Law of Mass Action and Equilibrium Constant K_c and K_p

The law of mass action states that K_c is defined as the concentrations of products divided by the concentrations of reactants raised to correct powers. K_p is defined as the partial pressures of products divided by the partial pressures of reactants, raised to respective powers. The chemical coefficients become exponents in the expressions for K_c and K_p.

In the late 19th century, Norwegian chemists Cato Guldberg and Peter Waage proposed the law of mass action, a chemical law that states that when a reaction reaches equilibrium, the product concentrations multiplied together divided by the reactant concentrations multiplied together, with each term raised to the power of its coefficient in the balanced equation, is a constant.

Consider the chemical reaction

{a{\rm{A}}+b{\rm{B}}}\rightleftarrows{c{\rm{C}}+d{\rm{D}}}

in which a, b, c, and d are the coefficients for a balanced chemical equation. For a chemical system at equilibrium and at a certain temperature, the equilibrium constant (K_c) is the product concentrations multiplied together divided by the reactant concentrations multiplied together, with each term raised to the power of its coefficient in the balanced equation.

{K_{\rm{c}}} = \frac{{{{\left[ {\rm{C}} \right]}^c}{{\left[ {\rm{D}} \right]}^d}}}{{{{\left[ {\rm{A}} \right]}^a}{{\left[ {\rm{B}} \right]}^b}}}

In this formula, the square brackets represent molar concentrations of the compounds. The equilibrium constant, K_c, can be dimensionless or can have a unit, depending on whether the solution (mixture of aqueous or gaseous reactants and products) is highly concentrated, and the unit used for K_c depends on the degree of concentration. For calculations in practice, K_c is often treated as dimensionless. Note that the equilibrium constant, K_c, is calculated using concentrations. Solids and pure liquids do not have a concentration and are not used in calculating the equilibrium constant. K_c describes the forward reaction in which A and B are the reactants. In the reverse reaction, C and D are the reactants and A and B are the products. The equilibrium constant for the reverse reaction, K_c', is the reciprocal of K_c.

{K'_{\rm{c}}} = \frac{{{{\left[ {\rm{A}} \right]}^a}{{\left[ {\rm{B}} \right]}^b}}}{{{{\left[ {\rm{C}} \right]}^c}{{\left[ {\rm{D}} \right]}^d}}}=\frac{1}{K_{\rm{c}}}

When the chemical system is not in equilibrium, a useful expression is the reaction quotient (Q), a ratio that can be used to predict how a system not in equilibrium will change to reach equilibrium. The reaction quotient is calculated in the same way as the equilibrium constant, as the product concentrations multiplied together divided by the reactant concentrations multiplied together, with each term raised to the power of its coefficient in the balanced equation. The reaction quotient, however, uses concentrations for the system not at equilibrium. If Q is less than K_c, the reaction will continue in the forward direction. If Q is greater than K_c, the reaction will proceed in the reverse direction. Consider the following example. This equilibrium reaction forms the basis of the Haber-Bosch process, used for the synthesis of ammonia.

{{\rm{N}}_2}(g) + 3{{\rm{H}}_2}(g) \rightleftharpoons 2{\rm{N}}{{\rm{H}}_3}(g)

The equilibrium constant for the reaction is calculated as

{K_{\rm{c}}} = \frac{{{{\left[ {{\rm{N}}{{\rm{H}}_3}} \right]}^2}}}{{\left[ {{{\rm{N}}_2}} \right]{{\left[ {{{\rm{H}}_2}} \right]}^3}}}

When both the reactants and the products of a reversible reaction are gases, there is another way of calculating an equilibrium constant. This formula uses partial pressures instead of concentrations. If a mixture of gases is present in a closed container, the partial pressure of a gas is the individual pressure exerted by it. If A and B are the only two gases in a system, the total pressure in the system is given by

P_{\rm{total}}=P_{\rm{A}}+P_{\rm{B}}

Similarly, consider a chemical reaction in which all of the reactants and products are gases.

a{\rm{A}} + b{\rm{B}} \rightleftarrows c{\rm{C}} + d{\rm{D}}

The total pressure of the system is

{P_{{\rm{total}}}} = {P_{\rm{A}}} + {P_{\rm{B}}} + {P_{\rm{C}}} + {P_{\rm{D}}}

At constant pressure, another equilibrium constant, K_p, can be defined based on the partial pressure of each gas.

K_{\rm{p}}=\frac{({P_{\rm{C}}}^c)({P_{\rm{D}}}^d)}{({P_{\rm{A}}}^a)({P_{\rm{B}}}^b)}

Note that square brackets are not present in this formula. This helps differentiate K_p from K_c. The temperature must be a constant, or the value of K_p will change. Also, this relation holds true only at equilibrium. Just as

K^{\prime}_{\rm{c}}=K^{-1}_{\rm{c}}

for the reverse reaction, the equilibrium constant based on pressure for the reverse reaction is the reciprocal of the equilibrium constant based on pressure for the forward reaction.

K^\prime_{\rm{p}}=\frac{{({{P_{\rm{A}}}^a})({{P_{\rm{B}}}^b} )}}{{({{P_{\rm{C}}}^c})({{P_{\rm{D}}}^d})}}=\frac{1}{K_{\rm{p}}}

The partial pressure of a gas in a container can be calculated by

\begin{aligned}P_{\rm{A}}&=(\text{mole fraction of }{\rm{A})\text{(total pressure)}}\\&=(\chi_{\rm{A}})(P_{\rm{total}})\end{aligned}

Using this equation, the partial pressure terms in the equilibrium constant can be replaced.

{K_{\rm{p}}} = \frac{{{{{\rm{(}}{\chi _{\rm{C}}}{P_{{\rm{total}}}}{\rm{)}}}^c}{{{\rm{(}}{\chi _{\rm{D}}}{P_{{\rm{total}}}}{\rm{)}}}^d}}}{{{{{\rm{(}}{\chi _{\rm{A}}}{P_{{\rm{total}}}}{\rm{)}}}^a}{{{\rm{(}}{\chi _{\rm{B}}}{P_{{\rm{total}}}}{\rm{)}}}^b}}}

In this version of the equation, the P terms may or may not cancel, depending on the coefficients of the balanced chemical equation. Once again consider the Haber-Bosch process for the synthesis of ammonia.

{{\rm{N}}_2}(g) + 3{{\rm{H}}_2}(g) \rightleftharpoons 2{\rm{N}}{{\rm{H}}_3}(g)

For this reaction, the equilibrium constant in terms of pressure is

{K_{\rm{p}}}=\frac{{{P_{\rm{NH_3}}}^2}}{{P_{\rm{N_2}}}\times{{P_{\rm{H_2}}}^3}}

<Reversible Reactions and Equilibrium >Properties of Chemical Systems and Equilibrium

Equilibrium Concepts

Contents

Law of Mass Action and Equilibrium Constant K_c and K_p

Equilibrium Concepts

Contents

Law of Mass Action and Equilibrium Constant Kc and Kp

Law of Mass Action and Equilibrium Constant K_c and K_p