The effect of addition of helium gas to the following reaction in equilibrium state at constant volume, is :
$$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$

We need to determine the effect of adding helium gas at constant volume to the equilibrium:

$$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$

Understand what determines the position of equilibrium

For a gaseous equilibrium, the equilibrium constant $$K_p$$ depends on the partial pressures of the reacting gases (not the total pressure). The equilibrium position shifts only if the partial pressures (or equivalently, concentrations) of the reactants or products change.

Analyse the effect of adding an inert gas at constant volume

When helium (an inert, non-reacting gas) is added at constant volume:

- The total pressure of the system increases (because there are more gas molecules in the same volume).

- However, the partial pressures of $$PCl_5$$, $$PCl_3$$, and $$Cl_2$$ remain unchanged. This is because partial pressure depends on the number of moles of each gas and the volume, neither of which has changed for the reacting species.

- Since $$p_i = \frac{n_i RT}{V}$$, and $$n_i$$ and $$V$$ are unchanged for each reacting species, the partial pressures are unchanged.

Apply Le Chatelier's Principle

Since the partial pressures (and concentrations) of all the reacting species remain the same, the reaction quotient $$Q_p$$ equals $$K_p$$, and there is no shift in equilibrium.

Important note: If the inert gas were added at constant pressure (instead of constant volume), the volume would increase, reducing the partial pressures of all gases. This would shift the equilibrium in the direction of more moles of gas (forward direction in this case). But at constant volume, there is no effect.

The correct answer is Option (4): Addition of helium will not affect the equilibrium.