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# Calculating Enthalpy Change

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Which fuel provides the most energy - one mole of ammonia, hydrogen gas, methane, or glucose? You could try and work this out by burning all the fuels and measuring the temperature change of the surroundings. This gives you an idea of the energy each fuel releases. However, this particular method has lots of room for error. Instead, why not try mathematically calculating enthalpy change using Hess' law?

• We'll start by defining enthalpy change before giving its units.
• We'll then recap certain types of enthalpy change.
• After that, we'll learn how you calculate enthalpy change using Hess' law.
• You'll first find out the method, then you'll be able to apply your knowledge with our worked examples.
• We include questions that use enthalpy of formation, enthalpy of combustion, and mean bond energy.

## Calculating enthalpy change: definition, units, and types

Before we go any further, there are two important questions we need to answer:

1. What is enthalpy change?
2. What are standard enthalpy changes?

Enthalpy change (∆H) is the amount of heat energy transferred during a chemical reaction at constant pressure. We measure it kJ mol-1.

Standard enthalpy changes (∆Hº) are enthalpy changes measured under standard conditions with the reactants and products in their standard states.

(Units: kJ mol-1)

### Types of enthalpy change

Although there are many types of enthalpy change, today we will discuss about standard enthalpy changes of reaction, formation, and combustion.

The standard enthalpy change of the reaction (∆rHº) is the enthalpy change when reactants form products in quantities given in the balanced chemical equation, under standard conditions and with all species in their standard states. Simply put, it is the enthalpy change that accompanies any standard reaction.

The standard enthalpy change of formation (∆fH°) is the enthalpy change when one mole of a species is formed from its elements, under standard conditions and with all species in their standard states.

The standard enthalpy change of combustion (∆cH°) is the enthalpy change when one mole of a substance burns completely in oxygen under standard conditions, with all species in their standard states.

Find out more about enthalpy changes, further definitions and examples, over at Enthalpy Changes.

## Calculating enthalpy change using Hess' law

It is impossible to measure enthalpy change directly. Instead, we can calculate it in several ways. One way of doing this is by using Hess' law.

Hess' law states that the enthalpy change of a reaction is independent of the route taken.

Simply put, the enthalpy change of a reaction is always the same, no matter how many steps the reaction takes, which other species you go by, or how you get from start to finish.

Consider a reactant A whose product is B. You can start with A and end up with B in just one simple reaction-the direct route, or you can go through several different reactions ( A to Z, Z to Y, Y to X and X to B) to arrive at B- the indirect route. Either way, the enthalpy change remains the same since both the routes start with the same reactant and end with the same product.

We can represent the two routes on a diagram known as a Hess' cycle.

Fig. 1 - An example of Hess' cycle. The direct and indirect routes are labelled accordingly.StudySmarter Originals

How does this help us?

We may not know the enthalpy change of the direct route, but we know all the enthalpy changes involved in the indirect route which can be used to calculate the enthalpy change of the direct route

In the above example, the direct route (going from A to B) is represented by the enthalpy change ΔH1. The indirect route (going via Z, Y, and X) is represented by the enthalpy changes ΔH2, ΔH3, ΔH4, and ΔH5. Hess' law tells us that ΔH1 = ΔH2 + ΔH3 + ΔH4 + ΔH5.

Visit Hess' Law for a comprehensive exploration of the topic. You can also learn about another method of calculating enthalpy change in the article Measuring Enthalpy Changes. This involves using a technique known as calorimetry to measure the heat energy transferred during a reaction, which we referred to in the introduction.

## Calculating enthalpy change of reaction

We will now see how Hess' law is used in calculating enthalpy change.

Hess's law allows us to calculate the unknown enthalpy change of any reaction, provided we can represent it in terms of enthalpy changes that we do know. At AS-level chemistry, this is typically limited to working out the standard enthalpy of reaction (∆rHº) using standard enthalpy of formation (∆fHº) and standard enthalpy of combustion (∆cHº).

