# What expression represents the DeltaH for a chemical reaction in terms of the potential energy, E, of its products and reactants?

Jul 18, 2017

There isn't one I can think of off-hand, so we'll have to derive it. I get:

$\overline{\underline{| \text{ "stackrel(" ")(DeltaH_(rxn) = DeltaE_(rxn) + PDeltaV_(rxn))" } |}}$

Consider a general potential energy diagram for an endothermic reaction

$A + B \to C + D$, Let the reactants $A$ and $B$ at potential energies ${E}_{A}$ and ${E}_{B}$ form $C$ and $D$ at potential energies ${E}_{C}$ and ${E}_{D}$. The change in internal energy for the reaction is given by

$\Delta {E}_{r x n} = \left({E}_{C} + {E}_{D}\right) - \left({E}_{A} + {E}_{B}\right)$

So, if we know the ground-state energies of the reactants and products, we can calculate $\Delta {E}_{r x n}$.

From the Maxwell Relation for the enthalpy $H$ of a reversible process in a thermodynamically-closed system (conservation of mass),

$\mathrm{dH} = T \mathrm{dS} + V \mathrm{dP}$

Now, consider adding and subtracting reversible pressure-volume work ${w}_{r e v}$ (done from the perspective of the system), $- P \mathrm{dV}$, so that

$\mathrm{dH} = T \mathrm{dS} - P \mathrm{dV} + P \mathrm{dV} + V \mathrm{dP}$

The first two terms are given in the first law of thermodynamics, i.e. conservation of energy for a reversible process in a thermodynamically-closed system:

$\mathrm{dE} = {q}_{r e v} + {w}_{r e v} = T \mathrm{dS} - P \mathrm{dV}$,

where ${q}_{r e v}$ is the reversible heat flow and $S$ is the entropy.

Thus, we can rewrite this as:

$\mathrm{dH} = \mathrm{dE} + P \mathrm{dV} + V \mathrm{dP}$

Now, in ordinary reactions, we are at a constant atmospheric pressure, so $\mathrm{dP} = 0$, and $P$ as shown is in a sense the initial pressure:

$\mathrm{dH} = \mathrm{dE} + P \mathrm{dV}$

By integrating this from an initial state to a final state, we obtain:

$\Delta H = \Delta E + P \Delta V$,

which should be familiar from general chemistry. For a reaction, we can write this as:

$\textcolor{b l u e}{\overline{\underline{| \text{ "stackrel(" ")(DeltaH_(rxn) = DeltaE_(rxn) + PDeltaV_(rxn))" } |}}}$

Thus, if we also know the pressure and the change in volume (through knowing the masses and the densities of each substance), we can then calculate $\Delta {H}_{r x n}$.