Equilibrium shifts and inert gases in reversible reactions: Given Δx = (moles of gaseous products) − (moles of gaseous reactants), identify the correct statements about adding an inert gas and about pressure effects on equilibrium.

Introduction:
Chemical reaction equilibria depend on temperature, pressure, and composition. The effect of adding an inert gas (a diluent that does not participate in the reaction) depends critically on whether the addition occurs at constant pressure or constant volume, as well as on the change in total moles (Δx) between products and reactants.

Given Data / Assumptions:

• Reversible gas-phase reaction at fixed temperature.
• Δx = moles of gaseous products − moles of gaseous reactants.
• Inert gas does not react and has no direct role in the equilibrium constant K(T).

Concept / Approach:
At constant pressure, adding inert gas increases the total volume to maintain P, thereby decreasing partial pressures. If Δx > 0 (more moles on the product side), the system shifts toward products to counter the dilution (Le Châtelier principle). If Δx < 0, it shifts toward reactants. At constant volume, adding inert gas does not change the partial pressures of reacting species (mole fractions change, but total pressure rises correspondingly), so the equilibrium composition is unaltered. The equilibrium constant K depends only on temperature, not on the presence of inert species or total pressure, hence remains unchanged.

Step-by-Step Solution:
Evaluate (a): Constant-pressure dilution favors the side with higher moles when Δx > 0 → true.Evaluate (b): When Δx = 0, changing pressure does not shift equilibrium → true.Evaluate (c): At constant volume, adding inert gas does not alter equilibrium composition; K(T) unchanged → true.Therefore, all three statements are correct.

Verification / Alternative check:
Deriving equilibrium from minimizing Gibbs free energy under appropriate constraints reproduces these conclusions.

Why Other Options Are Wrong:

• Option (e) asserts a universal decrease in conversion, which is false; the direction depends on Δx and constraints.

Common Pitfalls:
Forgetting to specify whether the system is at constant pressure or constant volume; the effect of inert gas differs drastically.

Final Answer:
all 'a', 'b' & 'c'.