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Now that we understand that chemical reactions occur with a simultaneous change in energy, we can apply the concept more broadly. To start, remember that some chemical reactions are rather difficult to perform. For example, consider the combustion of carbon to make carbon monoxide:

[latex]\ce{2C(s)}+\ce{O2(g)}\rightarrow \ce{2CO(g)}\quad \Delta H=?[/latex]

In reality, this is extremely difficult to do; given the opportunity, carbon will react to make another compound, carbon dioxide:

[latex]\ce{2C(s)}+\ce{O2(g)}\rightarrow \ce{2CO2(g)}\quad \Delta H=-393.5\text{ kJ}[/latex]

Is there a way around this? Yes. It comes from the understanding that chemical equations can be treated like algebraic equations, with the arrow acting like the equals sign. Like algebraic equations, chemical equations can be combined, and if the same substance appears on both sides of the arrow, it can be cancelled out (much like a spectator ion in ionic equations). For example, consider these two reactions:

[latex]\begin{array}{rrl} \ce{2C(s)}+\ce{2O2(g)}&\rightarrow&\ce{2CO2(g)} \\ \\ \ce{2CO2(g)}&\rightarrow&\ce{2CO(g)}+\ce{O2(g)} \end{array}[/latex]

If we added these two equations by combining all the reactants together and all the products together, we would get:

[latex]\ce{2C(s)}+\ce{2O2(g)}+\ce{2CO2(g)}\rightarrow \ce{2CO2(g)}+\ce{2CO(g)}+\ce{O2(g)}[/latex]

We note that 2CO₂(g) appears on both sides of the arrow, so they cancel:

[latex]\ce{2C(s)}+\ce{2O2(g)}+\cancel{\ce{2CO2(g)}}\rightarrow \cancel{\ce{2CO2(g)}}+\ce{2CO(g)}+\ce{O2(g)}[/latex]

We also note that there are 2 mol of O₂ on the reactant side, and 1 mol of O₂ on the product side. We can cancel 1 mol of O₂ from both sides:

[latex]\ce{2C(s)}+\ce{\cancel{2}O2(g)}\rightarrow \ce{2CO(g)}+\cancel{\ce{O2(g)}}[/latex]

What do we have left?

[latex]\ce{2C(s)}+\ce{O2(g)}\rightarrow \ce{2CO(g)}[/latex]

This is the reaction we are looking for! So by algebraically combining chemical equations, we can generate new chemical equations that may not be feasible to perform.

What about the enthalpy changes? Hess’s law states that when chemical equations are combined algebraically, their enthalpies can be combined in exactly the same way. Two corollaries immediately present themselves:

1. If a chemical reaction is reversed, the sign on ΔH is changed.
2. If a multiple of a chemical reaction is taken, the same multiple of the ΔH is taken as well.

What are the equations being combined? The first chemical equation is the combustion of C, which produces CO₂:

[latex]\ce{2C(s)}+\ce{2O2(g)}\rightarrow \ce{2CO2(g)}[/latex]

This reaction is two times the reaction to make CO₂ from C(s) and O₂(g), whose enthalpy change is known:

[latex]\ce{C(s)}+\ce{O2(g)}\rightarrow \ce{CO2(g)}\quad \Delta H=-393.5\text{ kJ}[/latex]

According to the first corollary, the first reaction has an energy change of two times −393.5 kJ, or −787.0 kJ:

[latex]\ce{2C(s)}+\ce{2O2(g)}\rightarrow \ce{2CO2(g)}\quad \Delta H=-787.0\text{ kJ}[/latex]

The second reaction in the combination is related to the combustion of CO(g):

[latex]\ce{2CO(g)}+\ce{O2(g)}\rightarrow \ce{2CO2(g)}\quad \Delta H=-566.0\text{ kJ}[/latex]

The second reaction in our combination is the reverse of the combustion of CO. When we reverse the reaction, we change the sign on the ΔH:

[latex]\ce{2CO2(g)}\rightarrow \ce{2CO(g)}+\ce{O2(g)}\quad \Delta H=+566.0\text{ kJ}[/latex]

Now that we have identified the enthalpy changes of the two component chemical equations, we can combine the ΔH values and add them:

[latex]\begin{array}{ll} \begin{array}{rrl} \ce{2C(s)}+\ce{\cancel{2}O2(g)}&\rightarrow&\cancel{\ce{2CO2(g)}} \\ \cancel{\ce{2CO2(g)}}&\rightarrow&\ce{2CO(g)}+\cancel{\ce{O2(g)}} \\ \midrule \ce{2C(s)}+\ce{O2(g)}&\rightarrow&\ce{2CO(g)} \end{array} & \begin{array}{rrl} \Delta H&=&-787.0\text{ kJ} \\ \Delta H&=&+566.0\text{ kJ} \\ \Delta H&=&-787.0+566.0\text{ kJ}=-221.0\text{ kJ} \end{array} \end{array}[/latex]

Hess’s law is very powerful. It allows us to combine equations to generate new chemical reactions whose enthalpy changes can be calculated, rather than directly measured.