Calculating the Equilibrium Constant (College Board AP® Chemistry): Revision Note

H₂ (g) + CO₂ (g) ⇌ H₂O (g) + CO (g)

A mixture of H₂, CO₂, H₂O, and CO gases is combined in a 1.0 L previously evacuated, rigid sealed container. The gases are allowed to react, according to the equation above, at a constant temperature of 700°C. A graph of the concentration of each gas as a function of time is shown below.

What is the value of K_c for this reaction at 700°C?

Answer:

• For the generic equation

aA + bB ⇌ cC + dD

The equilibrium expression for K_c is

 ${\mathrm{K}}_{\mathrm{c}}=\frac{{\left[\mathrm{C}\right]}^{\mathrm{c}}{\left[\mathrm{D}\right]}^{\mathrm{d}}}{{\left[\mathrm{A}\right]}^{\mathrm{a}}{\left[\mathrm{B}\right]}^{\mathrm{b}}}$

• So, for the given reaction the equilibrium expression is

${\mathrm{K}}_{\mathrm{c}}=\frac{\left[{\mathrm{H}}_2\mathrm{O}\right]\left[\mathrm{CO}\right]}{\left[{\mathrm{H}}_2\right]\left[{\mathrm{CO}}_2\right]}$

• To solve for K_c we substitute the concentration of each species at equilibrium

• Equilibrium is established when the concentrations of each species remain constant

• From the graph, the equilibrium concentrations are

[H₂O] = 0.8 M

[CO] = 0.2 M

[H₂] = 0.6 M

[CO₂] = 0.5 M

• Substituting the concentrations into the equation for K_c 

${\mathrm{K}}_{\mathrm{c}}=\frac{0.8\times 0.2}{0.6\times 0.5}$

${\mathrm{K}}_{\mathrm{c}}=\frac{0.16}{0.3}$

K_c = 0.5

• Note that K_c is a unitless value

2NO₂ (g) ⇌ 2NO (g) + O₂ (g)

A sample of pure NO₂ gas in a sealed rigid container decomposes at a constant temperature of 1000 K and a constant pressure of 6.00 atm according to the equation above. At equilibrium, the vessel is found to contain 0.200 mol of NO₂, 1.700 mol of NO, and 1.100 mol of O₂. Calculate the equilibrium constant for partial pressure, K_P, for this reaction.

Answer:

• For the generic equation

aA + bB ⇌ cC + dD

The equilibrium expression for K_P is

$K_p=\frac{{\left(P_C\right)}^c{\left(P_D\right)}^d}{{\left(P_A\right)}^a{\left(P_B\right)}^b}$

• For the given reaction the equilibrium expression is

       $K_p=\frac{{\left(P_{NO}\right)}^2\left(P_{O_2}\right)}{{\left(P_{N{O}_2}\right)}^2}$

• To solve for K_P we substitute the partial pressure of each species at equilibrium

• Determining the total number of moles of the gas mixture

${\mathrm{n}}_{\mathrm{total}}={\mathrm{n}}_{\left({\mathrm{NO}}_2\right)}+{\mathrm{n}}_{\mathrm{NO}}+{\mathrm{n}}_{\left({\mathrm{O}}_2\right)}$

${\mathrm{n}}_{\mathrm{total}}=0.200\mathrm{mol}+1.700\mathrm{mol}+1.100\mathrm{mol}$

${\mathrm{n}}_{\mathrm{total}}=3.000\mathrm{mol}$

• Determining the partial pressure for each gas using its mole fraction and total pressure

NO₂ 

${\mathrm{P}}_{{\mathrm{NO}}_2}=\frac{{\mathrm{n}}_{{\mathrm{NO}}_2}}{{\mathrm{n}}_{\mathrm{total}}}\times {\mathrm{P}}_{\mathrm{total}}$

${\mathrm{P}}_{{\mathrm{NO}}_2}=\frac{0.200\mathrm{mol}}{3.000\mathrm{mol}}\times 6.00\mathrm{atm}$

${\mathrm{P}}_{{\mathrm{NO}}_2}=0.400\mathrm{atm}$

NO

${\mathrm{P}}_{\mathrm{NO}}=\frac{{\mathrm{n}}_{\mathrm{NO}}}{{\mathrm{n}}_{\mathrm{total}}}\times {\mathrm{P}}_{\mathrm{total}}$

${\mathrm{P}}_{\mathrm{NO}}=\frac{1.700\mathrm{mol}}{3.000\mathrm{mol}}\times 6.00\mathrm{atm}$

${\mathrm{P}}_{\mathrm{NO}}=3.40\mathrm{atm}$

O₂

${\mathrm{P}}_{{\mathrm{O}}_2}=\frac{{\mathrm{n}}_{{\mathrm{O}}_2}}{{\mathrm{n}}_{\mathrm{total}}}\times {\mathrm{P}}_{\mathrm{total}}$

${\mathrm{P}}_{{\mathrm{O}}_2}=\frac{1.100\mathrm{mol}}{3.000\mathrm{mol}}\times 6.00\mathrm{atm}$

${\mathrm{P}}_{{\mathrm{O}}_2}=2.20\mathrm{atm}$

• Substituting the partial pressures into the equation for K_P 

${\mathrm{K}}_{\mathrm{P}}=\frac{{\left(3.40\right)}^2\times 2.20}{{\left(0.400\right)}^2}$

${\mathrm{K}}_{\mathrm{P}}=\frac{11.56\times 2.20}{0.16}$

KP=25.4320.16

KP=159

• Note that K_P is a unitless value

When writing a K_c expression, always be sure to include the brackets, as brackets around a species represent the concentration of that species. However, when writing a K_P expression never use brackets as this expression involves partial pressures not concentrations.

Make sure that the values substituted into a K_c expression have the unit mol/L and values substituted into a K_P expression have the unit atm.

Remember that although the values substituted into a K_c or K_P expression have units, the equilibrium constant is a unitless value.