The direct route of your Hess's cycle is always the reaction that you want to find the enthalpy change of. However, your indirect route looks slightly different depending on whether you use ∆fHº or ∆cHº:

• If you wish to calculate ∆rHº using ∆fHº, your indirect route goes via the standard state elements that make up all the species in the reaction.
• If you wish to calculate ∆rHº using ∆cHº, your indirect route goes via the combustion products of all the species in the reaction.

For example, consider the hydrogenation of ethene: $$C_2H_4(g)+H_2(g)\rightarrow C_2H_6(g)$$

Here's what the two Hess's cycles look like:

Fig. 2 - Hess' cycles for the hydrogenation of ethene. The left-hand cycle uses enthalpy of formation whilst the right-hand cycle uses enthalpy of combustion.StudySmarter Originals

Note the differences between the two cycles. For example, pay attention to the directions of the arrows. Also, note that we don't need to include the enthalpy of formation of H2(g). This is because H2 is elemental and already in its standard state, so its ∆fHº has a value of +0.0 kJ mol-1.

### Method for calculating enthalpy change

Calculating an unknown enthalpy change of reaction using Hess' cycles:

1. Decide whether you are going to use enthalpy of formation or enthalpy of combustion to find the unknown enthalpy change of reaction.
2. Construct a Hess' cycle using the reaction's balanced chemical equation. This involves the direct route, which is the unknown enthalpy change of reaction you are trying to find out, and the indirect route, which goes via elemental forms or combustion products.
3. Using either ∆fHº or ∆cHº given in the question or in a data table, fill in the enthalpy changes involved in the indirect route.
4. Use the indirect route to calculate the enthalpy change of the direct route.

Here's a formula that might help you. However, some people prefer to draw a Hess' cycle out each time they try a question - experiment with different methods to find out what works for you.

In general:

$$\Delta _rH^\circ = -\Delta _fH^\circ (reactants) + -\Delta _fH^\circ (products)$$

$$\Delta _rH^\circ = \Delta _cH^\circ (reactants) -\Delta _cH^\circ (products)$$

### Example

Ready to give calculating enthalpy change of the reaction- ∆rHº a go?

Try this first example. It uses standard enthalpies of formation to find the standard enthalpy of reaction.

Use the following information to calculate ∆rHº of the forward reaction for the equilibrium given below:

$$CO(g)+2H_2(g)\rightleftharpoons CH_3OH(l)$$

Solution:

The question asks us to calculate the enthalpy of reaction for a hypothetical case of methanol production. It gives us some enthalpies of formation, and we'll use these to find the unknown enthalpy change.

Let's draw a Hess' cycle.

The direct route : CO and H2 to CH3OH.

The indirect route: CO and H2 to their standard state elements, and from the elements to CH3OH.

Note that this cycle features just one mole each of CO and CH3OH. In addition, note that we don't have to include the ∆fHº of H2 - it is an elemental species in its standard state, and so has an enthalpy of formation of +0 kJ mol-1.

Fig. 3: Hess' cycle for Methanol production | StudySmarter Originals

We can now fill in the enthalpy changes of the indirect route:

Fig. 4: Hess' Cycle for methanol production with enthalpy of formations for indirect route filled in | StudySmarter Originals

Now, calculate the unknown enthalpy change of the direct route by adding together the enthalpy changes of the indirect route. This is the reverse of the enthalpy of formation of CO, added to the enthalpy of formation of CH3OH:

$$-(-110.5)+(-238.0)=-90.5\space kJ\space mol^{-1}$$

Our final answer is -127.5 kJ mol-1.

The next problem instead uses enthalpies of combustion to find an enthalpy of formation. We solve it using the same method as for our first example problem.

Use the following data to calculate the standard enthalpy of formation of propanol (C3H7OH):

Solution:We're given standard enthalpies of combustion and told to use them to find the standard enthalpy of formation of propane. First, we need to represent the unknown enthalpy change with a chemical equation, which we can then find the enthalpy change of reaction for:$$3C(s)+4H_2(g)+\frac{1}{2} O_2(g)\rightarrow C_3H_7OH(l)$$ Now, we can draw a Hess' cycle using the enthalpy changes of combustion given in the question:

Fig. 5: Hess' cycle for the formation of propanol using enthalpy of combustion, with the enthalpy changes within the indirect route filled in. StudySmarter Originals

If we follow the indirect route along, we can calculate ∆rHº, the enthalpy change of the direct route:

$$(3\times -393.5)+(4\times -285.8)-(-2021)=-302.7\space kJ\space mol^{-1}$$

Therefore, the enthalpy of formation of propanol is -302.7 kJ mol-1.

