Question

Suppose \(K=4.5 \times 10^{-3}\) at a certain temperature for the reaction $$ \mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) $$ If it is found that the concentration of \(\mathrm{PCl}_{5}\) is twice the concentration of \(\mathrm{PCl}_{3}\), what must be the concentration of \(\mathrm{Cl}_{2}\) under these conditions?

Short answer

The concentration of Cl2 under these conditions is \(9\times10^{-3}\) M.

Step-by-step solution

Write the given information as algebraic expressions

Given that the concentration of PCl5 is twice the concentration of PCl3, we express it as: $$ [\mathrm{PCl_5}] = 2 [\mathrm{PCl_3}] $$ Now we need to find the concentration of Cl2.

Substitute the given information into the equilibrium expression

Insert the given relationship and the unknown concentration of Cl2 into the equilibrium expression: $$ K=\frac{[\mathrm{PCl_3}][\mathrm{Cl_2}]}{(2 [\mathrm{PCl_3}])} $$

Solve the expression for the unknown concentration of Cl2

We want to isolate the unknown concentration of Cl2, so we will first multiply both sides of the equation by \(2[\mathrm{PCl_{3}}]\): $$ 2 [\mathrm{PCl_{3}}]K = [\mathrm{PCl{3}}][\mathrm{Cl_2}] $$ Now, divide both sides of the equation by \([\mathrm{PCl_3}]\): $$ 2K = [\mathrm{Cl_2}] $$

Substitute the given value of K

Substitute the given value of K into the equation: $$ 2(4.5\times10^{-3}) = [\mathrm{Cl_2}] $$

Calculate the concentration of Cl2

Multiply to solve for the concentration of Cl2: $$ [\mathrm{Cl_2}] = 9\times10^{-3}\mathrm{M} $$ So, the concentration of Cl2 under these conditions is \(9\times10^{-3}\) M.