5.7.2: Free Energy and Temperature

So we have two very clear-cut cases that allow us to predict whether a process will occur: where the enthalpy and entropy predict the same outcome. But there are two possible situations where the enthalpy change and the entropy term (TΔS) “point” in different directions. When ΔH is positive and ΔS is positive, and when ΔH is negative while ΔS is negative. When this happens, we need to use the fact that the free energy change is temperature-dependent in order to predict the outcome. Recall that the expression ΔG = ΔH – TΔS depends upon temperature. For a system where the entropy change is positive (+ΔS), an increase in temperature will lead to an increasingly negative contribution to ΔG. In other words, as the temperature rises, a process that involves an increase in entropy becomes more favorable. Conversely, if the system change involves a decrease in entropy, (ΔS is negative), ΔG becomes more positive (and less favorable) as the temperature increases.

ΔH	ΔS	ΔG
Negative (exothermic)	Positive (entropy increases)	Negative at all temperatures (always thermodynamically favored)
Positive (endothermic)	Negative (entropy decreases)	Positive at all temperatures (never thermodynamically favored)
Negative (exothermic)	Negative (entropy decreases)	Temperature dependent: as the temperature increases ΔG will become more positive and the reaction will become less favored (go backwards)
Positive (endothermic)	Positive (entropy increases)	Temperature dependent: as the temperature increases ΔG will become more negative and the reaction will become favored (go forwards)

The idea that temperature affects the direction of some processes is perhaps a little disconcerting. It goes against common-sense that if you heat something up a reaction might actually stop and go backwards (rest assured we will come back to this point later). But in fact there are a number of common processes where we can apply this kind of reasoning and find that they make perfect sense.

Up to this point, we have been considering physical changes to a system—populations of molecules going from solid to liquid or liquid to gaseous states (and back). Not really what one commonly thinks of as chemistry, but the fact is that these transformations involve the making and breaking of interactions between molecules. We can therefore consider phase transitions as analogous to chemical reactions, and because they are somewhat simpler, develop a logic that applies to both processes. So let us begin by considering the phase change/reaction system

H2O (liquid) ⇌ H2O (gas).

We use a double arrow ⇌ to indicate that, depending upon the conditions the reaction

could go either to the right (boiling) or to the left (condensing). So, let us assume for the moment that we do not already know that water boils (changes from liquid to gas) at 100 ^oC. What factors

would determine whether the reaction H2O (liquid) ⇌ H2O (gas) favors the liquid or the gaseous state at a particular temperature? As we have seen, the criterion for whether a process will “go” at a particular temperature is ΔG. We also know that the free energy change for a reaction going in one direction is the negative of the ΔG for the reaction going in the opposite direction. So that the ΔG for the reaction: H2O (liquid) ⇌ H2O (gas) is –ΔG for the reaction H2O (gas) ⇌ H2O (liquid).

When water boils, all the intermolecular attractions between the water molecules must be overcome, allowing the water molecules to fly off into the gaseous phase. Therefore, the process of water boiling is endothermic (ΔHvaporization = +40.65 kJ/mol); it requires an energy input from the surroundings (when you put a pot of water on the stove you have to turn on the burner for it to boil). When the water boils, the entropy change is quite large (ΔSvaporization = 109 J/mol K), as the molecules go from being relatively constrained in the liquid to gas molecules that can fly around. At temperatures lower than the boiling point, the enthalpy term predominates and ΔG is positive. As you increase the temperature in your pan of water, eventually it reaches a point where the contributions to ΔG of ΔH and TΔS are equal. That is, ΔG goes from being positive to negative and the process becomes favorable. At the temperature where this crossover occurs ΔG = 0 and ΔH = TΔS. At this temperature (373 K, 100 oC) water boils (at 1 atmosphere). At temperatures above the boiling point, ΔG is always negative and water exists predominantly in the gas phase. If we let the temperature drop below the boiling point, the enthalpy term becomes predominant again and ΔG for boiling is positive. Water does not boil at temperatures below 100 oC at one atmosphere.¹¹⁰

Let us now consider a different phase change, such as water freezing. When water freezes, the molecules in the liquid start to aggregate and form hydrogen bonding interactions with each other, and energy is released to the surroundings (remember it is this energy that is responsible for increasing the entropy of the surroundings). Therefore ΔH is negative: freezing is an exothermic process (ΔHfusion = – 6 kJ/mol).¹¹¹ Freezing is also a process that reduces the system’s entropy. When water molecules are constrained, as in ice, their positional entropy is reduced. So, water freezing is a process that is favored by the change in enthalpy and disfavored by the change in entropy. As the temperature falls, the entropy term contributes less to ΔG, and eventually (at the crossover point) ΔGfusion goes to zero and then becomes negative. The process becomes thermodynamically favored. Water freezes at temperatures below 0 oC. At temperatures where phase changes take place (boiling point, melting point), ΔG = 0. Furthermore, if the temperature were kept constant, there would be no observable change in the system. We say that the system is at equilibrium; for any system at equilibrium, ΔG = 0.

Questions to Answer

• For each of these processes, give the change in entropy of the system, the direction of thermal energy transfer (the sign of ΔH), the change in entropy of the surroundings, and the change in entropy of the universe:

• Water freezing at -10 °C

• Water boiling at 110 °C

• For each of these processes predict the sign of change in entropy (ΔS) of the system, the direction of thermal energy transfer (the sign of ΔH), and the sign of the Gibbs free energy change, ΔG. What does the sign of ΔG tell you?
• Water freezing at -10 °C; Water boiling at 110 °C
• Water boiling at -10 °C; Water freezing at 110 °C

Questions to Ponder

• Why do we denote excess energy as a product of this equation? captured energy + 6 CO2(g) + 6 H2O(g) ⇆ C6H12O6(s) + 6 O2(g) + excess energy.

• What other processes do you know that must be coupled to external energy sources to make them go?

References

¹¹⁰ So why, you might ask, does water evaporate at temperatures lower than 100 oC? We will come to that soon.

¹¹¹ This is another rather counterintuitive idea, but remember that to freeze something you have to take heat away (for example, in a refrigerator).