Hess's law

Use of enthalpies of formation

Hess's law states that enthalpy changes are additive. Thus the value of the standard enthalpy of reaction can be calculated from standard enthalpies of formation of products and reactants as follows:

${\displaystyle \mathrm{\Delta }{H}_{\text{reaction}}^{\text{⦵}}=\sum _i{a}_i{\mathrm{\Delta }}_{\text{f}}{H}_{\text{products}}^{\text{⦵}}-\sum _i{b}_i{\mathrm{\Delta }}_{\text{f}}{H}_{\text{reactants}}^{\text{⦵}}}$

Here, the first sum is over all products and the second over all reactants, ${\displaystyle a_i}$ and ${\displaystyle b_i}$ are the stoichiometric coefficients of products and reactants respectively, ${\displaystyle {\mathrm{\Delta }}_{\text{f}}{H}_{\text{products}}^{\text{⦵}}}$ and${\displaystyle {\mathrm{\Delta }}_{\text{f}}{H}_{\text{reactants}}^{\text{⦵}}}$ are the standard enthalpies of formation of products and reactants respectively, and the ^o superscript indicates standard state values. This may be considered as the sum of two (real or fictitious) reactions:

Reactants → Elements (in their standard states)

${\displaystyle \mathrm{\Delta }{H}_{\text{RE}}^{\text{⦵}}=-\sum _i{b}_i{\mathrm{\Delta }}_{\text{f}}{H}_{\text{reactants}}^{\text{⦵}}}$

and Elements → Products

${\displaystyle \mathrm{\Delta }{H}_{\text{EP}}^{\text{⦵}}=\sum _i{a}_i{\mathrm{\Delta }}_{\text{f}}{H}_{\text{products}}^{\text{⦵}}}$

Examples

1. Given:
   1. C_graphite + O₂ → CO₂(g) ( ΔH = −393.5 kJ/mol) (direct step)
   2. C_graphite + 1/2 O₂ → CO(g) (ΔH = −110.5 kJ/mol)
   3. CO(g) +1/2 O₂ → CO₂(g) (ΔH = −283.0 kJ/mol)

   Reaction (a) is the sum of reactions (b) and (c), for which the total ΔH = −393.5 kJ/mol, which is equal to ΔH in (a).

2. Given:
   • B₂O₃(s) + 3H₂O(g) → 3O₂(g) + B₂H₆(g) (ΔH = 2035 kJ/mol)
   • H₂O(l) → H₂O(g) (ΔH = 44 kJ/mol)
   • H₂(g) + 1/2 O₂(g) → H₂O(l) (ΔH = −286 kJ/mol)
   • 2B(s) + 3H₂(g) → B₂H₆(g) (ΔH = 36 kJ/mol)

   Find the Δ_fH of:

   • 2B(s) + 3/2 O₂(g) → B₂O₃(s)

   After multiplying the equations (and their enthalpy changes) by appropriate factors and reversing the direction when necessary, the result is:

   • B₂H₆(g) + 3O₂(g) → B₂O₃(s) + 3H₂O(g) (ΔH = 2035 × (−1) = −2035 kJ/mol)
   • 3H₂O(g) → 3H₂O(l) (ΔH = 44 × (−3) = −132 kJ/mol)
   • 3H₂O(l) → 3H₂(g) + (3/2) O₂(g) (ΔH = −286 × (−3) = 858 kJ/mol)
   • 2B(s) + 3H₂(g) → B₂H₆(g) (ΔH = 36 kJ/mol)

   Adding these equations and canceling out the common terms on both sides, we obtain

   • 2B(s) + 3/2 O₂(g) → B₂O₃(s) (ΔH = −1273 kJ/mol)

Extension to free energy and entropy

The concepts of Hess's law can be expanded to include changes in entropy and in Gibbs free energy, since these are also state functions. The Bordwell thermodynamic cycle is an example of such an extension that takes advantage of easily measured equilibria and redox potentials to determine experimentally inaccessible Gibbs free energy values. Combining ΔG^o values from Bordwell thermodynamic cycles and ΔH^o values found with Hess's law can be helpful in determining entropy values that have not been measured directly and therefore need to be calculated through alternative paths.

For the free energy:

${\displaystyle \mathrm{\Delta }{G}_{\text{reaction}}^{\text{⦵}}=\sum {\nu }_{\text{p}}\mathrm{\Delta }{G}_{\mathrm{f}(\text{p})}^{\text{⦵}}-\sum {\nu }_{\text{r}}\mathrm{\Delta }{G}_{\mathrm{f}(\text{r})}^{\text{⦵}}.}$

For entropy, the situation is a little different. Because entropy can be measured as an absolute value, not relative to those of the elements in their reference states (as with ΔH^o and ΔG^o), there is no need to use the entropy of formation; one simply uses the absolute entropies for products and reactants:

${\displaystyle \mathrm{\Delta }{S}_{\text{reaction}}^{\text{⦵}}=\sum {\nu }_{\text{p}}{S}_{(\text{p})}^{\text{⦵}}-\sum {\nu }_{\text{r}}{S}_{(\text{r})}^{\text{⦵}}.}$

Applications

Hess's law is useful in the determination of enthalpies of the following:

1. Heats of formation of unstable intermediates like CO_(g) and NO_(g).
2. Heat changes in phase transitions and allotropic transitions.
3. Lattice energies of ionic substances by constructing Born–Haber cycles if the electron affinity to form the anion is known, or
4. Electron affinities using a Born–Haber cycle with a theoretical lattice energy.