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# 20.2: Spontaneous Reactions and Free Energy

Difficulty Level: At Grade Created by: CK-12

## Lesson Objectives

• Describe the meaning of a spontaneous reaction in terms of enthalpy and entropy changes.
• Define free energy and calculate the change in free energy for a reaction using known values of the changes in enthalpy and entropy.
• Determine the spontaneity of a reaction based on the value of its change in free energy at various temperatures.

## Lesson Vocabulary

• free energy
• non-spontaneous reaction
• spontaneous reaction

### Recalling Prior Knowledge

• Is an increase or decrease in enthalpy more favorable for a reaction?
• Is an increase or decrease in entropy more favorable for a reaction?

The change in enthalpy and change in entropy of a reaction are the driving forces behind all chemical reactions. In this lesson, we will examine a new function called free energy, which combines enthalpy and entropy and can be used to determine whether or not a given reaction will occur.

## Spontaneous Reactions

Reactions are favorable when they result in a decrease in the enthalpy and an increase in the entropy of the system. When both of these conditions are met, the reaction is said to be spontaneous at all temperatures. A spontaneous reaction is a reaction that favors the formation of products at the conditions under which the reaction is occurring. A roaring bonfire (Figure below) is an example of a spontaneous reaction. A fire is certainly exothermic, which means a decrease in the energy of the system as energy is released to the surroundings as heat. The products of a fire are composed mostly of gases such as carbon dioxide and water vapor, so the entropy of the system increases during most combustion reactions. This combination of a decrease in energy and an increase in entropy means that combustion reactions occur spontaneously.

Combustion reactions such as this fire are spontaneous reactions. Once the reaction begins, it continues on its own until one of the reactants (fuel or oxygen) is gone.

A nonspontaneous reaction is a reaction that does not favor the formation of products at the given set of conditions. In order for a reaction to be nonspontaneous, one or both of the driving forces must favor the reactants over the products. In other words, the reaction is endothermic, is accompanied by a decrease in entropy, or both. Our atmosphere is composed primarily of a mixture of nitrogen and oxygen gases. One could write an equation showing these gases undergoing a chemical reaction to form nitrogen monoxide.

N2(g) + O2(g) → 2NO(g)

Fortunately, this reaction is nonspontaneous at normal temperatures and pressures. It is a highly endothermic reaction with a slightly positive entropy change (ΔS). However, nitrogen monoxide is capable of being produced at very high temperatures, and this reaction has been observed to occur as a result of lightning strikes.

One must be careful not to confuse the term spontaneous with the notion that a reaction occurs rapidly. A spontaneous reaction is one in which product formation is favored, even if the reaction is extremely slow. You do not have to worry about a piece of paper on your desk suddenly bursting into flames, although its combustion is a spontaneous reaction. What is missing is the required activation energy to get the reaction started. If the paper were to be heated to a high enough temperature, it would begin to burn, at which point the reaction would proceed spontaneously until completion.

In a reversible reaction, one reaction direction may be favored over the other. Carbonic acid is present in carbonated beverages. It decomposes spontaneously to carbon dioxide and water according to the following reaction.

H2CO3(aq) CO2(g) + H2O(l)

If you were to start with pure carbonic acid in water and allow the system to come to equilibrium, more than 99% of the carbonic acid would be converted into carbon dioxide and water. The forward reaction is spontaneous because the products of the forward reaction are favored at equilibrium. In the reverse reaction, carbon dioxide and water are the reactants and carbonic acid is the product. When carbon dioxide is bubbled into water (Figure below), less than 1% is converted to carbonic acid when the reaction reaches equilibrium. The reverse reaction as written above is not spontaneous. This illustrates another important point about spontaneity. Just because a reaction is not spontaneous does not mean that it does not occur at all. Rather, it means that the reactants will be favored over the products at equilibrium, even though some products may indeed form.

A home soda making machine is shown with a bottle of water and a CO2 cartridge. When the water is carbonated, only a small amount of carbonic acid is formed because the reaction is nonspontaneous.

## Gibbs Free Energy

Many chemical reactions and physical processes release energy that can be used to do other things. When the fuel in a car is burned, some of the released energy is used to power the vehicle. Free energy is energy that is available to do work. Spontaneous reactions release free energy as they proceed. Recall that the determining factors for spontaneity of a reaction are the enthalpy and entropy changes that occur for the system. The free energy change of a reaction is a mathematical combination of the enthalpy change and the entropy change.

ΔG° = ΔH°-TΔS°

The symbol for free energy is G, in honor of American scientist Josiah Gibbs (1839-1903), who made many contributions to thermodynamics. The change in Gibbs free energy is equal to the change in enthalpy minus the mathematical product of the change in entropy multiplied by the Kelvin temperature. Each thermodynamic quantity in the equation is for substances in their standard states, as indicated by the ° superscripts. While ΔH and ΔG values are generally reported in units of kJ/mol, ΔS is often reported in J/K•mol. Before using this equation, the value for ΔS must be converted to kJ/K•mol so that the units work out correctly.

A spontaneous reaction is one that releases free energy, and so the sign of ΔG must be negative. Since both ΔH and ΔS can be either positive or negative, depending on the characteristics of the particular reaction, there are four different possible combinations. The outcomes for ΔG based on the signs of ΔH and ΔS are outlined in Table below.

