Questions: Assume that the change in concentration of COCl2 is small enough to be neglected in the following problem. (a) Calculate the equilibrium concentration of all species in an equilibrium mixture that results from the decomposition of COCl2 with an initial concentration of 0.3166 M. COCl2(g) ⇌ CO(g) + Cl2(g) Kc=2.2 × 10^-10 (b) Confirm that the change is small enough to be neglected.

Solution Steps

Given the reaction: \[ \mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g) + \mathrm{Cl}_{2}(g) \] The equilibrium constant expression \( K_c \) is: \[ K_c = \frac{[\mathrm{CO}][\mathrm{Cl}_2]}{[\mathrm{COCl}_2]} \]

Let the initial concentration of \(\mathrm{COCl}_2\) be \( [\mathrm{COCl}_2]_0 = 0.3166 \) M. At equilibrium, let the change in concentration of \(\mathrm{COCl}_2\) be \( x \). Therefore, the equilibrium concentrations are: \[ [\mathrm{COCl}_2] = 0.3166 - x \] \[ [\mathrm{CO}] = x \] \[ [\mathrm{Cl}_2] = x \]

Substitute the equilibrium concentrations into the equilibrium expression: \[ K_c = \frac{[\mathrm{CO}][\mathrm{Cl}_2]}{[\mathrm{COCl}_2]} = \frac{x \cdot x}{0.3166 - x} = \frac{x^2}{0.3166 - x} \] Given \( K_c = 2.2 \times 10^{-10} \): \[ 2.2 \times 10^{-10} = \frac{x^2}{0.3166 - x} \]

Since \( K_c \) is very small, we assume \( x \) is very small compared to 0.3166, so \( 0.3166 - x \approx 0.3166 \): \[ 2.2 \times 10^{-10} = \frac{x^2}{0.3166} \] \[ x^2 = 2.2 \times 10^{-10} \times 0.3166 \] \[ x^2 = 6.9652 \times 10^{-11} \] \[ x = \sqrt{6.9652 \times 10^{-11}} = 8.347 \times 10^{-6} \]

Using \( x = 8.347 \times 10^{-6} \): \[ [\mathrm{COCl}_2] = 0.3166 - 8.347 \times 10^{-6} \approx 0.3166 \, \text{M} \] \[ [\mathrm{CO}] = 8.347 \times 10^{-6} \, \text{M} \] \[ [\mathrm{Cl}_2] = 8.347 \times 10^{-6} \, \text{M} \]

To confirm the change is negligible, compare \( x \) to the initial concentration: \[ \frac{x}{0.3166} = \frac{8.347 \times 10^{-6}}{0.3166} \approx 2.637 \times 10^{-5} \] Since \( 2.637 \times 10^{-5} \) is much less than 5%, the change is indeed negligible.

Final Answer