Chemistry (9701) | Hess’s law

🔥 Hess's Law: The Ultimate Shortcut for Energy Changes

Welcome to the exciting world of Chemical Energetics! Calculating the energy released or absorbed ($\Delta H$) in a chemical reaction is crucial. But what if a reaction is impossible to measure directly? Maybe it's too slow, too fast (explosive!), or produces unwanted side products.

This is where Hess's Law steps in. It's the chemist's ultimate shortcut, allowing us to calculate tricky enthalpy changes indirectly, by using data from reactions that are easy to measure.

What you will learn in this chapter:

• The fundamental principle behind Hess's Law.
• How to construct and use energy cycles (maps of enthalpy change).
• How to calculate unknown enthalpy changes using standard enthalpy data (formation and combustion).

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1. The Foundational Principle: Enthalpy as a State Function

1.1 Defining Hess’s Law

Hess’s Law states that the total enthalpy change for a reaction is independent of the pathway taken, provided the initial and final conditions are the same.

Think of it like walking from the ground floor (Initial State) to the third floor (Final State) of a building.

• Path 1: Take the stairs directly. The total potential energy gained is \(X\).
• Path 2: Take the elevator to the fifth floor, then walk back down two floors. The total potential energy gained is still \(X\).

In both paths, the net change in energy is the same because enthalpy ($\Delta H$) is a State Function.

Key Term: A State Function is a property (like enthalpy or temperature) whose value depends only on the current state of the system, not on how that state was reached.

1.2 Why We Need Hess’s Law (Syllabus 5.2(2a))

Hess's Law allows us to determine enthalpy changes that cannot be found by direct experiment.

Example: We want to find the enthalpy of formation of methane (\(\Delta H_{\text{f}}^{\theta}\) for \(\text{C} + 2\text{H}_2 \rightarrow \text{CH}_4\)).
If you tried mixing carbon (as graphite) and hydrogen gas, the reaction is far too slow and complex to measure the heat change accurately, and it certainly won't yield pure methane easily.

Instead, we use easily measurable reactions (like the combustion of C, \(\text{H}_2\), and \(\text{CH}_4\)) and arrange them using Hess's Law.

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2. Constructing Energy Cycles (Hess Cycles)

To apply Hess’s Law, we map out the different pathways using a diagram called an Energy Cycle (or Hess Cycle). This cycle must always link the initial reactants to the final products via a common intermediate state (usually the elements or the combustion products).

2.1 The Two Essential Types of Cycles

Type A: Using Enthalpies of Formation ($\Delta H_{\text{f}}^{\theta}$)

This cycle is used when you are given (or calculating) the standard enthalpy of formation data ($\Delta H_{\text{f}}^{\theta}$).

The common intermediate state is the elements involved in their standard states.

• Reactants and Products are both formed from their elements.
• The arrows for \(\Delta H_{\text{f}}^{\theta}\) always point DOWN (or AWAY from the elements).

Analogy Tip: Think "F-R-C" - Formation starts at the Floor (elements) and goes Reactants down to the Compound.

Example Reaction: \(\text{A} + \text{B} \xrightarrow{\Delta H_{\text{R}}^{\theta}} \text{C}\)

Type B: Using Enthalpies of Combustion ($\Delta H_{\text{c}}^{\theta}$)

This cycle is used when you are given (or calculating) the standard enthalpy of combustion data ($\Delta H_{\text{c}}^{\theta}$).

The common intermediate state is the combustion products (e.g., \(\text{CO}_2\) and \(\text{H}_2\text{O}\)).

• Reactants and Products are both combusted to form this intermediate state.
• The arrows for \(\Delta H_{\text{c}}^{\theta}\) always point DOWN (or AWAY from the burning substance).

Example Reaction: \(\text{A} + \text{B} \xrightarrow{\Delta H_{\text{R}}^{\theta}} \text{C}\)

Key Takeaway for Cycles: Draw your equation horizontally. Place the common intermediate state (elements or combustion products) on the third corner. Make sure all known enthalpy changes point towards the common state.

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3. Performing Calculations Using Energy Cycles (Syllabus 5.2(1))

The core mathematical rule of Hess's law is: The sum of the enthalpy changes following one path must equal the sum of the enthalpy changes following another path between the same start and end points.

