Question

Derive the relationship between $\Delta H$ and $\Delta U$ for an ideal gas. Explain each term involved in the equation.

Answer 1

• $\Delta H=q_p$ i.e. heat absorbed by the system at constant pressure
• Let us consider a reaction involving gases.
• Let $V_A$ be the total volume of gaseous reactants
• $V_B$ be the total volume of gaseous products.
• $n_B$ be the number of moles of gaseous products.
• At constant temperature and pressure using ideal gas equation

$P{V}_A=n_{A}RT$ ....$\left(1\right)$

$P{V}_B=n_{B}RT$ ....$\left(2\right)$

Subtracting $\left(1\right)$ from $\left(1\right)$:

$P(V_B-V_A)=RT(n_B-n_A)$

$P\Delta V=\Delta n_{g}RT$

Where $\Delta n_g$ refers to the number of moles of gaseous product minus the number of moles of gaseous reactant.

As, $\Delta H=q_p$; at constant pressure
and

$q_p=\Delta U+P\Delta V$

$\therefore \Delta H=\Delta U+\Delta n_{g}RT$

Where $\Delta H=H_2-H_1=$ Change in enthalpy of the system

$\Delta U=U_2-U_1=$ Change in internal energy

$R=$ Gas constant

$T=$ Temperature in Kelvin

$\therefore$ Reactions involving ideal gases in a close container at constant pressure,

$\Delta H=\Delta U+\Delta n_{g}RT$