Derive the relation between K_p and K_c for the following reactions.
$\text{ i) }2{\mathrm{HI}}_{\left(\mathrm{g}\right)}\underset{ }{\rightleftharpoons }{\mathrm{H}}_{2\left(\mathrm{g}\right)}+{\mathrm{I}}_{2\left(\mathrm{g}\right)}$
$\text{ ii) }{\mathrm{N}}_{2\left(\mathrm{g}\right)}+3{\mathrm{H}}_{2\left(\mathrm{g}\right)}\underset{ }{\rightleftharpoons }2{\mathrm{NH}}_{3\left(\mathrm{g}\right)}$
$\text{ iii) }2{\mathrm{SO}}_{2\left(\mathrm{g}\right)}+{\mathrm{O}}_{2\left(\mathrm{g}\right)}\underset{ }{\overset{ }{\rightleftharpoons }}2{\mathrm{SO}}_{3\left(\mathrm{g}\right)}$
$\text{ iv) }{\mathrm{CaCO}}_{3\left(\mathrm{s}\right)}\underset{ }{\rightleftharpoons }{\mathrm{CaO}}_{\left(\mathrm{s}\right)}+{\mathrm{CO}}_{2\left(\mathrm{g}\right)}$

In chemical equilibrium involving gases, the equilibrium constants K_p and K_c are related by the equation:

K_p = K_c (RT)^Δn

Where:

• K_p = equilibrium constant in terms of partial pressures
• K_c = equilibrium constant in terms of concentrations
• R = universal gas constant
• T = temperature in Kelvin
• Δn = change in the number of moles of gas, calculated as n_products − n_reactants

i) Reaction: 2 HI (g) ⇌ H₂ (g) + I₂ (g)

For this reaction, the change in the number of moles (Δn) is calculated as:

Δn = n_products − n_reactants = (1 + 1) − 2 = 0

Substituting into the K_p and K_c relation formula:

K_p = K_c (RT)⁰

Since any number raised to the power of 0 is 1:

K_p = K_c

ii) Reaction: N₂ (g) + 3 H₂ (g) ⇌ 2 NH₃ (g)

For this reaction, the change in the number of moles (Δn) is:

Δn = n_products − n_reactants = 2 − (1 + 3) = −2

Substituting into the K_p and K_c relation formula:

K_p = K_c (RT)⁻²

Since (RT)⁻² is less than 1:

K_p < K_c

iii) Reaction: 2 SO₂ (g) + O₂ (g) ⇌ 2 SO₃ (g)

For this reaction, the change in the number of moles (Δn) is:

Δn = n_products − n_reactants = 2 − (2 + 1) = −1

Substituting into the K_p and K_c relation formula:

K_p = K_c (RT)⁻¹

Since (RT)⁻¹ is less than 1:

K_p < K_c

iv) Reaction: CaCO₃ (s) ⇌ CaO (s) + CO₂ (g)

For this reaction, the change in the number of moles (Δn) is:

Δn = n_products − n_reactants = 1 − 0 = 1

Substituting into the K_p and K_c relation formula:

K_p = K_c (RT)¹

Since (RT)¹ is greater than 1:

K_p > K_c