This is often simplified using the "Clockwise/Anti-clockwise" rule.

3.1 The Clockwise/Anti-clockwise Rule

When calculating the unknown $\Delta H$ (\(\Delta H_{\text{R}}^{\theta}\)):

1. Start at the beginning of the unknown $\Delta H_{\text{R}}^{\theta}$ (the reactants).
2. Find a path around the cycle that finishes at the end of $\Delta H_{\text{R}}^{\theta}$ (the products).
3. Sum the values following the direction of the arrows.

The Calculation Rule:
Sum of Clockwise Enthalpy Changes = Sum of Anti-clockwise Enthalpy Changes

If you follow an arrow in the opposite direction, you reverse the sign of the enthalpy change.

3.2 Calculation Type 1: Using Enthalpies of Formation ($\Delta H_{\text{f}}^{\theta}$)

We typically use this to find the enthalpy change of reaction, $\Delta H_{\text{R}}^{\theta}$, when $\Delta H_{\text{f}}^{\theta}$ values are known.

The Formula:
$$\Delta H_{\text{R}}^{\theta} = \sum \Delta H_{\text{f}}^{\theta} (\text{Products}) - \sum \Delta H_{\text{f}}^{\theta} (\text{Reactants})$$

Step-by-Step Example: Finding $\Delta H_{\text{R}}^{\theta}$ for: $$\text{CH}_4(\text{g}) + 2\text{O}_2(\text{g}) \xrightarrow{\Delta H_{\text{R}}^{\theta}} \text{CO}_2(\text{g}) + 2\text{H}_2\text{O}(\text{l})$$

1. Identify the known values: You will be given \(\Delta H_{\text{f}}^{\theta}\) for \(\text{CH}_4\), \(\text{CO}_2\), and \(\text{H}_2\text{O}\).
   Important: The \(\Delta H_{\text{f}}^{\theta}\) for elements in their standard state (\(\text{O}_2\)(\text{g}) here) is always zero.
2. Set up the cycle: The reactants are \(\text{CH}_4 + 2\text{O}_2\) and the products are \(\text{CO}_2 + 2\text{H}_2\text{O}\). The common state is \(\text{C} + 2\text{H}_2 + 2\text{O}_2\).
3. Apply the principle (Clockwise Path = Anti-clockwise Path):
   Path 1 (Anti-clockwise): Go from Reactants UP to the Elements. This is the reverse of forming \(\text{CH}_4\).
   Path 2 (Clockwise): Go from Elements DOWN to the Products. This is the formation of \(\text{CO}_2\) and \(2\text{H}_2\text{O}\).
   $$\Delta H_{\text{R}}^{\theta} + \Delta H_{\text{f}}^{\theta} (\text{Reactants}) = \Delta H_{\text{f}}^{\theta} (\text{Products})$$
   Rearranging gives the formula above.
4. Substitute and Calculate: Remember to multiply $\Delta H_{\text{f}}^{\theta}$ values by the stoichiometric coefficients from the balanced equation (e.g., \(2 \times \Delta H_{\text{f}}^{\theta} (\text{H}_2\text{O})\)).

3.3 Calculation Type 2: Using Enthalpies of Combustion ($\Delta H_{\text{c}}^{\theta}$)

We typically use this to find the enthalpy change of reaction, $\Delta H_{\text{R}}^{\theta}$, when $\Delta H_{\text{c}}^{\theta}$ values are known. This is very common for finding the \(\Delta H_{\text{f}}^{\theta}\) of fuels or organic compounds.

The Formula:
$$\Delta H_{\text{R}}^{\theta} = \sum \Delta H_{\text{c}}^{\theta} (\text{Reactants}) - \sum \Delta H_{\text{c}}^{\theta} (\text{Products})$$

Step-by-Step Example: Finding $\Delta H_{\text{R}}^{\theta}$ for: $$\text{C}(\text{s}) + 2\text{H}_2(\text{g}) \xrightarrow{\Delta H_{\text{f}}^{\theta}} \text{CH}_4(\text{g})$$

