Laws of Thermodynamics and Free Energy

Equilibrium constant from standard free energy

The standard Gibbs energy change of a reaction fixes its equilibrium constant through \Delta G^\circ = -RT\ln K, so K = \exp(-\Delta G^\circ/RT). A negative \Delta G^\circ gives K > 1 and favours the products. Chemical equations can be combined algebraically: reversing a reaction changes the sign of \Delta G^\circ, and multiplying all coefficients by a factor multiplies \Delta G^\circ by the same factor. The \Delta G^\circ and K of a target reaction are therefore obtained from the component values.

In writing K, pure solids and liquids have unit activity and are omitted, while gaseous species enter as partial pressures in atm, each raised to its stoichiometric coefficient. For \text{CO} + \tfrac{1}{2}\text{O}_2 \rightleftharpoons \text{CO}_2 the expression is K_p = p_{\text{CO}_2}/(p_{\text{CO}}\,p_{\text{O}_2}^{1/2}), so fractional exponents appear whenever the coefficients are fractional. K can be very large when the reactant partial pressures are very small.

A condensed phase that is not pure carries its activity a as a factor in K. For example, K = p_{\text{CO}_2}^2/(p_{\text{CO}}^2\,a) when the solid oxide has a \neq 1. Solving K for a partial-pressure ratio gives the gas composition at equilibrium, which is the usual form of such problems.

Ellingham diagram for metal oxides

The Ellingham diagram plots the standard Gibbs energy of formation of a metal oxide, written per mole of O_2, against temperature. Each line is nearly straight because \Delta H^\circ changes little with temperature, so its slope is -\Delta S^\circ. Oxidation of a metal consumes gas and gives a negative \Delta S^\circ, so these lines slope upward. The lines bend where a metal or an oxide melts or boils, because \Delta S changes at that point. A lower line marks a more stable oxide.

A metal reduces the oxide of another metal when its own oxide line lies below that oxide's line at the temperature of interest. The reduction reaction is obtained by subtracting the two oxidation reactions, so \Delta G^\circ_{\text{red}} = \Delta G^\circ_{\text{product oxide}} - \Delta G^\circ_{\text{reactant oxide}}. If the two lines do not cross, the sign of this difference is the same at every temperature, and reduction is either always or never spontaneous.

For a gaseous reductant such as H_2, the oxide is reduced only when the ratio of product to reactant gas partial pressures stays below the equilibrium constant. For reduction of FeO by H_2, this means p_{\text{H}_2\text{O}}/p_{\text{H}_2} < K.

In electrolysis, the decomposition potential is E = \Delta G^\circ/(nF), where n is the number of electrons transferred per reaction as written. Replacing an inert anode with a carbon anode lowers the voltage required, because CO_2 formation at the anode has a more negative \Delta G^\circ than O_2 release.

Enthalpy as heat at constant pressure

Enthalpy is defined as H = U + PV. For a process at constant pressure in which the only work is pressure-volume work, dU = \delta q - P\,dV, so dH = \delta q_P and the heat exchanged equals the change in enthalpy. This is why calorimetry carried out at constant pressure reports \Delta H directly.

The heat capacity at constant pressure is C_P = (\partial H/\partial T)_P. For n moles, the enthalpy change between two temperatures is \Delta H = n\int_{T_1}^{T_2} C_P\,dT. When C_P is given as a polynomial in T, such as a + bT, the integral is evaluated term by term. For example, heating 1 mol with C_P = 25 + 0.01T J/mol K from 300 K to 400 K gives \Delta H = 25(100) + 0.005(400^2 - 300^2) = 2850 J.

Free energy of solidification

At the equilibrium melting temperature T_m, liquid and solid coexist, so \Delta G = 0 and the entropy of fusion is \Delta S_m = \Delta H_m/T_m. If the heat capacities of the two phases are equal, \Delta H and \Delta S of the transformation are independent of temperature. For solidification, \Delta G = -\Delta H_m(1 - T/T_m).

Below T_m this value is negative, so solidification is spontaneous. The driving force grows with the undercooling \Delta T = T_m - T, and for small undercooling \Delta G \approx -\Delta H_m\,\Delta T/T_m.

Reversible adiabatic process of ideal gas

In a reversible adiabatic process no heat enters or leaves, so \delta q = 0 and the work is supplied entirely from the internal energy. For an ideal gas, dU = C_V\,dT and \delta w = -P\,dV, which combine into C_V\,dT = -P\,dV. With \gamma = C_P/C_V and C_P - C_V = R, integration gives TV^{\gamma-1} = constant and TP^{(1-\gamma)/\gamma} = constant. A monatomic ideal gas has C_V = \tfrac{3}{2}R and \gamma = 5/3.

The work done by the gas in an expansion is W = -\Delta U = nC_V(T_1 - T_2). This is positive because the temperature falls during the expansion, and the result is found without computing the heat.

Carnot engine efficiency

A Carnot cycle consists of two reversible isotherms and two reversible adiabats. On a T–S diagram the adiabats appear as vertical lines and the isotherms as horizontal lines, so the cycle is a rectangle. Heat is absorbed along the upper isotherm as Q_H = T_H\,\Delta S and rejected along the lower isotherm as Q_C = T_C\,\Delta S. The enclosed area equals the net work.

The efficiency is therefore \eta = 1 - T_C/T_H, with both temperatures in kelvin. Any reversible engine working between the same two reservoirs has the same efficiency, and no irreversible engine can exceed it.

Cp–Cv relation for solids

For any substance the difference between the two heat capacities is C_P - C_V = TV\alpha^2/\kappa_T, where \alpha = (1/V)(\partial V/\partial T)_P is the isobaric expansivity and \kappa_T = -(1/V)(\partial V/\partial P)_T is the isothermal compressibility. Solids have small \alpha and small \kappa_T, so C_P and C_V differ only slightly and their ratio stays close to unity.

The Dulong–Petit law gives C_V \approx 3R per mole of atoms for a monatomic solid at high temperature. Combining this value with the correction term above gives C_P from the measured volume, expansivity and compressibility. All quantities must be entered in consistent SI units.