## Calculating enthalpy change from bond energies

Hess' cycles don't just have to involve calculating ∆rHº from ∆fHº or ∆cHº. We can also calculate unknown enthalpy changes using mean bond energy.

Mean bond energy (also known as mean bond enthalpy) is the average energy required to break one mole of a specific covalent bond in the gaseous phase, averaged over a range of molecules.

Once again, the direct route in our Hess' cycle goes directly from reactants to products and represents the unknown enthalpy change. But this time, the indirect route goes via all the unbonded gaseous atoms that make up the species in the equation. When filling in the enthalpy changes of the indirect route, we consider the bond energy of all the covalent bonds in the reactants and products, and calculate the enthalpy change of the direct route accordingly.

Consider the hydrogenation of ethene once again. Here's our Hess' cycle:

Fig. 6 - Hess' cycle for the hydrogenation of ethene using mean bond energy.StudySmarter Originals

### Method

Here's how you calculate enthalpy change using bond energy:

• Construct a Hess's cycle using the reaction's balanced chemical equation. This involves the direct route, which is the unknown enthalpy change of reaction you are trying to find out, and the indirect route, which goes via unbonded atoms.
• Using mean bond energies given in the question or in a data table, fill in the enthalpy changes involved in the indirect route.
• Use the indirect route to calculate the direct route: the unknown enthalpy change of reaction.

### Example

Let's put the method into practice. Try the following question and compare your answer to our worked solution.

Calculate the enthalpy change of the hydrogenation of ethene, shown below:

$$C_2H_4(g)+H_2(g)\rightarrow C_2H_6(g)$$

Solution:

You'll notice that this is the same reaction that we considered when we introduced the topic of mean bond energy. We've already drawn a Hess' cycle - we just need to fill in the enthalpy changes using the data table given:

Fig. 7 - Hess' cycle for the hydrogenation of ethene using mean bond energy, with the known enthalpy changes within the indirect route filled in.StudySmarter Originals

Now, we can follow the indirect route along to calculate the enthalpy change of the direct route. We need the bond energies of the reactants, added to the reverse of the bond energies of the products:

$$(4\times 412)+614+436-(348+(6\times 412))=-122\space kJ\space mol^{-1}$$

Our final answer is -122 kJ mol-1.

This next problem is slightly different. Here, we work backward and calculate one particular bond energy using a known enthalpy change of reaction, and other mean bond energies.

Calculate the mean bond energy of the N-H bond, using the following information:

$$N_2(g)+3H_2(g)\rightarrow 2NH_3(g)\qquad \Delta H^\circ =-91.8\space kJ\space mol^{-1}$$

Solution:

We start by drawing a Hess' cycle. But instead of trying to find an enthalpy change of reaction, here we want to find a particular bond energy: the N-H bond energy. We are given the enthalpy change of reaction for the production of NH3, and the N≡N and H-H bond energies. The direct route of our Hess' cycle goes from NH3 to unbonded N and H atoms. The indirect route goes from NH3 to N2 and H2 (using ∆rHº), and from N2 and H2 to unbonded atoms (using N≡N and H-H bond energies). Here is our Hess' cycle, with known enthalpy changes filled in:

Fig. 8 - Hess' cycle for ammonia production using mean bond energy. Known enthalpy changes are filled in.StudySmarter Originals

$$-(-91.8)+(941+3(436))=2340.8\space kJ\space mol^{-1}$$

This gives us the enthalpy change for the direct route. However, note that the indirect route tells us the enthalpy change when six moles of N-H bonds are broken. The question asks us to find the bond energy of N-H, which is the enthalpy change when just one mole of N-H bonds is broken. Therefore, we divide the enthalpy change of the direct route by 6:

$$\frac{2340.8}{6} =390.13\space kJ\space mol^{-1}$$

Our final answer is 390 kJ mol-1.

