D21.2 Equilibrium Constant and Partial Pressure

John Moore; Jia Zhou; Etienne Garand

D21.2 Equilibrium Constant and Partial Pressure

For reactions in which all reactants and products are in the gas phase, the partial pressure of each gas can be used instead of its concentration in the equilibrium constant equation. At constant temperature, the partial pressure of a gas is directly proportional to its molar concentration (c), which is the amount of a substance (n, moles) per unit volume (V, liters): c = n/V. This proportionality of pressure and concentration can be derived from the ideal gas equation:

$\begin{array}{rcl} PV & = & nRT \\[0.5em] P & = & \left(\dfrac{n}{V}\right)RT \\[1em] P & = & cRT \end{array}$

For an example, consider the following reaction:

C₂H₆(g) ⇌ C₂H₄(g) + H₂(g)

We can write the equilibrium constant, K_p, using the equilibrium partial pressures of the gases, by following the same guidelines as for K_c expressions:

$K_{\text{p}} = \dfrac{P_{\text{C}_2\text{H}_4}P_{\text{H}_2}}{P_{\text{C}_2\text{H}_6}}$

The two equilibrium constants, K_c and K_p, are directly related to each other. For the generic gas-phase reaction:

m A + n B ⇌ x C + y D

$\begin{array}{rcl} K_{\text{p}} & = & \dfrac{(P_C)^x(P_D)^y}{(P_A)^m(P_B)^n} \\[1em] & = & \dfrac{([\text{C}]\;\times\;RT)^x([\text{D}]\;\times\;RT)^y}{([\text{A}]\;\times\;RT)^m([\text{B}]\;\times\;RT)^n} \\[1em] & = & \dfrac{[\text{C}]^x[\text{D}]^y}{[\text{A}]^m[\text{B}]^n}\;\times\;\dfrac{(RT)^{x+y}}{(RT)^{m+n}} \\[1em] & = & K_{\text{c}}(RT)^{(x+y)\;-\;(m+n)} \\[0.5em] K_{\text{p}} & = & K_{\text{c}}(RT)^{{\Delta}n} \end{array}$

where Δn is the difference between the sum of the coefficients of the gaseous products and the sum of the coefficients of the gaseous reactants in the reaction (that is, the change in amount of gas between the reactants and the products).

Note that the gas constant, R, can be expressed in different units (for example, R = 8.314 $\frac{\text{J}}{\text{K} \cdot \text{mol}}$ = 0.08314 $\frac{\text{L} \cdot \text{bar}}{\text{K} \cdot \text{mol}}$ = 0.08206 $\frac{\text{L} \cdot \text{atm}}{\text{K} \cdot \text{mol}}$ ). Use the R value and associated units that match the partial pressure units used in the K_p expression.

For some heterogeneous equilibria that involve gases, equilibrium constants can be expressed using partial pressures instead of concentrations. Two examples are:

CaO(s) + CO₂(g) ⇌ CaCO₃(s) $K_{\text{p}} = \dfrac{1}{P_{\text{CO}_2}}$

C(s) + 2 S(g) ⇌ CS₂(g) $K_{\text{p}} = \dfrac{P_{\text{CS}_2}}{(P_{\text{S}})^2}$

Exercise: Pressure and Concentration Equilibrium Constant Expressions

Exercise: Converting Pressure and Concentration Equilibrium Constants

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Chemistry 109 Fall 2021 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.