Equilibrium Constant in Gaseous Systems

From Concentration to Partial Pressure

So far, every equilibrium constant you have seen has been written using molar concentrations — the familiar square-bracket notation $[A]$, with the constant called $K_c$. For reactions that involve gases, however, there is a more natural way to express the composition of the mixture: by using the partial pressure of each gas. The reason is simple: when you work with gases in a closed container, you measure pressure, not concentration. The equilibrium constant expressed in partial pressures is given the symbol $K_p$.

The bridge between concentration and pressure is the ideal gas law. For a gas:

Rearrange it:

The quantity $n/V$ is the concentration in moles per unit volume. If you measure concentration $c$ in mol/L (or mol/dm$^3$) and pressure $p$ in bar, the relation becomes:

where $R = 0.0831\ \text{bar L mol}^{-1}\text{K}^{-1}$. You can also write this as $p = [\text{gas}]\,RT$, where $[\text{gas}]$ is the molar concentration. At a fixed temperature, $R$ and $T$ are constants, so the pressure of a gas is directly proportional to its concentration:

This proportionality is the key that lets us convert between $K_c$ and $K_p$.

The First Example: $H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$

For this reaction, you can write the equilibrium constant in two ways. Using concentrations:

Using partial pressures:

Now substitute $p = [\text{gas}]RT$ for each gas:

The $(RT)$ factors cancel completely. For this particular reaction, $K_p = K_c$.

The Second Example: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

Here the story is different. Write $K_p$:

Substitute $p = [\text{gas}]RT$:

So:

Or, more neatly:

The exponent $-2$ is not arbitrary. Count the moles of gas on each side:

• Products: $2$ moles of $NH_3$
• Reactants: $1 + 3 = 4$ moles
• Difference: $\Delta n = 2 - 4 = -2$

That difference appears as the exponent of $(RT)$.

The General Relation

For any gaseous reaction:

Write $K_p$:

Substitute $p = [\text{gas}]RT$ for every gas:

The fraction of concentrations is exactly $K_c$. The exponent of $(RT)$ is $(c+d) - (a+b)$, which is the change in the number of moles of gas, $\Delta n$: