CHEMISTRY THE CENTRAL SCIENCE

5 THERMOCHEMISTRY

5.6 HESS'S LAW

It is often possible to calculate the ΔH for a reaction from the tabulated ΔH values of other reactions. Thus, it is not necessary to make calorimetric measurements for all reactions.

Because enthalpy is a state function, the enthalpy change, ΔH, associated with any chemical process depends only on the amount of matter that undergoes change and on the nature of the initial state of the reactants and the final state of the products. This means that whether a particular reaction is carried out in one step or in a series of steps, the sum of the enthalpy changes associated with the individual steps must be the same as the enthalpy change associated with the one-step process. As an example, combustion of methane gas, CH₄(g), to form CO₂(g) and H₂O(l) can be thought of as occurring in one step, as represented on the left in FIGURE 5.21, or in two steps, as represented on the right in Figure 5.21: (1) combustion of CH₄(g) to form CO₂(g) and H₂O(g) and (2) condensation of H₂O(g) to form H₂O(l). The enthalpy change for the overall process is the sum of the enthalpy changes for these two steps:

GO FIGURE

What process corresponds to the –88 kJ enthalpy change?

FIGURE 5.21 Enthalpy diagram for combustion of 1 mol of methane. The enthalpy change of the one-step reaction equals the sum of the enthalpy changes of the reaction run in two steps: –890 kJ = –802 kJ + (–88 kJ).

The net equation is

Hess's law states that if a reaction is carried out in a series of steps, ΔH for the overall reaction equals the sum of the enthalpy changes for the individual steps. The overall enthalpy change for the process is independent of the number of steps and independent of the path by which the reaction is carried out. This law is a consequence of the fact that enthalpy is a state function. We can therefore calculate ΔH for any process as long as we find a route for which ΔH is known for each step. This means that a relatively small number of experimental measurements can be used to calculate ΔH for a vast number of reactions.

Hess's law provides a useful means of calculating energy changes that are difficult to measure directly. For instance, it is impossible to measure directly the enthalpy for the combustion of carbon to form carbon monoxide. Combustion of 1 mol of carbon with 0.5 mol of O₂ produces both CO and CO₂, leaving some carbon unreacted. However, solid carbon and carbon monoxide can both be completely burned in O₂ to produce CO₂. We can therefore use the enthalpy changes of these reactions to calculate the heat of combustion of carbon.

GIVE IT SOME THOUGHT

What effect do these changes have on ΔH for a reaction:

a. reversing the reaction,

b. multiplying the coefficients of the equation for the reaction by 2?

SAMPLE EXERCISE 5.8 Using Hess's Law to Calculate Δ H

The enthalpy of reaction for the combustion of C to CO₂ is −393.5 kJ/mol C, and the enthalpy for the combustion of CO to CO₂ is −283.0 kJ/mol CO:

Using these data, calculate the enthalpy for the combustion of C to CO:

SOLUTION

Analyze We are given two thermochemical equations, and our goal is to combine them in such a way as to obtain the third equation and its enthalpy change.

Plan We will use Hess's law. In doing so, we first note the numbers of moles of substances among the reactants and products in the target equation, (3). We then manipulate equations (1) and (2) to give the same number of moles of these substances, so that when the resulting equations are added, we obtain the target equation. At the same time, we keep track of the enthalpy changes, which we add.

Solve To use equations (1) and (2), we arrange them so that C(s) is on the reactant side and CO(g) is on the product side of the arrow, as in the target reaction, equation (3). Because equation (1) has C(s) as a reactant, we can use that equation just as it is. We need to turn equation (2) around, however, so that CO(g) is a product. Remember that when reactions are turned around, the sign of ΔH is reversed. We arrange the two equations so that they can be added to give the desired equation:

When we add the two equations, CO₂(g) appears on both sides of the arrow and therefore cancels out. Likewise, is eliminated from each side.

PRACTICE EXERCISE

Carbon occurs in two forms, graphite and diamond. The enthalpy of the combustion of graphite is –393.5 kJ/mol, and that of diamond is −395.4 kJ/mol:

Calculate ΔH for the conversion of graphite to diamond:

Answer: +1.9 kJ

SAMPLE EXERCISE 5.9 Using Three Equations with Hess's Law to Calculate ΔH

Calculate ΔH for the reaction

given the following chemical equations and their respective enthalpy changes:

SOLUTION

Analyze We are given a chemical equation and asked to calculate its ΔH using three chemical equations and their associated enthalpy changes.

Plan We will use Hess's law, summing the three equations or their reverses and multiplying each by an appropriate coefficient so that they add to give the net equation for the reaction of interest. At the same time, we keep track of the ΔH values, reversing their signs if the reactions are reversed and multiplying them by whatever coefficient is employed in the equation.

Solve Because the target equation has C₂H₂ as a product, we turn the first equation around; the sign of ΔH is therefore changed. The desired equation has 2 C(s) as a reactant, so we multiply the second equation and its ΔH by 2. Because the target equation has H₂ as a reactant, we keep the third equation as it is. We then add the three equations and their enthalpy changes in accordance with Hess's law:

When the equations are added, there are 2 CO₂, O₂, and H₂O on both sides of the arrow. These are canceled in writing the net equation.

Check The procedure must be correct because we obtained the correct net equation. In cases like this you should go back over the numerical manipulations of the ΔH values to ensure that you did not make an inadvertent error with signs.

PRACTICE EXERCISE

Calculate ΔH for the reaction

given the following information:

Answer: –304.1 kj

The key point of these examples is that H is a state function, so for a particular set ofreactants and products, ΔH is the same whether the reaction takes place in one step or in a series of steps. We reinforce this point by giving one more example of an enthalpy diagram and Hess's law. Again we use combustion of methane to form CO₂ and H₂O, our reaction from Figure 5.21. This time we envision a different two-step path, with the initial formation of CO, which is then combusted to CO₂ (FIGURE 5.22). Even though the two-step path is different from that in Figure 5.21, the overall reaction again has ΔH₁ = – 890 kJ. Because H is a state function, both paths must produce the same value of ΔH. In Figure 5.22, that means ΔH₁ = ΔH₂ + ΔH₃. We will soon see that breaking up reactions in this way allows us to derive the enthalpy changes for reactions that are hard to carry out in the laboratory.

GO FIGURE

Suppose the overall reaction were modified to produce 2 H₂O(g) rather than 2 H₂O(l). Would any of the values of ΔH in the diagram stay the same?

FIGURE 5.22 Enthalpy diagram illustrating Hess's law. The net reaction is the same as in Figure 5.21, but here we imagine different reactions in our two-step version. As long as we can write a series of equations that add up to the equation we need, and as long as we know a value for ΔH for all intermediate reactions, we can calculate the overall ΔH.