General Chemistry/Thermodynamics/Free Energy

Introduction

What's the point of entropy and enthalpy? So far, you have studied equilibrium to tell you how far a reaction occurs and kinetics to tell you how fast a reaction occurs. Thermodynamics can tell you if a reaction will occur, and at what temperatures. It may seem too obvious, but why does an ice cube spontaneously melt when it is at 30 °C? Melting is endothermic, so it would seem that the reverse reaction (freezing) is favored. After all, reactions that release heat are usually more favorable than those that absorb heat.

The answer is free energy (also called Gibbs Free Energy). The change in free energy for a reaction ultimately determines if it can occur spontaneously or not. Free energy is a combination of entropy and enthalpy, and when a reaction decreases the free energy, it will occur spontaneously.

The Equation

Free energy is defined by $\Delta G=\Delta H-T\Delta S$. By measuring or calculating the entropy change and enthalpy change of a reaction, you can determine the change in free energy. Notice that free energy depends on temperature as well. In this equation (as with all other thermo equations) $T$ must be an absolute temperature, measured in Kelvin. So the freezing point of water is not zero but rather 273 K. By using the Kelvin temperature scale, all temperatures will be greater than zero.

When you have solved for $\Delta G$ for a particular reaction at a certain temperature, you will find one of three possible outcomes:

Reaction type		Means...
$\Delta G<0$	exergonic	the reaction occurs spontaneously.
$\Delta G=0$		the reaction is at equilibrium.
$\Delta G>0$	endergonic	the reaction will not occur.

Equipped with this knowledge, you can determine the temperature at which a reaction will be at equilibrium (by setting G to 0 and solving for T). If a reaction is endergonic, it will not occur spontaneously. However, at a different temperature, the reaction might occur. Also, the reverse reaction will have an opposite value for G. For example, the combustion of propane will have a large negative value for its change in free energy. The reverse reaction would have the same large value, but positive. This makes sense, knowing that propane does not spontaneously precipitate from the smokey exhaust of a grill.

Spontaneity

Let's return to the example of a melting ice cube. At T = 273 K (0 °C) the processes of freezing and melting are at equilibrium. ΔG must equal zero. At higher temperatures, the melting process occurs spontaneously, so we can deduce that ΔH is positive and ΔS is also positive. We already know that melting is endothermic and increases entropy, so it seems the free energy equation works.

Between entropy and enthalpy, there can be four possible outcomes:

$\Delta H$	$\Delta S$	$\Delta G$	Result
+	-	always +	never spontaneous
-	+	always -	always spontaneous
+	+	+ for low T, and - for high T	depends on T
-	-	- for low T, and + for high T	depends on T

We can see that the melting of an ice cube is spontaneous for high temperatures.

Relating Free Energy

Free energy is related to equilibrium, as we have already seen. There is an equation that will allow you to convert between reaction free energy and the equilibrium constant for a particular reaction at a given temperature:

$\Delta G=-RT\times \ln K_{eq}$

$R$ is the Universal Gas constant, and $\ln$ is the natural logarithm. A scientific calculator will have a [LN] to calculate logarithms and a [e^] button to calculate anti-logarithms.