Equilibrium Chemistry - Equilibrium Constant

Equilibrium Constant

Chemical potential is the partial molar free energy. The potential, μ_i, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, N_i

A general chemical equilibrium can be written as

n_j are the stoichiometric coefficients of the reactants in the equilibrium equation, and m_j are the coefficients of the products. The value of δG_r for these reactions is a function of the chemical potentials of all the species.

The chemical potential, μ_i, of the ith species can be calculated in terms of its activity, a_i.

μ_i is the standard chemical potential of the species, R is the gas constant and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δG_r (Eq) = 0

Rearranging the terms,

This relates the standard Gibbs free energy change, ΔG to an equilibrium constant, K, the reaction quotient of activity values at equilibrium.

$\ln K= \sum_k \ln {a_k}^{m_k}-\sum_j \ln {a_j}^{n_j}; K=\frac{\prod_k {a_k}^{m_k}}{\prod_j {a_j}^{n_j}}$

It follows that any equilibrium of this kind can be characterized either by the standard free energy change or by the equilibrium constant. In practice concentrations are more useful than activities. Activities can be calculated from concentrations if the activity coefficient are known, but this is rarely the case. Sometimes activity coefficients can be calculated using, for example, Pitzer equations or Specific ion interaction theory. Otherwise conditions must be adjusted so that activity coefficients do not vary much. For ionic solutions this is achieved by using a background ionic medium at a high concentration relative to the concentrations of the species in equilibrium.

If activity coefficients are unknown they may be subsumed into the equilibrium constant, which becomes a concentration quotient. Each activity a_i is assumed to be the product of a concentration, and an activity coefficient, γ_i

This expression for activity is placed in the expression defining the equilibrium constant.

$K=\frac{\prod_k {a_k}^{m_k}}{\prod_j {a_j}^{n_j}} =\frac{\prod_k \left(\gamma_k\right)^{m_k}}{\prod_j \left(\gamma_j\right)^{n_j}} =\frac{\prod_k ^{m_k}}{\prod_j ^{n_j}}\times \frac{\prod_k {\gamma_k}^{m_k}}{\prod_j {\gamma_j}^{n_j}} =\frac{\prod_k ^{m_k}}{\prod_j ^{n_j}}\times \Gamma$

By setting the quotient of activity coefficients, Γ, equal to one the equilibrium constant is defined as a quotient of concentrations.

In more familiar notation, for a general equilibrium

This definition is much more practical, but an equilibrium constant defined in terms of concentrations is dependent on conditions. In particular, equilibrium constants for species in aqueous solution are dependent on ionic strength, as the quotient of activity coefficients varies with the ionic strength of the solution.

The values of the standard free energy change and of the equilibrium constant are temperature dependent. To a first approximation, the van 't Hoff equation may be used.

This shows that when the reaction is exothermic (ΔH, the standard enthalpy change, is negative), then K decreases with increasing temperature, in accordance with Le Chatelier's principle. The approximation involved is that the standard enthalpy change, ΔH, is independent of temperature, which is a good approximation only over a small temperature range. Thermodynamic arguments can be used to show that

where C_p is the heat capacity at constant pressure.

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