Q.Derive the relation between $K_P$ and $K_C$.

For N2(g) + 3H2(g) <=> 2NH3(g), the relation Kp = Kc(RT)^Δn with Δn = -2 gives Kc/Kp = (RT)^2.

The general relation between Kp (equilibrium constant in terms of partial pressures) and Kc (equilibrium constant in terms of molar concentrations) for a gas-phase reaction is:

Kp = Kc (RT)^Δn

where Δn = (total moles of gaseous products) - (total moles of gaseous reactants).

For the reaction N2(g) + 3H2(g) <=> 2NH3(g):

• moles of gaseous products = 2 (from 2NH3) …

Delta-ng for 2SO2(g) + O2(g) <=> 2SO3(g) equals (gaseous product moles) - (gaseous reactant moles) = 2 - 3 = -1.

Delta-ng is used to relate the equilibrium constants Kp and Kc via Kp = Kc(RT)^(Delta-ng), and is defined purely from the stoichiometric coefficients of GASEOUS species in the balanced chemical equation:

Delta-ng = (sum of stoichiometric coefficients of gaseous products) - (sum of stoichiometric coefficients of gaseous reactants)

For the reaction 2SO2(g) + O2(g) <=> 2SO3(g):

• Gaseous reactants: 2 mol SO2 + 1 mol O2 = 3 mol total
• Gaseous products: 2 mol SO3 = 2 mol total

Delta-ng = 2 - 3 = -1

…

For a negative Δng, Kp is smaller than Kc.

For a gaseous equilibrium, Kp and Kc are related by:

Kp = Kc(RT)^Δng

where Δng = (moles of gaseous products) − (moles of gaseous reactants).

When Δng is negative (fewer moles of gaseous product than reactant, e.g. N2 + 3H2 ⇌ 2NH3, Δng = 2−4 = −2), the exponent on (RT) is negative, so (RT)^Δng = 1/(RT)^|Δng|, which is a fraction less than 1 (since RT > 1 in the units normally used). Multiplying Kc by a factor less than 1 gives a smaller value, so:

Kp = Kc × (a fraction < 1) ⟹ Kp < Kc

…