Hess’s Law — ΔfH from Combustion Enthalpies
Hess’s law states that the enthalpy change of a process depends only on the
initial and final states. Therefore, if a target reaction can be written as the
algebraic sum of several known reactions, its enthalpy change is the corresponding
algebraic sum of their enthalpy changes.
Target: standard enthalpy of formation, ΔfH°
For a compound \( \mathrm{A} \), the formation reaction is the balanced equation that
forms exactly 1 mol of \( \mathrm{A} \) from its constituent elements in their
reference forms at 1 bar and \(25^\circ\text{C}\) (e.g., C(graphite), \( \mathrm{H_2(g)} \), \( \mathrm{O_2(g)} \)).
The enthalpy change of this reaction is \( \Delta_f H^\circ(\mathrm{A}) \) (units: kJ·mol⁻¹ of product).
Normalization: The product coefficient in the formation reaction must be 1.
If all coefficients are multiplied by a factor \(a\), the reaction enthalpy scales by the same factor \(a\).
Reversing any reaction changes the sign of its enthalpy.
Using combustion data
Many organic (C/H/O) compounds have well-tabulated standard enthalpies of combustion,
\( \Delta H_c^\circ \) (kJ·mol⁻¹ of substance burned). A combustion reaction consumes the substance
with \( \mathrm{O_2} \) and produces the corresponding oxides (typically \( \mathrm{CO_2} \) and \( \mathrm{H_2O} \)).
Strategy (what the calculator automates)
- Write the formation reaction for your product with its coefficient = 1,
e.g. \( \nu_1\,\mathrm{R_1} + \nu_2\,\mathrm{R_2} \to \mathrm{P}\).
-
List the combustion reactions for each reactant \( \mathrm{R_{i}} \) and for the product \( \mathrm{P} \).
(There is no combustion step for \( \mathrm{O_{2}} \).)
For C/H/O species the tool shows an auto-balanced preview like
\[
\mathrm{C_{x}H_{y}O_{z}} + a\,\mathrm{O_{2}} \longrightarrow x\,\mathrm{CO_{2}} + \tfrac{y}{2}\,\mathrm{H_{2}O}.
\]
These previews are for display only; you supply the tabulated \( \Delta H_{c}^{\circ} \) values.
- Scale the reactant combustions by the formation coefficients:
multiply reaction \(i\) by \( \nu_i \). Their enthalpies become \( \nu_i \Delta H_{c,i}^\circ \).
- Reverse the product’s combustion (so that \( \mathrm{CO_2} \) and \( \mathrm{H_2O} \) cancel when added):
the enthalpy becomes \( -\Delta H_{c,\mathrm{P}}^\circ \).
- Add the modified steps (Hess’s law). The oxides cancel and the
sum equals the formation reaction. Therefore,
\[
\boxed{\;\Delta_f H^\circ(\mathrm{P}) = \sum_i \nu_i\,\Delta H_{c,i}^\circ \;-\; \Delta H_{c,\mathrm{P}}^\circ\;}
\]
(units: kJ per mole of product formed).
Worked pattern (propane-like)
Target formation (product coefficient = 1):
\[
3\,\mathrm{C} + 4\,\mathrm{H_2} \longrightarrow \mathrm{C_3H_8}
\]
Combustion steps and modifications
\[
\begin{aligned}
&\text{(a) } \mathrm{C} + \mathrm{O_2} \to \mathrm{CO_2} \qquad &&\Delta H_c^\circ(\mathrm{C}) \\
&\text{(b) } \mathrm{H_2} + \tfrac{1}{2}\mathrm{O_2} \to \mathrm{H_2O} \qquad &&\Delta H_c^\circ(\mathrm{H_2}) \\
&\text{(c) } \mathrm{C_3H_8} + 5\,\mathrm{O_2} \to 3\,\mathrm{CO_2} + 4\,\mathrm{H_2O} \qquad &&\Delta H_c^\circ(\mathrm{C_3H_8})
\end{aligned}
\]
Scale (a) by 3 and (b) by 4; reverse (c):
\[
\begin{aligned}
&3\times\text{(a)} \qquad &&\Delta H = 3\,\Delta H_c^\circ(\mathrm{C}) \\
&4\times\text{(b)} \qquad &&\Delta H = 4\,\Delta H_c^\circ(\mathrm{H_2}) \\
&-\text{(c)} \qquad &&\Delta H = -\,\Delta H_c^\circ(\mathrm{C_3H_8})
\end{aligned}
\]
Adding the three gives the formation equation above, so
\[
\Delta_f H^\circ(\mathrm{C_3H_8}) = 3\,\Delta H_c^\circ(\mathrm{C}) + 4\,\Delta H_c^\circ(\mathrm{H_2})
- \Delta H_c^\circ(\mathrm{C_3H_8}).
\]
Signs, units, and cautions
- Units: Enter \( \Delta H_c^\circ \) in kJ·mol⁻¹ of the substance combusted; the tool returns \( \Delta_f H^\circ \) in kJ·mol⁻¹ of product.
- Signs: Combustions are typically negative (exothermic). Reversing a reaction changes the sign.
- Oxygen has no combustion reaction. Do not provide \( \Delta H_c^\circ \) for \( \mathrm{O_2} \).
- Non-C/H/O species: The Hess addition still works with your tabulated \( \Delta H_c^\circ \) values, but the auto-balanced preview is shown only for C/H/O formulae.
- Fractional coefficients: Allowed in intermediate steps; the final displayed formation equation is balanced with whole-number stoichiometry and the product normalized to 1.
Why this works: Summing the scaled reactant combustions and the reversed product combustion cancels the oxide
intermediates (e.g., \( \mathrm{CO_2} \), \( \mathrm{H_2O} \)), leaving the pure elements → product pathway.
By the state-function nature of enthalpy, this sum equals the desired \( \Delta_f H^\circ \).
