The Equilibrium Constant

The equilibrium constant

Consider the reversible chemical reaction:

\[{\text{a}}\color{orange}{\text{A}} + {\text{b}}\color{orange}{\text{B}} \leftrightharpoons {\text{c}}\color{purple}{\text{C}} + {\text{d}}\color{purple}{\text{D}}\]

\(\color{orange}{\text{A}}\) and \(\color{orange}{\text{B}}\) are the reactants, \(\color{purple}{\text{C}}\) and \(\color{purple}{\text{D}}\) are the products and a, b, c, and d are the coefficients from the balanced reaction.

When the rate of the forward reaction equals the rate of the reverse reaction, the system is in chemical equilibrium. It is useful to know how much of each substance is in the container - in particular the amount of products compared to the amount of reactants. A simple ratio from the balanced chemical equation gives us a number called the equilibrium constant (\(\text{K}_{\text{c}}\)).

\[\text{K}_{\text{c}} = \dfrac{{[\color{purple}{\text{C}}\text]}^{c}{[\color{purple}{\text{D}}\text]}^{d}}{{[\color{orange}{\text{A}}\text]}^{a}{[\color{orange}{\text{B}}\text]}^{b}}\]

\([\color{orange}{\text{A}}\text], [\color{orange}{\text{B}}\text], [\color{purple}{\text{C}}\text] \text{and} [\color{purple}{\text{D}}\text]\) are the molar concentrations of each substance present at equilibrium.

When the concentration of the reactants is much larger than the concentration of the products \(\text{K}_{\text{c}}\) will be small (normally less than one). When the concentration of the reactants is much less than that of the products \(\text{K}_{\text{c}}\) will be large (normally greater than one).

For example:

\(2\color{orange}{\text{NO}{\text{(g)}}} + 2\color{orange}{\text{H}_{2}{\text{(g)}}} \rightleftharpoons \color{purple}{\text{N}_{2}{\text{(g)}}} + 2\color{purple}{\text{H}_{2}{\text{O(g)}}}\)

The \(\color{orange}{\textbf{reactants}}\) are \(\color{orange}{\text{NO}}\) and \(\color{orange}{\text{H}_{2}}\). The \(\color{purple}{\textbf{products}}\) are \(\color{purple}{\text{N}_{2}}\) and \(\color{purple}{\text{H}_{2}{\text{O}}}\).

It is important to look at the coefficients of the equation as well:

\(\color{orange}{\textbf{2}}\)\(\text{NO}(\text{g})\) + \(\color{orange}{\textbf{2}}\)\(\text{H}_{2}(\text{g})\) \(\rightleftharpoons \color{purple}{\textbf{1}}\)\(\text{N}_{2}(\text{g})\) + \(\color{purple}{\textbf{2}}\)\(\text{H}_{2}\text{O}(\text{g})\)

\(\text{K}_{\text{c}}\) for this equation will therefore be written as follows:

\[\text{K}_{\text{c}}=\dfrac{\text{[N}_{2}{\text{]}}{\color{purple}{^{1}}}{\text{[H}}_{2}{\text{O]}}{\color{purple}{^{2}}}}{\text{[NO]}{\color{orange}{^{2}}}{\text{[H}}_{2}{\text{]}}{\color{orange}{^{2}}}}\]

In the expression for \(\text{K}_{\text{c}}\) the concentration of a product or reactant is taken to the power of its coefficient in the balanced reaction. So if the coefficient of a product (C) is 3 in the balanced equation, then the concentration of C ([C]) will be written \([\text{C}]^{3}\) in the expression for \(\text{K}_{\text{c}}\).

We leave out reactants or products that are either pure liquids or in the solid phase when calculating \(\text{K}_{\text{c}}\). For example:

\(\color{red}{\text{C(s)}}\) \(+ \text{H}_{2}\text{O}(\text{g})\) \(\leftrightharpoons\) \(\text{CO}(\text{g}) + \text{H}_{2}(\text{g})\)\(\text{K}_{\text{c}}=\dfrac{{\text{[CO][H}}_{2}\text{]}}{{\text{[H}}_{2}\text{O]}}\)