1. Identify the known values: You will be given \(\Delta H_{\text{c}}^{\theta}\) for \(\text{C}\), \(\text{H}_2\), and \(\text{CH}_4\).
2. Set up the cycle: The reactants are the elements (\(\text{C} + 2\text{H}_2\)) and the product is \(\text{CH}_4\). The common state is the combustion products (\(\text{CO}_2 + 2\text{H}_2\text{O}\)).
3. Apply the principle (Clockwise Path = Anti-clockwise Path):
   Path 1 (Clockwise): Go from Reactants DOWN to the Combustion Products. This is the sum of \(\Delta H_{\text{c}}^{\theta}\) of the reactants.
   Path 2 (Anti-clockwise): Go from Product UP from the Combustion Products. This is the reverse of the combustion of \(\text{CH}_4\).
   $$\Delta H_{\text{R}}^{\theta} + \Delta H_{\text{c}}^{\theta} (\text{Product}) = \Delta H_{\text{c}}^{\theta} (\text{Reactants})$$
   Rearranging gives the formula above.
4. Substitute and Calculate: Again, use stoichiometric coefficients.

Key Takeaway for Calculations: Whether you use formation or combustion data, Hess's law states that the enthalpy changes around the cycle must sum to zero, or, more simply, the total energy of one route must equal the total energy of the other route.

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4. Using Bond Energies (Syllabus 5.2(2b))

In addition to using Hess cycles with formation or combustion data, we can also calculate $\Delta H_{\text{R}}^{\theta}$ using average bond energies. This method is fundamentally an application of Hess's Law, as it treats the reaction as a two-step process:

1. All bonds in the reactants are broken (requires energy, \(\Delta H\) is positive).
2. New bonds in the products are formed (releases energy, \(\Delta H\) is negative).

4.1 Bond Energy Formula

The standard formula derived from this application of Hess's Law is:
$$\Delta H_{\text{reaction}} = \sum (\text{Energy required to break bonds}) - \sum (\text{Energy released forming bonds})$$

Or simply:
$$\Delta H_{\text{reaction}} = \sum \Delta H (\text{Bonds Broken}) - \sum \Delta H (\text{Bonds Formed})$$

Remember: Bond energies are always positive values (energy is required to break a bond). The sign difference in the calculation handles the fact that bond formation is exothermic.

Did you know?
Most bond energy data provided are average bond energies (Syllabus 5.1(6)). These are calculated from a range of different molecules. This is why calculations using bond energies are often less accurate than those using standard enthalpies of formation or combustion, which are specific values for that exact reaction.

Key Takeaway for Bond Energy: If you are asked to calculate $\Delta H_{\text{reaction}}$ using bond data, you must draw out the structure of the molecules to count all the bonds involved in both the reactants (energy in) and products (energy out).

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Summary: Applying Hess's Law

Step-by-Step Guide to Solving Hess's Law Problems

Don't worry about memorizing both calculation formulas immediately! Just follow these mechanical steps for any cycle question:

1. Write the Target Equation: Write the reaction you are trying to find $\Delta H$ for horizontally (Reactants $\rightarrow$ Products).
2. Identify Data Type: Are you given $\Delta H_{\text{f}}^{\theta}$ (Formation) or $\Delta H_{\text{c}}^{\theta}$ (Combustion)?
3. Draw the Cycle: Place the common intermediate (Elements for formation data; Products of combustion for combustion data) below or above the main reaction. Draw arrows for all known $\Delta H$ values pointing correctly (e.g., $\Delta H_{\text{f}}^{\theta}$ arrows always point away from elements).
4. Balance and Sum: Use the stoichiometric coefficients to multiply the given $\Delta H$ values.
5. Set up the Equation: Follow the path around the cycle. If you trace the unknown $\Delta H$ arrow from left to right, the total enthalpy change going down and back up (or around the other way) must be equal.
   $$ \text{Path 1 (Target } \Delta H) = \text{Path 2 (Sum of Known } \Delta H\text{'s)}$$
6. Calculate and Check Sign: Ensure your final answer has the correct sign (+ for endothermic, – for exothermic) and units (\(\text{kJ mol}^{-1}\)).

Hess's Law is a vital tool in energetics because it frees us from the constraints of direct measurement, allowing us to quantify energy changes for complex or impractical chemical processes. Master the cycles, and you master the calculations!