Question

What are the equilibrium concentrations of all the gases for the reaction $PC{l_3}(g) + C{l_2}(g) \rightleftharpoons PC{l_5}(g)$ ${K_c} = 0.18$ in an initial mixture of $0.3M$ $PC{l_3}(g)$ and $0.4M$ $C{l_2}(g)$ ?

Answer 1

The equilibrium established between the phosphorus trichloride gas and chlorine gas to produce phosphorus pentachloride gas has a fixed equilibrium constant that can be expressed as the ratio of products of concentration of product and reactants.

Complete Step By Step Answer:
A dynamic equilibrium is said to exist when the rate of formation of products is the same as that of rate of formation of reactants. As the equilibrium is achieved the concentration of all the reactants and products attain a constant value which are known as the equilibrium concentrations.
The relationship between the initial concentrations and the equilibrium concentration can be established by taking into account the amount of each reactant that gets consumed and the amount of products formed.
Initially only the reactants are present in the reaction flask and the product has a zero concentration. If we assume that $x$ is the concentration that gets consumed, then we subtract the same value from the initial concentration of both the reactants and add this value to the concentration of products.
${\text{ }}PC{l_3}(g) + C{l_2}(g) \rightleftharpoons PC{l_5}(g)$
$t = 0{\text{ 0}}{\text{.3M 0}}{\text{.4M 0 }}$
$t = eq{\text{ 0}}{\text{.3}} - x{\text{ 0}}{\text{.4}} - x{\text{ }}x$
The equilibrium constant can be expressed as the ratio of product equilibrium concentration divided by the product of reactant equilibrium concentrations raised to the power of their stoichiometric coefficients:
${K_c} = \dfrac{{(x)}}{{(0.3 - x)(0.4 - x)}} = 0.18$
The exact values of equilibrium concentration can be determined by solving the above equation for $x$
The quadratic equation comes out to $0.18{x^2} - 1.126x + 0.0216 = 0$ which gives $x = 0.02$ as the solution. This value can be inserted into the expression for equilibrium concentration.
The final values of equilibrium concentration are:
$[PC{l_3}(g)] = {\text{0}}{\text{.3}} - 0.02 = 0.28M$
$[C{l_2}(g)] = 0.4 - 0.02 = 0.38M$
$[PC{l_5}(g)] = 0.02M$ .

Note:
The amount consumed and the amount of product formed is same for all the reactants and products because of the stoichiometric coefficients. Since all three gases have a coefficient equal to unity, the same amount is consumed and formed in the reaction.