Kp: Mole Fractions and Partial Pressures

Learning Objectives (OCR Spec 5.1.2)

By the end of this lesson you should be able to:

• Define mole fraction x_A and partial pressure p_A of a gas
• State and apply the relationship p_A = x_A x p_total
• Write a Kp expression from a balanced gas equation
• Calculate Kp from equilibrium data, with correct units

Recap: Kc vs Kp

We have seen Kc expressed using equilibrium concentrations for a reaction in solution or gas:

aA + bB <-> cC + dD, Kc = [C]^c [D]^d / ([A]^a [B]^b)

For gas-phase equilibria, it is often more useful to work with partial pressures instead, because partial pressures are what we can actually measure for gases. The equivalent expression is Kp, in which each [species] is replaced by the partial pressure of that species:

Kp = p_C^c x p_D^d / (p_A^a x p_B^b)

Kp is only defined for reactions involving gases (and only the gas-phase species appear; solids and liquids are omitted, as with Kc).

Mole Fraction

The mole fraction x_A of a component A in a mixture is:

x_A = n_A / n_total

where n_A is the number of moles of A and n_total is the total moles of all gases in the mixture. Mole fractions are dimensionless and sum to 1:

x_A + x_B + x_C + ... = 1

Partial Pressure

Dalton's Law of partial pressures states that the pressure exerted by a gas in a mixture is the pressure it would exert if it alone occupied the container. The partial pressure of A is:

p_A = x_A x p_total

Therefore:

p_A + p_B + p_C + ... = p_total

Partial pressures have the same units as total pressure — commonly Pa, kPa or atm.

Writing a Kp Expression

1. Write the balanced equation.
2. Identify the gas-phase species only.
3. Put products over reactants.
4. Raise each partial pressure to the power of the stoichiometric coefficient.

Example 1 — N2O4 / NO2

N2O4(g) <-> 2NO2(g)

Kp = p_NO2^2 / p_N2O4

Example 2 — Haber process

N2(g) + 3H2(g) <-> 2NH3(g)

Kp = p_NH3^2 / (p_N2 x p_H2^3)

Example 3 — Contact process

2SO2(g) + O2(g) <-> 2SO3(g)

Kp = p_SO3^2 / (p_SO2^2 x p_O2)

Example 4 — Reaction with a Solid

CaCO3(s) <-> CaO(s) + CO2(g)

Only the gas appears: Kp = p_CO2 (a simple expression — the solids are omitted).

Units of Kp

As with Kc, the units depend on the exponent difference between product and reactant gases (sum of coefficients):

• If sum of coefficients on left = sum on right, Kp is dimensionless.
• Otherwise, each excess power of atm (or kPa) contributes to the units.

Example 1 — Units for N2O4 / NO2

Kp = p_NO2^2 / p_N2O4. Total power on top = 2. Total power on bottom = 1. Net = +1. Units = atm (or kPa).

Example 2 — Units for Haber

Kp = p_NH3^2 / (p_N2 x p_H2^3). Top power = 2. Bottom power = 4. Net = -2. Units = atm^-2 (or kPa^-2).

Example 3 — Units for H2 + I2 <-> 2HI

Kp = p_HI^2 / (p_H2 x p_I2). Top = 2, bottom = 2. Net = 0. Kp is dimensionless.

Worked Example — Calculating Kp

At equilibrium in a sealed container at 500 K, the mixture contains: