1.8.2: Enthalpy
There is considerable merit in identifying an extensive property of a closed system called the enthalpy, \(\mathrm{H}\). The enthalpy of a closed system is a state variable and defined by equation (a).
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
We identify a given state by the symbol I having enthalpy \(\mathrm{H}[\mathrm{I}]\), energy \(\mathrm{U}[\mathrm{I}]\) and volume \(\mathrm{V}[\mathrm{I}]\) at pressure \(\mathrm{p}\).
\[\mathrm{H}[\mathrm{I}]=\mathrm{U}[\mathrm{I}]+\mathrm{p} \, \mathrm{V}[\mathrm{I}] \nonumber \]
This system is displaced to a neighbouring state such that the differential change in enthalpy is \(\mathrm{dH}\). Using equation (a),
\[\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
But according to the first law of thermodynamics, the differential change in thermodynamic energy \(\mathrm{dU}\) is given by ‘\(q-p \, d V\)’ where \(\mathrm{q}\) is the heat accompanying the change. Then,
\[\mathrm{dH}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
or,
\[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp} \nonumber \]
At constant pressure,
\[\mathrm{dH}=\mathrm{q} \nonumber \]
For a change from state I to state II the change in enthalpy is given by equation (g).
\[\Delta \mathrm{H}=\int_{\mathrm{I}}^{\mathrm{II}} \mathrm{dH}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q} \nonumber \]
In equation (g) we replace the integral of dH by the difference \(\mathrm{H}(\mathrm{II}) - \mathrm{H}(\mathrm{I})\) because enthalpy is a state variable and so \(\Delta \mathrm{H}\) is independent of the path between the two states and hence so is \(\mathrm{q}\). In liquid solutions, the recorded heat is also independent of the rate of change in chemical composition between state I and state II.