14.10 : 小さい x の仮定

The small x assumption is an approximation that can be used under certain conditions to simplify solving the equilibrium expression and to avoid using the quadratic formula. 

If the equilibrium constant for a reaction is small and the initial concentrations of reactants are sufficiently high, it is possible that only a small amount of reactants will be converted into products.

When these conditions are met, the equilibrium concentrations can be approximated using the assumption that the change in the initial concentrations is negligible. However, it must be confirmed that the change in concentration, x, is less than 5% of the initial concentrations to show the the assumption was valid.

The decomposition of 0.66 molar carbonyl dichloride gas produces carbon monoxide and chlorine gas, and the equilibrium constant for this reaction is 2.2 × 10⁻¹⁰.

To calculate the equilibrium concentrations, the values are tabulated in the ICE table along with the change and equilibrium concentrations.

Substituting the equilibrium concentrations in the expression, K equals x squared divided by 0.66 minus x. As K is very small, the change in the initial concentration, x, of carbonyl dichloride is expected to be negligible.

Therefore, 0.66 minus x can be approximated to 0.66. When solved, x equals 1.2 × 10⁻⁵ molar.

Here, x is only 0.0018 percent of the initial 0.66 molar concentration of carbonyl dichloride, which is far less than the 5% maximum allowed. Thus, the small x assumption is valid here.

Using the value for x in the ICE table, the equilibrium concentration of carbonyl dichloride is still 0.66 molar with significant figures, while the concentrations of carbon monoxide and chlorine are both 1.2 × 10⁻⁵ molar.