Make sure to give your answer to the correct number of significant figures. Here, the smallest number of significant figures used in the question is three, and so we give our answer to three significant figures .

## Calculating Enthalpy Change - Key takeaways

• Enthalpy change (∆H) is the amount of heat energy transferred during a chemical reaction at constant pressure. We typically measure it kJ mol-1.
• We can calculate enthalpy change using Hess' law. This states that the enthalpy change of a reaction is independent of the route taken. It allows us to calculate an unknown enthalpy change using known enthalpy changes.
• To calculate an unknown enthalpy change:
• Construct a Hess' cycle using the reaction's balanced chemical equation. This involves the direct route, which is the unknown enthalpy change of reaction you are trying to find out, and the indirect route, which involves known enthalpy changes.
• Use data given in the question to fill in the enthalpy changes involved in the indirect route.
• Use the indirect route to calculate the enthalpy change of the direct route.
• Calculating enthalpy change often involves working with standard enthalpy of reaction, formation, and combustion, as well as mean bond energy.

You calculate enthalpy change using either a calorimetry experiment or Hess' law. Hess' law tells you that the enthalpy change of a reaction is always the same, independent of the route taken. This means that you can use an indirect route involving known enthalpy changes to calculate the unknown enthalpy change of the direct route of a reaction.

Hess' law calculations are mathematical calculations used to find an unknown enthalpy change. You do this by representing a chemical reaction using a direct route and an indirect route. Hess' law tells you that the enthalpy change of a reaction is always the same, independent of the route taken. This means that if you know all the enthalpy changes involved in the indirect route, you can use them to calculate the unknown enthalpy change of the direct route.

You calculate enthalpy change in an experiment using calorimetry. This involves measuring the heat energy transferred during a certain chemical reaction. Check out the article Calorimetry for a more detailed look at the technique.

Hess' law tells you that the enthalpy change of a reaction is always the same, independent of the route taken. You calculate enthalpy changes using Hess' law by representing a reaction with an unknown enthalpy change using a direct route and an indirect route. Provided that you know all the enthalpy changes involved in the indirect route, you can use them to calculate the unknown enthalpy change of the direct route.

The unit of enthalpy change is kJ mol-1.

## Calculating Enthalpy Change Quiz - Teste dein Wissen

Question

What is enthalpy change?

The amount of heat energy transferred during a chemical reaction at constant pressure.

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Question

What is the symbol for enthalpy change?

∆H

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Question

What are the units of enthalpy change?

kJ mol-1

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Question

What is Hess' law?

A law used in chemistry which states that the enthalpy change of a reaction is independent of the route taken.

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Question

Outline how you can use Hess' law to calculate an unknown enthalpy change.

• Construct a Hess' cycle using the reaction's balanced chemical equation. This involves the direct route, which is the unknown enthalpy change of reaction you are trying to find out, and the indirect route, which involves known enthalpy changes.
• Use data given in the question to fill in the enthalpy changes involved in the indirect route.
• Use the indirect route to calculate the enthalpy change of the direct route.

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Question

Which of the following are true about Hess' cycles involving standard enthalpy of formation?

The indirect route features standard state elements.

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Question

Which of the following are true about Hess' cycles involving standard enthalpy of combustion?

The arrows showing c point away from the direct route's reactants and products.

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Question

A reaction can be represented by a direct route and an indirect route:

• The enthalpy change of the direct route is H1.
• The sum of the enthalpy changes of the indirect route is +145 kJ mol-1.

What can you conclude?

H1 = +145 kJ mol-1

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Question

Write an expression for r using fHº(reactants) and rHº(products).

r = -fHº(reactants) + rHº(products)

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Question

Write an expression for r using cHº(reactants) and cHº(products).

r = cHº(reactants) - cHº(products)

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Question

Write an expression for r using mean bond energy (reactants) and mean bond energy (products).

r = mean bond energy (reactants) - mean bond energy (products)

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Question

True or false? To calculate enthalpy change using Hess' law, you need to know all the enthalpy changes involved in the indirect route.

True

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