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# 20.5: Calculating Free Energy Change (ΔG°)

Difficulty Level: At Grade Created by: CK-12
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Practice Calculating Free Energy Change

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#### Time for dessert!

When you are baking something, you heat the oven to the temperature indicated in the recipe. Then you mix all the ingredients, put them in the proper baking dish, and place them in the oven for a specified amount of time. If you had mixed the ingredients and left them out at room temperature, not much would change. The materials need to be heated to a given temperature for a set time in order for the ingredients to react with one another and produce a delicious final product.

### Calculating Free Energy(ΔG∘)\begin{align*}(\Delta G^\circ)\end{align*}

The free energy change of a reaction can be calculated using the following expression:

ΔG=ΔHTΔS\begin{align*}\Delta G^\circ=\Delta H^\circ - T\Delta S^\circ\end{align*}

where ΔG=free energy change (kJ/mol)\begin{align*}\Delta G = \text{free energy change (kJ/mol)}\end{align*}

ΔH=change in enthalpy (kJ/mol)\begin{align*}\Delta H = \text{change in enthalpy (kJ/mol)}\end{align*}

ΔS=change in entropy (J/Kmol)\begin{align*}\Delta S = \text{change in entropy (J/K} \cdot \text{mol)}\end{align*}

T=temperature (Kelvin)\begin{align*}T = \text{temperature (Kelvin)}\end{align*}

Note that all values are for substances in their standard state. In performing calculations, it is necessary to change the units for ΔS\begin{align*}\Delta S\end{align*} to kJ/K • mol, so that the calculation of ΔG\begin{align*}\Delta G\end{align*} is in kJ/mol.

#### Sample Problem: Gibbs Free Energy

Methane gas reacts with water vapor to produce a mixture of carbon monoxide and hydrogen according to the balanced equation below.

CH4(g)+H2O(g)CO(g)+3H2(g)\begin{align*}\text{CH}_4(g)+\text{H}_2\text{O}(g) \rightarrow \text{CO}(g)+3\text{H}_2(g)\end{align*}

The ΔH\begin{align*}\Delta H^\circ\end{align*} for the reaction is +206.1 kJ/mol, while the ΔS\begin{align*}\Delta S^\circ\end{align*} is +215 J/K • mol. Calculate the ΔG\begin{align*}\Delta G^\circ\end{align*} at 25°C and determine if the reaction is spontaneous at that temperature.

Step 1: List the known values and plan the problem.

Known

• ΔH=206.1 kJ/mol\begin{align*}\Delta H^\circ =206.1 \ \text{kJ/mol}\end{align*}
• ΔS=215 J/Kmol=0.215 kJ/Kmol\begin{align*}\Delta S^\circ = 215 \ \text{J/K}\cdot \text{mol}=0.215 \ \text{kJ/K}\cdot \text{mol}\end{align*}
• T=25C=298 K\begin{align*}T =25^\circ \text{C}=298 \ \text{K}\end{align*}

Unknown

• ΔG=? kJ/mol\begin{align*}\Delta G^\circ =? \ \text{kJ/mol}\end{align*}

Prior to substitution into the Gibbs free energy equation, the entropy change is converted to kJ/K • mol and the temperature to Kelvins.

Step 2: Solve.

ΔG=ΔHTΔS=206.1 kJ/mol298 K(0.215 kJ/Kmol)=+142.0 kJ/mol\begin{align*}\Delta G^\circ =\Delta H^\circ - T\Delta S^\circ = 206.1 \ \text{kJ/mol} - 298 \ \text{K}(0.215 \ \text{kJ/K}\cdot \text{mol})=+142.0 \ \text{kJ/mol}\end{align*}

The resulting positive value of ΔG\begin{align*}\Delta G\end{align*} indicates that the reaction is not spontaneous at 25°C.

The unfavorable driving force of increasing enthalpy outweighed the favorable increase in entropy. The reaction will be spontaneous only at some elevated temperature.

Available values for enthalpy and entropy changes are generally measured at the standard conditions of 25°C and 1 atm pressure. The values are slightly temperature dependent and so we must use caution when calculating specific ΔG\begin{align*}\Delta G\end{align*} values at temperatures other than 25°C. However, since the values for ΔH\begin{align*}\Delta H\end{align*} and ΔS\begin{align*}\Delta S\end{align*} do not change a great deal, the tabulated values can safely be used when making general predictions about the spontaneity of a reaction at various temperatures.

### Summary

• Calculations of free energy changes are described.

### Review

1. What would happen to ΔH\begin{align*}\Delta H\end{align*} if you forgot to change the units for ΔS\begin{align*}\Delta S\end{align*} to kJ/K • mol?
2. What are standard conditions for enthalpy and entropy changes?
3. At what temperature would the reaction become spontaneous?

### Explore More

Use the resource below to answer the questions that follow.

1. Why is ΔH\begin{align*}\Delta H\end{align*} negative in this example?
2. What would happen if you forgot to change the sign of the TΔS\begin{align*}T\Delta S\end{align*} value in the first calculation?
3. What indicates that the reaction is spontaneous?

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