Enthalpy, Entropy, and Free Energy Changes
ΔH ΔS ΔG
− value (exothermic) + value (disordering) always negative
+ value (endothermic) + value (disordering) negative at higher temperatures
− value (exothermic) − value (ordering) negative at lower temperatures
+ value (endothermic) − value (ordering) never negative

Keep in mind that the temperature in the Gibbs free energy equation is the Kelvin temperature, so it can only have a positive value. When ΔH is negative and ΔS is positive, the sign of ΔG will always be negative, and the reaction will be spontaneous at all temperatures. This corresponds to both driving forces being in favor of product formation. When ΔH is positive and ΔS is negative, the sign of ΔG will always be positive, and the reaction can never be spontaneous. This corresponds to both driving forces working against product formation.

When one driving force favors the reaction, but the other does not, it is the temperature that determines the sign of ΔG. Consider first an endothermic reaction (positive ΔH) that also displays an increase in entropy (positive ΔS). It is the entropy term that favors the reaction. Therefore, as the temperature increases, the TΔS term in the Gibbs free energy equation will begin to predominate and ΔG will become negative. A common example of a process which falls into this category is the melting of ice (Figure below). At a relatively low temperature (below 273 K), the melting is not spontaneous because the positive ΔH term “outweighs” the TΔS term. When the temperature rises above 273 K, the process becomes spontaneous because the larger T value has tipped the sign of ΔG over to being negative.

Ice melts spontaneously only when the temperature is above 0°C. The increase in entropy is then able to drive the unfavorable endothermic process.

When the reaction is exothermic (negative ΔH), but undergoes a decrease in entropy (negative ΔS), it is the enthalpy term which favors the reaction. In this case, a spontaneous reaction is dependent upon the TΔS term being small relative to the ΔH term so that ΔG is negative. The freezing of water is an example of this type of process. It is spontaneous only at a relatively low temperature. Above 273 K, the larger TΔS value causes the sign of ΔG to be positive and freezing does not occur.

Sample Problem 20.1: 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)

ΔH° for the reaction is +206.1 kJ/mol, while ΔS° is +215 J/K•mol. Calculate ΔG° for this reaction at 25°C and determine whether it is spontaneous at that temperature.

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

Known

• ΔH° = 206.1 kJ/mol
• ΔS° = 215 J/K•mol = 0.215 kJ/K•mol
• T = 25°C = 298 K

Unknown

• ΔG° = ? kJ/mol

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° = ΔH° - TΔS° = 206.1 kJ/mol - 298 K(0.215 kJ/K•mol)
ΔG° = +142.0 kJ/mol

The resulting positive value of ΔG indicates that the reaction is not spontaneous at 25°C.

The unfavorable increase in enthalpy outweighed the favorable increase in entropy. The reaction will be spontaneous only at a more elevated temperature.

Practice Problem
1. For the reaction in Sample Problem 20.1, calculate ΔG at a temperature of 1200. K. Is the reaction spontaneous at that temperature?

Available values for enthalpy and entropy changes are generally measured at the standard conditions of 25°C and 1 atm pressure. They are slightly temperature dependent, so we must use caution when calculating specific ΔG values at temperatures other than 25°C, as in the practice problem above. However, since the values for ΔH and ΔS 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.

## Lesson Summary

• A reaction or process is considered to be spontaneous when the formation of products is favored at a given set of conditions.
• Spontaneous reactions release free energy, which can be used to do work.
• A mathematical combination of enthalpy change and entropy change allows the change in free energy to be calculated. A reaction with a negative value for ΔG releases free energy and is thus spontaneous. A reaction with a positive ΔG is nonspontaneous and will not favor the products.
• Some reactions may be spontaneous at some temperatures and nonspontaneous at other temperatures.

## Lesson Review Questions

### Reviewing Concepts

1. What is true about the relative amounts of reactants and products at the end of a spontaneous reaction?
2. Can a proposed reaction be spontaneous and yet still not be observed to occur? Explain.
3. The forward reaction is spontaneous for a particular reversible reaction. What can you conclude about the reverse reaction?
4. Explain how free energy is used to determine whether or not a reaction is spontaneous.
5. Under what conditions of enthalpy and entropy change is a reaction always spontaneous? Under what conditions is a reaction never spontaneous?
6. If the entropy change is unfavorable for a certain reaction, is the reaction more likely to be spontaneous at a high temperature or a low temperature?
7. If the enthalpy change is unfavorable, but the entropy change is favorable, would a high temperature or a low temperature be more likely to lead to a spontaneous reaction?

### Problems

1. Based on the values of ΔH, ΔS, and T shown below, calculate ΔG and predict whether the reaction will occur spontaneously.
1. ΔH = −245 kJ/mol; ΔS = −361 J/K•mol; T = 325 K
2. ΔH = +87.6 kJ/mol; ΔS = −112 J/K•mol; T = 295 K
3. ΔH = −95.5 kJ/mol; ΔS = +21.9 J/K•mol; T= 15°C
4. ΔH = +104.9 kJ/mol; ΔS = +177 J/K•mol; T = 246°C
2. Referring to problem number 8: Which reaction will be spontaneous at any temperature? Which will be nonspontaneous at any temperature?
3. One mole of mercury(II) oxide decomposes according to the following reaction: .
1. Calculate ΔG for the reaction at 25°C and predict whether the reaction is spontaneous or not.
2. Calculate ΔG at 800°C and predict whether the reaction is spontaneous at this temperature.
3. Assuming the values for ΔH and ΔS do not vary with temperature, calculate the lowest Kelvin temperature at which the reaction would be spontaneous.

## Points to Consider

When a reversible reaction is at equilibrium, the concentrations of the reactants and products are constant, so neither the forward nor reverse reaction is spontaneous (favored).

• How can we determine the conditions under which a certain reaction will be at equilibrium?
• What is the relationship of the free energy change, ΔG, to the equilibrium constant, Keq?

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