Law Of Chemical Equilibrium And Equilibrium Constant

Let the initial moles of PCl₅ be 1 and the degree of dissociation be α. We can set up an ICE table based on moles:

Total moles at equilibrium = (1-α) + α + α = 1+α.
The total pressure at equilibrium is given as P. Now, calculate the partial pressures using p_i = X_i × P_total, where X_i is the mole fraction. p_PCl5 = ( (1-α) / (1+α) ) P
p_PCl3 = ( α / (1+α) ) P
p_Cl2 = ( α / (1+α) ) P The expression for the equilibrium constant K_p is:
K_p = (p_PCl3 × p_Cl2) / p_PCl5 Substitute the partial pressures:
K_p = (( αP / (1+α) ) ( αP / (1+α) )) / (( (1-α)P ) / (1+α) ) K_p = (α²P² / (1+α)²) / (( (1-α)P ) / (1+α) ) K_p
= (α²P² / (1+α)²) × ((1+α) / ((1-α)P)) K_p = (α²P) / ((1+α)(1-α)) K_p = (α²P) / (1-α²)
Now, rearrange to solve for α:
K_p (1-α²) = α² P K_p - K_p α² = α² P K_p
= α² P + K_p α² K_p = α² (P + K_p) α²
= K_p / (P + K_p) α = √( K_p / (K_p + P) ) = ( K_p / (K_p + P) )^1/2
This corresponds to option (C).