Question

$0.1mole$ of $PC{l_5}$ is heated in a litre vessel at $533K$. Determine the concentration of various species present at equilibrium, if the equilibrium constant for the dissociation of $PC{l_5}$ at $533K$ is $0.414$.

Answer 1

Hint: The equilibrium constant of a reaction is defined as the ratio of concentration of products raised to the power equal to the stoichiometric coefficient and the concentration of reactants raised to the power equal to the stoichiometric coefficient at equilibrium.

Complete answer:
The reaction of the dissociation of phosphorus pentachloride ($PC{l_5}$) at equilibrium is given as follows:

As per the question,$0.1mole$ of $PC{l_5}$ is heated in the vessel whose volume is equal to 1 litre. Let ‘x’ moles of $PC{l_5}$ get dissociated when heated. Initially, the moles of $PC{l_3}$ and $C{l_2}$ are zero.

Initially: ($0.1mole$) , 0 , 0
Finally: ($0.1 - x$) , $x$ , $x$
The equilibrium concentrations of the individual components are:
$\left[ {PC{l_5}} \right] = \dfrac{{0.1 - x}}{1} = (0.1 - x)mol/L$
$\left[ {PC{l_3}} \right] = \dfrac{x}{1} = (x)mol/L$
$\left[ {C{l_2}} \right] = \dfrac{x}{1} = (x)mol/L$
The equilibrium constant of the following reaction is equal to:
${K_C} = \dfrac{{\left[ {PC{l_3}} \right]\left[ {C{l_2}} \right]}}{{\left[ {PC{l_5}} \right]}}$
As per the question, ${K_C} = 0.414$
Now, substituting the values and solving, we have:
$0.414 = \dfrac{{x \times x}}{{0.1 - x}}$
$ \Rightarrow 0.414 \times \left( {0.1 - x} \right) = {x^2}$
$ \Rightarrow {x^2} + 0.414x - 0.0414 = 0$
On solving the quadratic equation from the formula: $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Where, $a = 1,b = + 0.414,c = - 0.0414$
We have the value of $x$ = $0.08326$ or$ - 0.4973$.
As the value of concentration cannot be negative, thus the dissociated mole is equal to $0.08326$.
Now, substituting the value of $x$ in the individual concentrations of reactants and products at equilibrium, we have:
$PC{l_5} = 0.1 - 0.08326 = 0.0167moles$
$PC{l_3} = x = 0.08326moles$
$C{l_2} = x = 0.08326moles$
Thus, the concentrations of various species present at equilibrium have been determined.

Note:
The concentration of the reactants or products shall only be calculated when they are in their gaseous states and not in solid or liquid states. The application of reactions like above one can be achieved only when they are in equilibrium.