Q.On increasing the pressure, in which direction will the gas phase reaction proceed to re-establish equilibrium, is predicted by applying the Le Chatelier's principle. Consider the reaction.
N2

Concept: Equilibrium Constant and Le Chatelier's Principle

The equilibrium constant $K$ (whether $K_p$ or $K_c$) depends only on temperature for a given reaction. It is a thermodynamic quantity derived from the standard Gibbs energy change:

Since $\Delta G^\circ$ is fixed at a given temperature, $K$ remains constant regardless of changes in pressure, concentration, or volume. …

The equilibrium constant $K$ is a thermodynamic quantity that measures the ratio of products to reactants at equilibrium, and it is fundamentally tied to the standard Gibbs free energy change of the reaction through $\Delta G^\circ = -RT \ln K$. Because $\Delta G^\circ$ depends only on temperature (it reflects the intrinsic energy difference between products and reactants at standard conditions), $K$ itself is a function of temperature alone.

When we increase the total pressure on a gas-phase equilibrium without changing temperature, Le Chatelier's principle tells us the position of equilibrium will shift—the system responds by favoring the side with fewer moles of gas to partially counteract the pressure increase. But this shift does not alter the value of $K$; instead, the partial pressures (or concentrations) of all species adjust in such a way that their ratio still satisfies the same equilibrium constant.

Let's see why this is true for the ammonia synthesis reaction:

1. Count the moles of gas on each side.

   Reactants: $1 + 3 = 4$ moles of gas.

   Products: $2$ moles of gas.

   The forward reaction reduces the total number of moles.

2. Apply Le Chatelier's principle for a pressure increase.

   Increasing pressure at constant temperature favors the side with fewer moles of gas. Here, the equilibrium shifts to the right (toward $\ce{NH3}$), producing more ammonia and consuming nitrogen and hydrogen. The amounts of each species change.

3. Recognize that $K$ is defined in terms of activities (or partial pressures).

   For this reaction,

Even though the individual partial pressures $P_{\ce{NH3}}$, $P_{\ce{N2}}$, and $P_{\ce{H2}}$ all change when total pressure increases, they adjust in concert so that this ratio remains constant at a given temperature.

4. Understand the thermodynamic basis. …