Question

Derive the relation between ∆H and ∆U for an ideal gas. Explain each term involved in the equation.

Answer 1

When the system at constant pressure undergoes changes from an initial state with H₁, U₁, V₁ and P parameters to a final state with H₂, U₂, V₂ and P parameters, the change in enthalpy ∆H, is given by

H = U + PV

At initial state H₁ = U₁ + PV₁    ...(1)

At final state H₂ = U₂ + PV₂    ...(2)

change in enthalpy is (2) – (1)

⇒ (H₂ – H₁) = (U₂ – U₁) + P(V₂ – V₁)

∆H = ∆U + P∆V     ...(3)

As per first law of thermodynamics,

∆U = q + w; w = - P∆V 

Equation (3) becomes

∆H = q + w + P∆V

∆H = q_p – P∆V + P∆V

∆H = q_p    ...(4)

q_p is the heat absorbed at constant pressure and is considered as heat content.

Consider a closed system of gases which are chemically reacting to form gaseous products at constant temperature and pressure with V_i and V_f, as the total volumes of the reactant and product gases respectively, and n_i and n_f as the number of moles of gaseous reactants and products, then,

For reactants: P V_i = n_i RT   ...(5)

For products: P V_f = n_f RT   ...(6)

(6) - (5)

P (V_f – V_i) = (n_f – n_i) RT

P∆V = ∆n_g RT   ...(7)

Substituting in (7) in (3)

∆H = ∆U + P∆V

∆H = ∆U + ∆n_g RT