to -65 kJ/mol under cellular conditions) than the standard value.

Ellingham Diagrams

Ellingham diagrams plot $\Delta G^\circ$ for oxidation reactions vs temperature. Since $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$, each reaction appears as an approximately straight line with slope $-\Delta S^\circ$.

Metal oxidation: Lines slope upward because$\Delta S < 0$ (consuming gas O2 decreases entropy). At high enough temperature, the oxide becomes unstable.
Carbon reduction: The line for C + O2 → CO2 is nearly horizontal, while 2C + O2 → 2CO slopes downward. Below the crossover temperature, carbon cannot reduce the metal oxide; above it, carbon is a viable reductant.
Metallurgical application: To reduce a metal oxide with carbon, operate above the temperature where the carbon oxidation line falls below the metal oxidation line. This is the thermodynamic basis for smelting and steelmaking.

The Electrochemical Series

Standard reduction potentials $E^\circ$ rank the tendency of species to be reduced. A more positive $E^\circ$ means a stronger oxidizing agent.

Strong Oxidizers ($E^\circ > 0$)

F2/F-: +2.87 V

Au3+/Au: +1.50 V

Ag+/Ag: +0.80 V

Cu2+/Cu: +0.34 V

Strong Reducers ($E^\circ < 0$)

H+/H2: 0.00 V (reference)

Fe2+/Fe: -0.44 V

Zn2+/Zn: -0.76 V

Li+/Li: -3.04 V

For a galvanic cell: $E_{\text{cell}}^\circ = E_{\text{cathode}}^\circ - E_{\text{anode}}^\circ > 0$. The relationship $\Delta G^\circ = -nFE^\circ$ connects the electrochemical series directly to thermodynamic spontaneity.

Chapter Summary

• Equilibrium constant: $\Delta G^\circ = -RT\ln K$ connects standard Gibbs energy to the equilibrium composition.

• van't Hoff: $d\ln K/dT = \Delta H^\circ/(RT^2)$; endothermic reactions are favored at higher T.

• Le Chatelier: Systems at equilibrium shift to partially counteract perturbations in T, P, or concentration.

• Gibbs-Helmholtz: $\partial(G/T)/\partial T = -H/T^2$; connects enthalpy to temperature dependence of G.

• Nernst: $E = E^\circ - (RT/nF)\ln Q$; electrochemistry as applied thermodynamics.

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