Understanding the relationship between Kc and Kp

We can derive a value for the equilibrium constant in a gaseous equilibrium system either using partial pressures, K_p, or concentrations ,K_c, which means the two must be related.

N₂O_4(g) ⇌ 2NO_2(g)

The number of moles of a gas can be calculated in two ways:

1. Number of moles = concentration x volume

n_NO2 = [NO_2(g)] x V

n_N2O4 = [N₂O_4(g)] x V

2. Using the ideal gas equation (PV = nRT)

P_NO2 V = n_NO2 RT

P_N2O4 V = n_N2O4 RT

We can combine these two methods for each gas:

P_NO2 V = [NO_2(g)] x V x RT

P_N2O4 V = [N₂O_4(g)] x V x RT

In each equation, the volume, V, cancels out on each side.

P_NO2 = [NO_2(g)] x RT

P_N2O4 = [N₂O_4(g)] x RT

The equilibrium expressions, K_p and K_c, for this system are:

But we could also write the gaseous equilibrium expression as:

Which leaves us with …