A Useful New State Function

A Useful New State Function

When a chemical reaction is carried out in a beaker or flask open to the atmosphere, the pressure remains constant (atmospheric pressure) but the volume may change. Under these conditions, the heat absorbed by the system, $q_p$, is not equal to the change in internal energy $\Delta U$. We already know that at constant volume, $\Delta U = q_V$ because no work is done. But at constant pressure, the system can do expansion work (or have work done on it) as its volume changes.

From the first law of thermodynamics, $\Delta U = q + w$. If the only work is pressure-volume work, then at constant pressure $p$, the work done by the system is $w = -p\Delta V$. So we write:

Let the initial state be denoted by subscript 1 and the final state by subscript 2. Then:

Rearranging this equation gives:

The quantity $U + pV$ appears so naturally that it is given its own name and symbol. We define a new thermodynamic function called enthalpy, $H$ (from the Greek enthalpien, meaning "to warm" or "heat content"):

Using this definition, the equation above becomes:

This is a crucial result. Although $q$ (heat) is a path-dependent quantity, $H$ is a state function because it depends only on $U$, $p$, and $V$ — all of which are state functions. Therefore, $\Delta H$ is independent of the path taken, and consequently $q_p$ is also independent of path. This makes enthalpy an enormously useful quantity for studying reactions carried out under constant pressure — which is how most chemical reactions are performed.

$\Delta H$ is negative for exothermic reactions, which evolve heat during the reaction, and $\Delta H$ is positive for endothermic reactions, which absorb heat from the surroundings.

At constant volume ($\Delta V = 0$), the $p\Delta V$ term vanishes, so $\Delta H = \Delta U = q_V$.

For finite changes at constant pressure, we can write:

Since pressure is constant, $\Delta(pV) = p\Delta V$, so:

This relationship connects the enthalpy change to the internal energy change and the pressure-volume work done by the system.

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Relationship Between $\Delta H$ and $\Delta U$ for Reactions Involving Gases

For reactions involving gases, the $p\Delta V$ term can be expressed in terms of the change in the number of moles of gas. If we treat the gases as ideal, then $pV = nRT$. At constant temperature and pressure:

where $\Delta n_g$ is the change in the number of moles of gaseous species in the reaction (moles of gaseous products minus moles of gaseous reactants). Therefore:

This equation allows us to convert between $\Delta H$ and $\Delta U$ for any reaction, provided we know the temperature and the change in the number of moles of gas.