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

##### 1 Answer

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

#barul(|" "stackrel(" ")(DeltaH_(rxn) = DeltaE_(rxn) + PDeltaV_(rxn))" "|)#

Consider a general **potential energy diagram** for an endothermic reaction

#A + B -> C + D# ,

Let the reactants **change in internal energy** for the reaction is given by

#DeltaE_(rxn) = (E_C + E_D) - (E_A + E_B)#

So, if we know the ground-state energies of the reactants and products, we can calculate

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

#dH = TdS + VdP#

Now, consider adding and subtracting reversible pressure-volume work

#dH = TdS - PdV + PdV + VdP#

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:

#dE = q_(rev) + w_(rev) = TdS - PdV# ,where

#q_(rev)# is the reversible heat flow and#S# is the entropy.

Thus, we can rewrite this as:

#dH = dE + PdV + VdP#

Now, in ordinary reactions, we are at a **constant atmospheric pressure**, so

#dH = dE + PdV#

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

#DeltaH = DeltaE + PDeltaV# ,

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

#color(blue)(barul(|" "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