Question

Derive the relation ∆H − ∆U = ∆nRT.

Answer 1

Derivation for the relation ΔH − ΔU = ΔnRT:

1) ΔH and ΔU at constant pressure are related as,

ΔH = ΔU + PΔV

For reactions involving solids and liquids, ΔV is usually very small because solids and liquids do not expand or contract significantly as pressure changes. For such reactions neglecting PΔV, ΔH = ΔU

However, for reactions involving gases, ΔV cannot be neglected. The equation
ΔH = ΔU + PΔV

= ΔU + P (V₂ − V₁)
= ΔU + PV₂ − PV₁ ….. (1)

where, V1 is the volume of gas-phase reactants (initial state) and V2 is the volume
of gas-phase products (final-state).

If we assume that reactant and product gases are ideal, we can apply ideal gas equation, PV = nRT. Suppose that n1 moles of gaseous reactants produce n₂ moles of gaseous products. Then,

PV₁ = n₁RT and PV₂ = n₂RT ….. (2)

Substitution of equation (2) into equation(1) gives,

ΔH = ΔU + n₂RT − n₁RT

= ΔU + (n₂ − n₁)RT

= ΔU + ΔnRT

where, Δn is the difference between the number of moles of gaseous products and that of
gaseous reactants

∴ ΔH − ΔU = Δn RT