Relation between Enthalpy and Internal Energy

To relate enthalpy $H$ to internal energy $U$ concretely, consider a system at constant pressure moving from an initial state (enthalpy $H_1$, internal energy $U_1$, volume $V_1$) to a final state ($H_2$, $U_2$, $V_2$). Applying the definition $H=U+PV$ to each state separately:

Subtracting (7.10) from (7.11):

Now bring in the first law, $\Delta U=q+w$, and substitute into (7.12):

Since $w=-P\Delta V$ for ordinary expansion work, the $P\Delta V$ terms cancel exactly:

confirming that $q_p$ (heat absorbed at constant pressure) is the quantity often loosely called a system's "heat content" -- it IS $\Delta H$.

Reformulating for a gas-phase chemical reaction. For a closed system of gases reacting to give gaseous products at constant temperature and pressure, let $V_i$ and $V_f$ be the total volumes of reactant and product gases, with $n_i$ and $n_f$ their respective total moles. Applying the ideal gas law to each side separately:

Subtracting (7.14) from (7.15):

Substituting (7.16) into (7.12) gives the working formula chemists actually reach for when a reaction changes the number of moles of gas:

$\Delta H = \Delta U + \Delta n_{(g)}RT \qquad (7.17)$ …