Partial pressures and mole fractions, the equilibrium constant Kp for gaseous equilibria, calculating Kp and equilibrium amounts, and the effect of temperature, pressure and a catalyst on Kp and the position of equilibrium.

What this dot point is asking

CCEA wants you to define partial pressure and mole fraction, write $K_p$ for a gaseous equilibrium, calculate $K_p$ and equilibrium amounts, and explain how temperature, pressure and a catalyst affect $K_p$ and the position of equilibrium.

Partial pressures and mole fractions

In a mixture of ideal gases each gas behaves as if it alone occupied the container, so each contributes its own partial pressure and these sum to the total (Dalton's law): $P_{\text{total}} = p_A + p_B + p_C + \ldots$. The mole fractions of all the gases add up to $1$, which is the check you should run after working them out. Because partial pressure is proportional to the number of moles of that gas, the ratio of partial pressures equals the ratio of moles, and this is what makes $K_p$ a useful constant for gaseous systems where measuring individual concentrations would be awkward.

Writing and using Kp

The reliable five-step routine is: (1) write balanced equation and the $K_p$ expression; (2) build an ICE table (Initial, Change, Equilibrium) in moles, using the degree of dissociation or the change in one species to fill the others by stoichiometry; (3) add the equilibrium moles to get the total and divide each by it for mole fractions; (4) multiply each mole fraction by the total pressure for partial pressures; (5) substitute and work out the units. Note that $K_p$ uses partial pressures throughout, never concentrations; for a reaction in solution you would use $K_c$ instead. Solids and pure liquids are left out because their "pressure" (vapour pressure) is effectively constant and is absorbed into the value of the constant.

Working out the units is part of the answer. Write the units of every pressure term in the expression, then cancel. For $\text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g)$ the expression is $K_p = p(\text{NO}_2)^2 / p(\text{N}_2\text{O}_4)$, which leaves $\text{kPa}^2/\text{kPa} = \text{kPa}$. For $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$ there are two pressure terms top and bottom, so the units cancel and $K_p$ is dimensionless.

Effect of conditions

Examples in context

The Haber process, $\text{N}_2 + 3\text{H}_2 \rightleftharpoons 2\text{NH}_3$, shows the trade-offs $K_p$ analysis predicts: the forward reaction is exothermic, so a low temperature gives a high $K_p$ and high yield, but the rate is then too slow, so a compromise around $450\ \text{degrees C}$ is used with an iron catalyst (which raises rate without altering $K_p$) and a high pressure of about $200\ \text{atm}$ to push the equilibrium toward the fewer gas moles on the product side.

The Contact process for sulfuric acid, $2\text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2\text{SO}_3(g)$, is the same story: the forward reaction is exothermic and goes from three gas moles to two, so a moderate temperature (about $450\ \text{degrees C}$) with a vanadium(V) oxide catalyst gives a high yield without needing extreme pressure, since $K_p$ is already large at that temperature. In both processes the engineering choices follow directly from how temperature changes $K_p$ and how pressure shifts the position of equilibrium without touching $K_p$, which is exactly the reasoning CCEA asks you to reproduce.

Try this

Q1. Write the $K_p$ expression for $2\text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2\text{SO}_3(g)$. [1 mark]

• Cue. $K_p = p(\text{SO}_3)^2 / [p(\text{SO}_2)^2\, p(\text{O}_2)]$.

Q2. State the effect of adding a catalyst on the value of $K_p$. [1 mark]

• Cue. No effect; a catalyst changes only the rate, not $K_p$.

Q3. A gas mixture contains $0.30\ \text{mol}$ A and $0.70\ \text{mol}$ B at a total pressure of $200\ \text{kPa}$. Calculate the partial pressure of A. [1 mark]

• Cue. $x_A = 0.30/1.00 = 0.30$; $p_A = 0.30 \times 200 = 60\ \text{kPa}$.