Question

Given that ${K_c}$ = 13.7 at 546 $K$ for $PC{l_5}(g) \rightleftharpoons PC{l_3}(g) + C{l_2}(g)$ , calculate what pressure will develop in a 10 litre box at equilibrium at 546 $K$ when 1.00 mole of $PC{l_5}$ is injected into the empty box?

Answer 1

Hint: Various compounds break in to form more stable compounds. Some of such compounds exist in equilibrium with their products. At the equilibrium state, all the products and reactants coexist and simultaneously convert to reactant and product and vice versa.

Complete step by step answer:
Phosphorus pentachloride readily breaks into phosphorus trichloride releasing chlorine gas at equilibrium. The reaction can be represented as
$PC{l_5}(g) \rightleftharpoons PC{l_3}(g) + C{l_2}(g)$
Here all the reactants and the products are present in the gaseous phase.
The concentrations of $PC{l_5}$ can be represented 0.1 as 1 mole of the compound is present in the box of 10 litre. This concentration is represented in the form of moles per litre. Let the amount of the reactant that breaks into the product is x. So, the concentration of the reactions throughout the reactions can be shown as below with the final state be

	$PC{l_5}$	$PC{l_3}$	$C{l_2}$
Initial	0.1	0	0
Final	$0.1 - x$	x	x

From the above data we can write the equation for the equilibrium constant as
${K_c} = \dfrac{{[PC{l_3}] \times [C{l_2}]}}{{[PC{l_5}]}}$
Substituting the concentrations for the compounds we get
${K_c} = \dfrac{{x \times x}}{{0.1 - x}}$
Since the value of the equilibrium constant is given as 13.7 in the question itself, so equating it with the equation which we have got it can be inferred as
$\dfrac{{x \times x}}{{0.1 - x}} = 13.7$
So the value of x can be calculated as 0.0993
The total number of moles present can be seen as $(0.1 - x) + (x) + (x)$ multiplied with the volume of the container, which can be represented as n.
So, total number of moles, n = $10 \times (0.1 + x)$
$ \Rightarrow n = 10 \times (0.1 + 0.0993) = 1.993$
Now, using the ideal gas equation we can calculate the pressure as,
$P = \dfrac{{nRT}}{V}$,
Substituting the values in the above ideal gas equation we get,
$P = \dfrac{{1.993 \times 0.0821 \times 546}}{{10}}$
Therefore simplifying and calculating we get $P = 8.93atm$

Note: The ideal gas equation is the equation that states how the gas should act in ideal conditions and what the ideal characteristics are for the gas is also defined.
The ideal gas also draws heavily from the kinetic theory of gases but actually, the ideal gas is a theoretical concept and thus doesn't exist in the real world, the gases that exist are called real gases.