Question

One mole of \(\mathrm{PCl}_{5}\) is subjected to heating in a 1 1 vessel. The number of moles of \(\mathrm{PCl}_{3}\) formed at equilibrium is \(0.6 .\) Calculate the equilibrium constant for the dissociation of \(\mathrm{PCl}_{5}\).

Short answer

Answer: The equilibrium constant for the dissociation of \(\mathrm{PCl}_{5}\) is 0.9.

Step-by-step solution

Write the balanced chemical equation

The balanced equation for the dissociation of \(\mathrm{PCl}_{5}\) is: $$\mathrm{PCl_{5}} \rightleftharpoons \mathrm{PCl_{3}} + \mathrm{Cl_{2}}$$

Set up the ICE table

We will use the initial, change, and equilibrium concentrations of the reactants and products to find the equilibrium constant. | | \(\mathrm{PCl_{5}}\) | \(\mathrm{PCl_{3}}\) | \(\mathrm{Cl_{2}}\) | |---|---|---|---| | Initial (moles) | 1 | 0 | 0 | | Change | -\(x\) | +\(x\) | +\(x\) | | Equilibrium | \(1-x\) | \(0.6\) | \(x\) | Here, \(x\) is the amount of \(\mathrm{PCl_{5}}\) that dissociates at equilibrium.

Calculate the value of \(x\)

The number of moles of \(\mathrm{PCl}_{3}\) formed at equilibrium (\(0.6\) moles) resulted from the dissociation of \(\mathrm{PCl}_{5}\). Therefore, \(x = 0.6\).

Determine the equilibrium concentrations

We know that the initial volume of the reaction is \(1 L\). So, we can find the equilibrium concentrations of the reactants and products by dividing the equilibrium moles by this volume. For \(\mathrm{PCl_{5}}\), the equilibrium concentration is: \([PCl_{5}] = \frac{1-x}{1}\) = \(1 - 0.6 = 0.4\). For \(\mathrm{PCl_{3}}\), the equilibrium concentration is: \([PCl_{3}] = \frac{0.6}{1} = 0.6.\) For \(\mathrm{Cl_{2}}\), the equilibrium concentration is: \([Cl_{2}] = \frac{x}{1}\) = \(0.6.\)

Calculate the equilibrium constant using equilibrium concentrations

The equilibrium expression is: $$K_{c} = \frac{[\mathrm{PCl_{3}}][\mathrm{Cl_{2}}]}{[\mathrm{PCl_{5}}]}$$ Substituting the equilibrium concentrations, we get: $$K_{c} = \frac{(0.6)(0.6)}{0.4} = \frac{0.36}{0.4}$$

Calculate the equilibrium constant

Divide the numerator by the denominator to find the equilibrium constant: $$ K_{c} = \frac{0.36}{0.4} = 0.9 $$ The equilibrium constant for the dissociation of \(\mathrm{PCl}_{5}\) is \(0.9\).