What happens when an inert gas is added to
i. `PCl_(5)(g) hArr PCl_(3)(g) +Cl_(2)(g)`
ii. `2SO_(2)(g)+O_(2)(g) hArr 2SO_(3)(g)`
at equilibrium at : (a) constant pressure and temperature and temperature, and (b) at constant volume and temperature.

Explanation

Let us discuss what happens when inert gas is added to the following reaction at equilibrium stage:
`mA(g)+nB(g) hArr pC(g)+qD(g)`
If the reaction is at equilibrium, we have
`K_(eq)=Q` …(i)
Any factor (e.g change in the concentration of reactants`//` products or both) can have imapact (in terms of change in the concentration of reatants`//` products) on that reaction at equilibrium only if that factor changes the equation (i) so that `K_(eq) ne 0`
Two cases occur:
a. Inert gas addition at constant pressure:
`K_(eq)=Q=(p_(C)^(p) p_(D)^(q))/(p_(A)^(m) p_(B)^(n))` (at equilibrium)
and particle pressure `(p)=chiP_("Total")`
`["where" chi=n/n_("Total")="mole fraction"]`
Thus, when an inert gas is added, `chi` of each of the gas decreases and since `P_("Total")` is constant, the partial pressure decreases leading to a new equation:
`Q gt K_(eq)` or `Q lt K_(eq)`
which will depend on whether
`Deltan_(g)=Singma(n_(g))_(P)-Sigma(n_(g))_(R) lt 0 "or" gt 0`, respectively.
`rArr PCl_(5)(g) hArr PCl_(3)(g)+Cl_(2)(g)`
`Deltan_(g)=1 gt 0 rArr Q lt K_(eq)`
(Reaction will move in forward direction)
Similarly, `2SO_(2)(g)+O_(2)(g) hArr 2SO_(3)(g)`
`Deltan_(g)=-1 lt 0 rArr Q gt K_(eq)`
(Reaction will move in the backward direction)
b. Inert gas addition at constant volume:
`K_(eq)=Q=(p_(C)^(p) p_(D)^(q))/(p_(A)^(m) p_(B)^(n))` (at equilibrium)
and particle pressure `(p)=chiP_("Total")`
`["where" chi=n/n_("Total")="mole fraction"]`
Now, when inert gas is added at constant volume, `P_("Total") uarr` and `chi darr` but partial pressure of each gas remains same, since:
Partial pressure `(p)=n/n_("Total").P_("Total")` and using Dalton's Law and ideal gas equation:
`P_("Total").V=n_("Total")RT["check":P_("Total")/n_("Total")=(RT)/(V)="constant"]`
Thus, `K_(eq)=Q` even on adding inert gas at constant volume. Thus, adding inert gas at constant volume has no effect on a reaction at equilibrium.