Law of Chemical Equilibrium and Equilibrium Constant

The Law of Chemical Equilibrium

When a reversible reaction reaches equilibrium, the mixture of reactants and products that remains is called an equilibrium mixture. The central question is: what fixed relationship exists between the concentrations of the substances in that mixture?

Consider a general reversible reaction written in the form:

where A and B are reactants, C and D are products, and the lowercase letters are their stoichiometric coefficients from the balanced equation.

In 1864, the Norwegian chemists Cato Maximillian Guldberg and Peter Waage proposed, based on extensive experimental work, that at equilibrium the concentrations obey a simple mathematical relation. They called it the law of mass action — because in those days concentration was referred to as "active mass". Today we call it the law of chemical equilibrium.

The quantity $K_c$ is the equilibrium constant. The expression on the right is the equilibrium constant expression. The subscript $c$ indicates that concentrations are measured in moles per litre (mol L$^{-1}$). It is understood that every concentration in the expression is an equilibrium concentration, so the subscript "eq" is usually omitted.

---

How the Law Was Discovered: The H₂–I₂–HI System

To see how Guldberg and Waage arrived at this form, examine the reaction between hydrogen and iodine gases in a sealed vessel at 731 K:

Six experiments were performed. In experiments 1–4, the vessel was charged with only H₂ and I₂ in various amounts. In experiments 5 and 6, the vessel was charged with only HI. In every case the system was allowed to reach equilibrium, and the equilibrium concentrations of all three gases were measured.

The data obtained from all six experiments is given in Table 6.2:

The data revealed two important patterns:

1. In experiments 1–4, the number of moles of H₂ that reacted equalled the number of moles of I₂ that reacted, and this was exactly half the number of moles of HI formed.
2. In experiments 5 and 6 (starting from pure HI), at equilibrium $[\text{H}_2]_{\text{eq}} = [\text{I}_2]_{\text{eq}}$.

Now, to find a constant relationship, several trial expressions were tested. The simple expression $\frac{[\text{HI}]}{[\text{H}_2][\text{I}_2]}$ gave values that varied wildly from one experiment to the next — it was not constant at all.

But the expression $\frac{[\text{HI}]^2}{[\text{H}_2][\text{I}_2]}$ gave the same numerical value for all six experiments, regardless of the starting composition. Notice that the exponent on HI (2) is exactly its stoichiometric coefficient in the balanced equation. This is the key: the power to which each concentration is raised in the equilibrium expression is the stoichiometric coefficient of that substance in the balanced chemical equation.

Both ratios, side by side for all six experiments, are summarised in Table 6.3:

Thus for this reaction:

---

Writing Equilibrium Constant Expressions

For any balanced chemical equation, the equilibrium constant expression is written by placing the product concentrations in the numerator, each raised to its stoichiometric coefficient, and the reactant concentrations in the denominator, each raised to its stoichiometric coefficient.

Example: For the reaction

the equilibrium constant is:

The square brackets denote molar concentrations at equilibrium. Phase symbols (s, l, g) are generally omitted from the expression — they are not needed because the concentrations themselves already imply the physical state.

---

Properties of the Equilibrium Constant

The equilibrium constant has several important properties that follow directly from its definition. Each one is derived below.

Property 1: The Equilibrium Constant for the Reverse Reaction

If a reaction is written in the reverse direction, the equilibrium constant for the reverse reaction is the reciprocal of the equilibrium constant for the forward reaction.

Proof: Take the forward reaction

for which

Now write the reverse reaction:

Its equilibrium constant, call it $K'_c$, is:

But this is exactly $1/x$, which is $1/K_c$. Therefore:

Property 2: Multiplying the Equation by a Factor

If the stoichiometric coefficients in a balanced equation are multiplied by a factor $n$, the new equilibrium constant is the original constant raised to the power $n$.

Proof: Start again with

and $K_c = \frac{[\text{HI}]^2}{[\text{H}_2][\text{I}_2]} = x$.

First, multiply the equation by $\frac{1}{2}$:

The equilibrium constant for this version, call it $K''_c$, is:

Now square both sides:

Therefore $K''_c = x^{1/2} = K_c^{1/2}$.

Now multiply the original equation by a general factor $n$:

The equilibrium constant for this multiplied equation, $K'''_c$, is: …