4.2 Equilibrium Expression
For a general reaction
\[aA + bB \rightleftharpoons cC + dD\]
the equilibrium expression is written in terms of molar concentrations as
\[K_c = \dfrac{[C]^c[D]^d}{[A]^a[B]^b}\]
Note: Solids and pure liquids never appear in an equilibrium expression.
4.2.1 Gaseous Reactions
Equilibrium expressions can also be expressed as pressures of gases. For a general, gas-phase reaction
\[aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)\]
the equilibrium expression can be written as
\[K_p = \dfrac{P_C^cP_D^d}{P_A^aP_B^b}\]
4.2.2 Pressure to concentration relation
K expressed as pressures can be related to K expressed as concentrations via
\[K_p = K_c (RT)^{\Delta n}\]
where R is the gas constant (in L atm mol–1 K–1), T is temperature, and Δn is the change in moles of gas for a reaction given as
\[\Delta n = \sum(n_{\mathrm{gas~products}}) - \sum(n_{\mathrm{gas~reactants}})\]
Derivation
For a reaction
\[aA(g) \rightleftharpoons bB(g)\]
the equilibrium expression is
\[K_c = \dfrac{[\mathrm{B}]^b}{[\mathrm{A}]^a} ~~~\mathrm{or}~~~ K_P = \dfrac{(P_\mathrm{B})^b}{(P_\mathrm{A})^a}\]
Assuming an ideal gas, we can express each gas (A and B) in terms of their pressures such that
\[\begin{align*} PV = nRT ~~\rightarrow~~ &P_\mathrm{A}V = n_\mathrm{A}RT ~~\rightarrow~~ P_\mathrm{A} = \left(\frac{n_\mathrm{A}}{V}\right)RT \\[1.5ex] &P_\mathrm{B}V = n_\mathrm{B}RT ~~\rightarrow~~ P_\mathrm{B} = \left(\frac{n_\mathrm{B}}{V}\right)RT \end{align*}\]
Note that moles over volume is equal to molarity as shown below.
\[\dfrac{n}{V} = M\] We can therefore rewrite the ideal gas expressions with our square-bracket notation for molarity such as
\[\begin{align*} P_\mathrm{A} &= [\mathrm{A}]RT \\[1.5ex] P_\mathrm{B} &= [\mathrm{B}]RT \end{align*}\]
Now, substitute these terms into the KP equation to give
\[K_P = \dfrac{(P_\mathrm{B})^b}{(P_\mathrm{A})^a} = \dfrac{([\mathrm{B}]RT)^b}{([\mathrm{A}]RT)^a}\] and recognize that
\[\dfrac{[\mathrm{B}]^b}{[\mathrm{A}]^a} = K_c\] Therefore
\[\begin{align*} K_P &= \dfrac{([\mathrm{B}]RT)^b}{([\mathrm{A}]RT)^a} \\[2ex] &= K_c \dfrac{(RT)^b}{(RT)^a} \\[2ex] &= K_c (RT)^{b-a} \\[2ex] K_P &= K_c (RT)^{\Delta n} \end{align*}\]