Calculating the Equilibrium Constant (College Board AP® Chemistry): Study Guide

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. The initial and equilibrium concentrations of each gas are shown in the table below.

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

Answer:

• 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 table, 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 mixture of NO₂ and NO gases in a sealed rigid container reacts at a constant temperature of 1000 K according to the equation above. At equilibrium, the vessel contains 0.200 mol of NO₂, 1.700 mol of NO, and 1.100 mol of O₂, and the total pressure is 6.00 atm. Calculate the equilibrium constant for partial pressure, K_p, for this reaction.

Answer:

• 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 

K_p = $\frac{{\left(3.40\right)}^2\times 2.20}{{\left(0.400\right)}^2}$

K_p = $\frac{11.56\times 2.20}{0.16}$

K_p = 159

• Note that K_p is a unitless value

A rigid container initially holds SO₃ (g) only, at a partial pressure of 1.00 atm. It decomposes until equilibrium is reached:

2SO₃ (g) ⇌ 2SO₂ (g) + O₂ (g)

At equilibrium, the partial pressure of O₂ (g) is 0.150 atm. Calculate K_p.

Answer:

Step 1: Set up an ICE table in atm; the change row follows the coefficients (2 : 2 : 1)

Step 2: Use the equilibrium partial pressure of O₂ to find x, then the other partial pressures

x = P_O2 = 0.150 atm

P_SO2 = 2x = 0.300 atm

P_SO3 = 1.00 − 2x = 1.00 − 0.300 = 0.700 atm

Step 3: Write the K_p expression and substitute the equilibrium partial pressures

K_p = $\frac{{\left({P_{SO}}_2\right)}^2\left({P_O}_2\right)}{{\left({P_{SO}}_3\right)}^2}$

K_p = $\frac{{(0.300)}^2(0.150)}{{(0.700)}^2}$

K_p = 0.0276

In an ICE table, the change row follows the coefficients. So, if the coefficient is 2, the change is 2x. Always substitute the equilibrium row into the expression, never the initial values

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 square brackets. Write P with the formula as a subscript for each partial pressure, for example (P_NO)²

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.