R1.2.2 Hess’s Law

Hess's Law: The Conservation of Energy in Chemical Reactions

Imagine you’re navigating a city’s metro system—perhaps Tokyo or London.
To get from Station A to Station B, you have multiple route options.
Some may involve transfers, others may be direct, but no matter which path you take, the total distance between A and B remains the same.

This concept parallels Hess’s Law in chemistry: the enthalpy change of a reaction depends only on the starting and ending points, not the route taken.

What is Hess's Law?

Hess’s Law is an application of the law of conservation of energy, which states that energy cannot be created or destroyed—only transferred or transformed. In chemical reactions, this means:

Definition

Hess’s Law

The total enthalpy change for a reaction is the sum of the enthalpy changes for each step of the reaction pathway.

To illustrate, consider the oxidation of sulfur to sulfur trioxide:

Overall Reaction:

$S (s) + \frac{3}{2} O_{2} (g) \to S O_{3} (g) Δ H = - 396 kJ/mol$

This reaction can occur in two steps:

Step 1: $S (s) + O_{2} (g) \to S O_{2} (g) Δ H = - 296 kJ/mol$
Step 2: $S O_{2} (g) + \frac{1}{2} O_{2} (g) \to S O_{3} (g) Δ H = - 100 kJ/mol$

Adding the enthalpy changes for these steps gives the same total enthalpy change as the overall reaction:
$Δ H = (- 296) + (- 100) = - 396 kJ/mol$

This is the essence of Hess’s Law: the sum of the enthalpy changes for individual steps equals the enthalpy change of the overall reaction.

Note

Hess’s Law holds true because enthalpy ( $H$ ) is a state function, meaning its value depends only on the system’s current state (reactants and products), not on the process used to reach that state.

Applying Hess's Law: Calculating Enthalpy Changes

Hess's Law is particularly useful for calculating the enthalpy change ( $Δ H$ ) of reactions that are experimentally challenging to measure directly.
By combining known enthalpy changes of related reactions, we can deduce the enthalpy change for the target reaction.

There are two common methods for applying Hess’s Law:

Summation of Equations Method
Enthalpy Cycle Diagram Method

1. Summation of Equations Method

This method involves manipulating given chemical equations and their associated enthalpy changes to construct the target reaction. Key steps include:

Reversing equations (which changes the sign of $Δ H$ ).
Scaling equations by multiplying or dividing (which scales $Δ H$ proportionally).

Example question

Formation of Methanol

Calculate the enthalpy change for the formation of methanol ( $C H_{3} O H$ ):

$C (s) + 2 H_{2} (g) + \frac{1}{2} O_{2} (g) \to C H_{3} O H (l)$

Given:

$C H_{3} O H (l) + \frac{3}{2} O_{2} (g) \to C O_{2} (g) + 2 H_{2} O (l) Δ H = - 726 kJ/mol$
$C (s) + O_{2} (g) \to C O_{2} (g) Δ H = - 394 kJ/mol$
$H_{2} (g) + \frac{1}{2} O_{2} (g) \to H_{2} O (l) Δ H = - 286 kJ/mol$

Solution

Steps:

Reverse Equation 1 to make $C H_{3} O H (l)$ a product:
$C O_{2} (g) + 2 H_{2} O (l) \to C H_{3} O H (l) + \frac{3}{2} O_{2} (g) Δ H = + 726 kJ/mol$
Use Equation 2 as given:
$C (s) + O_{2} (g) \to C O_{2} (g) Δ H = - 394 kJ/mol$
Double Equation 3 to account for 2 moles of $H_{2} (g)$ :
$2 H_{2} (g) + O_{2} (g) \to 2 H_{2} O (l) Δ H = 2 (- 286) = - 572 kJ/mol$
Add the modified equations:
$C (s) + 2 H_{2} (g) + \frac{1}{2} O_{2} (g) \to C H_{3} O H (l) Δ H = (- 394) + (- 572) + (+ 726) = - 240 kJ/mol$

Result:

$Δ H = - 240 kJ/mol$

Tip

When combining equations, ensure intermediate species (e.g., $C O_{2} (g)$ ) cancel out, leaving only the reactants and products of the target reaction.

2. Enthalpy Cycle Diagram Method

This method visualizes the problem as a cycle, where the enthalpy change of the target reaction is calculated by following an alternative pathway.

Example question

Decomposition of Potassium Hydrogencarbonate

Find the enthalpy change for:
$2 K H C O_{3} (s) \to K_{2} C O_{3} (s) + C O_{2} (g) + H_{2} O (g)$

Given:

$K H C O_{3} (s) + H C l (a q) \to K C l (a q) + C O_{2} (g) + H_{2} O (l) Δ H_{1}$
$K_{2} C O_{3} (s) + 2 H C l (a q) \to 2 K C l (a q) + C O_{2} (g) + H_{2} O (l) Δ H_{2}$

Solution

Steps:

Place the target reaction at the top of the cycle.
Connect it to known reactions (e.g., reactions involving $H C l$ ) via intermediate states.
Use the alternative pathway to calculate $Δ H$ for the target reaction.

Hess's law for decomposition of potassium hydrogencarbonate.

Applications of Hess's Law

Hess’s Law has wide-ranging applications in chemistry, including:

1. Using Enthalpy of Formation Data (HL only)

The standard enthalpy change of a reaction ( $Δ H_{r}^{\circ}$ ) can be calculated using enthalpy of formation ( $Δ H_{f}^{\circ}$ ) values:
$Δ H_{r}^{\circ} = \sum (Δ H_{f}^{\circ} products) - \sum (Δ H_{f}^{\circ} reactants)$

2. Using Enthalpy of Combustion Data

Similarly, enthalpy changes can be calculated using enthalpy of combustion ( $Δ H_{c}^{\circ}$ ) values:
$Δ H_{r}^{\circ} = \sum (Δ H_{c}^{\circ} reactants) - \sum (Δ H_{c}^{\circ} products)$

Self review

Calculate the enthalpy change for the combustion of pentane, $C_{5} H_{12}$ , using enthalpy of formation data from the IB data booklet.

Common Mistakes and Tips

Common Mistake

Forgetting to reverse the sign of $Δ H$ when reversing a reaction is a common error.

Tip

Double-check that intermediate species cancel out when summing equations to ensure the final equation matches the target reaction.

Common Mistake

Failing to account for state changes (e.g., gas to liquid) when using average bond enthalpies to calculate $Δ H$ .

Reflection and Connections

Theory of Knowledge

How might the conservation of energy in Hess’s Law relate to other fields, such as physics or environmental science? Could this principle guide us in evaluating the sustainability of energy resources?