Hess's law — how to calculate ΔH using multiple reactions step by step

Question

State Hess’s law of constant heat summation. Using Hess’s law, calculate the enthalpy of formation of CO(g) given:

C(s) + O₂(g) → CO₂(g), $\Delta H_1 = -393.5$ kJ/mol
CO(g) + 1/2 O₂(g) → CO₂(g), $\Delta H_2 = -283.0$ kJ/mol

Solution — Step by Step

Hess’s law states that the total enthalpy change for a reaction is the same regardless of whether it occurs in one step or multiple steps — as long as the initial and final states are the same. This is because enthalpy is a state function (depends only on the state, not the path).

We want: C(s) + 1/2 O₂(g) → CO(g), $\Delta H_f = ?$

We have:

Reaction 1: C(s) + O₂(g) → CO₂(g), $\Delta H_1 = -393.5$ kJ/mol
Reaction 2: CO(g) + 1/2 O₂(g) → CO₂(g), $\Delta H_2 = -283.0$ kJ/mol

Notice that Reaction 1 goes from C to CO₂, and Reaction 2 goes from CO to CO₂. If we reverse Reaction 2, we get CO₂ → CO, and then Reaction 1 - reversed Reaction 2 gives us C → CO.

Target = Reaction 1 - Reaction 2:

\Delta H_f = \Delta H_1 - \Delta H_2 = (-393.5) - (-283.0) = -393.5 + 283.0 = \mathbf{-110.5 \text{ kJ/mol}}

Path: C(s) + O₂(g) → CO₂(g) → CO(g) + 1/2 O₂(g)

Step 1: C + O₂ → CO₂ ( $\Delta H = -393.5$ ) Step 2 (reversed): CO₂ → CO + 1/2 O₂ ( $\Delta H = +283.0$ )

Net: C + 1/2 O₂ → CO, $\Delta H = -393.5 + 283.0 = \mathbf{-110.5}$ kJ/mol

flowchart TD
    A[C + O₂] -->|Direct: ΔH₁ = -393.5| B[CO₂]
    A -->|Step 1: ΔHf = -110.5| C[CO + ½O₂]
    C -->|Step 2: ΔH₂ = -283.0| B

Why This Works

Since enthalpy is a state function, the total $\Delta H$ from reactants to products is path-independent. Whether carbon burns directly to CO₂ (one step) or first to CO then to CO₂ (two steps), the total $\Delta H$ is the same: $\Delta H_f(\text{CO}) + \Delta H_2 = \Delta H_1$ .

Common Mistake

When reversing a reaction, students forget to change the sign of $\Delta H$ . If the forward reaction has $\Delta H = -283.0$ kJ, the reverse has $\Delta H = +283.0$ kJ. Also, if you multiply a reaction by a factor, multiply $\Delta H$ by the same factor.

Algorithm for Hess’s law problems: (1) Write the target reaction. (2) Arrange given reactions so they add up to the target — reverse or multiply as needed. (3) Apply the same operations to $\Delta H$ values. (4) Add all $\Delta H$ values. Done.

Question

Solution — Step by Step

Why This Works

Common Mistake

Try These Next