Equilibrium Constants in Terms of Partial Pressure (Kp)
Equilibrium constants for gaseous reactions are expressed in terms of equilibrium partial pressures (\(K_{\text{p}}\)). Mole fractions and Dalton's law allow conversion between total pressure and individual partial pressures.
Key Definitions
| Term | Definition | Formula |
|---|---|---|
| Mole fraction (χ) | Ratio of moles of one gas to the total moles of all gases | \(\chi_A = \frac{n_A}{n_{total}}\) |
| Partial pressure | The pressure a gas would exert if it alone occupied the container | \(P_A = \chi_A \times P_{total}\) |
The Kp Expression
For a gaseous equilibrium: \(aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)\)
Gaseous Species Only in Kp Expressions
Only gaseous species (g) appear in \(K_{\text{p}}\) expressions. Pure solids (s) and pure liquids (l) have constant activities and are strictly omitted from \(K_{\text{p}}\).
Converting Between Kc and Kp
Where \(\Delta n = \text{total moles of gaseous products} - \text{total moles of gaseous reactants}\) and R = 8.314 J K⁻¹ mol⁻¹ (or 0.0821 atm L mol⁻¹ K⁻¹ depending on pressure units).
Worked Example: Calculating Kp
Converting Between Kc and Kp
Problem: For N₂(g) + 3H₂(g) ⇌ 2NH₃(g), Kc = 9.60 at 300 K. Calculate Kp.
Step 1: Find Δn = 2 − (1 + 3) = −2
Step 2: Apply the formula:
\(K_p = 9.60 \times (0.0821 \times 300)^{-2} = 9.60 \times (24.63)^{-2}\)
\(K_p = 9.60 \times 1.648 \times 10^{-3} = \textbf{0.0158}\)
Worked Example: Finding Mole Fractions & Partial Pressures
Calculating Kp from Partial Pressures
Problem: At equilibrium in the reaction N₂O₄(g) ⇌ 2NO₂(g) at 200 kPa total pressure, there are 0.40 mol N₂O₄ and 0.60 mol NO₂. Find Kp.
| N₂O₄ | NO₂ | |
|---|---|---|
| Moles | 0.40 | 0.60 |
| χ | 0.40/1.00 = 0.40 | 0.60/1.00 = 0.60 |
| P (kPa) | 0.40 × 200 = 80 | 0.60 × 200 = 120 |
\(K_p = \frac{(120)^2}{80} = \frac{14400}{80} = \textbf{180 kPa}\)
Checking Mole Fractions and Δn Sign
Always check that your calculated mole fractions sum to exactly 1 (\(\sum \chi_i = 1\)). When using \(K_{\text{p}} = K_{\text{c}}(RT)^{\Delta n}\), verify \(\Delta n = \text{moles gas (products)} - \text{moles gas (reactants)}\).
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