**Practice Problem 4**

Assume the following initial concentrations: (PCl_{5})
= 0.100 *M* and (Cl_{2}) = 0.020 *M*.
Calculate the equilibrium concentrations of PCl_{5}, PCl_{3},
and Cl_{2} if the equilibrium constant for the
decomposition of PCl_{5} is 0.030.

**Solution**

We start by representing the problem as follows.

Before doing anything else, we need to compare the reaction quotient for the initial conditions with the equilibrium constant for the reaction.

Once again, the reaction quotient (*Q*_{c}
= 0) is smaller than the equilibrium constant (*K*_{c}
= 0.030). Some of the PCl_{5} present initially therefore
has to decompose to PCl_{3} and Cl_{2} before the
reaction can come to equilibrium. Because of the stoichiometry of
this reaction, the magnitude of the change in the PCl_{5}
concentration as the reaction comes to equilibrium is equal to
the magnitude of the changes in the concentrations of PCl_{3}
and Cl_{2}. The problem can therefore be represented as
follows.

Substituting what we know about the concentrations of PCl_{5},
PCl_{3}, and Cl_{2} into the equilibrium constant
expression for the reaction gives the following result.

Expanding this gives the following quadratic equation.

^{2} + 0.050 - 0.0030 = 0

Solving this equation with the quadratic formula gives the following positive root.

= 0.035 *M*

This value of can be used to calculate the equilibrium concentrations of the three components of the reaction.

[PCl_{5}] = 0.100 - = 0.065 *M*

[PCl_{3}] = 0 + = 0.035 *M*

[Cl_{2}] = 0.020 + = 0.055 *M*

In this case, only about one-third of the PCl_{5}
present initially decomposes as the reaction comes to
equilibrium.

We can check these results by substituting them into the equilibrium constant expression.

The values for the equilibrium concentrations of PCl_{5},
PCl_{3}, and Cl_{2} in this calculation must be
legitimate because the results of this calculation agree with the
equilibrium constant for the reaction, within experimental error.