The Free Energy and the Chemical Equilibrium: Combining the Laws of Thermodynamics

In this section, we will build up our concepts about the 3 laws of thermodynamics (the 1st, 2nd, and the 3rd laws of thermodynamics) to bring the concept of spontaneity of reactions based on new parameter which is the free energy. In this section, there are 2 types of free energy which are the Helmholtz and Gibbs free energies, and we will have a deeper discussion in Gibbs free energy because it is used more common in chemistry. Furthermore, we will see briefly how the free energy is related to the equilibrium process.

Hermann Ludwig Ferdinand von Helmholtz (left) and Josiah Willard Gibbs (right)
The founders of the concept of free energies

To begin with let remind ourself about the laws of thermodynamics which are:

• The 1st Law of Thermodynamics (check this link):

The energy of an isolated system is constant;

• The 2nd Law of Thermodynamics (check this link):

The entropy of an isolated system increases in the course of a spontaneous change (ΔS_universe > 0);

• The 3rd Law of Thermodynamics (check this link):

All perfect crystalline materials have zero entropy at 0 K.

Besides that, in a reversible process the entropy can be defined as:

but when a process is a spontaneous one, so by definition the process is irreversible which means:

and this equation is for a system in thermal equilibrium with its surroundings at temperature T. Furthermore, this equation is known as Clausius inequality. Then, by rearranging the Clausius inequality and defining the system at constant volume where dU = dq, thus

and the equation above is called the Helmholtz free energy (dA). From the equation above, it is also shown that when dA < 0, the reaction is spontaneous whereas at equilibrium dA = 0. Besides that, dA is an exact differential, so

Therefore, we can define that the Helmholtz free energy (ΔA) is the maximum amount of work accompanying a process.

In the other sides, when we define our system at constant pressure where dH = dq, thus

and the equation above is called the Gibbs free energy (dG). From the equation above, it is also shown that when dG < 0, the reaction is spontaneous whereas at equilibrium dG = 0. Besides that, dG is an exact differential, so

Therefore, we can define the Gibbs free energy (ΔG) as the maximum amount of non-expansion work accompanying a process. Then, we can define the Gibbs energy for a reaction as

Besides that, the Gibbs energy for a reaction can be defined as well from the standard Gibbs energy of formation which is the standard reaction Gibbs energy for the formation of a compound from its element in their reference states, hence the Gibbs energy for a reaction can be calculated as:

When we look back about the 1st law of thermodynamics where internal energy can be defined as a function of volume and temperature as shown in figure below.

Internal energy as a function of volume and temperature

This can be rationalised by starting from its equation where dU = dq + dw, then from the first and second laws of thermodynamics we can define dU as

dU = TdS - pdV

This equation is known as the fundamental equation and it applies to any changes, either reversible or irreversible, of a closed system that does no additional (non-expansion) work and we can rewrite the equation as:

and by joining those equations we can see that:

From the fundamental equation, if we consider infinitesimal changes of G=H-TS:

Then, starting from

we can show that

From the equation above, we also shows that Gibbs energy depends on pressure and temperature and we can show it on a curve as shown below.

Gibbs energy as a function of temperature and pressure

When we analyse one by one the dependency of Gibbs energy with temperature and pressure, we need to keep one of the variable constant. Firstly, the Gibbs energy depends on temperature as shown on the graph below.

The Gibbs energy function of temperature

The variation of the Gibbs energy with the temperature is determined by the entropy. Because the entropy of the gaseous phase of a substance is greater than that of the liquid phase, and the entropy of the solid phase is smallest, the Gibbs energy changes most steeply for the gas phase, followed by the liquid phase, and then the solid phase of the substance. From the equation

we can show that

If we known the enthalpy of the system, we know how (G/T) varies with temperature.

Meanwhile, the dependence of Gibbs energy on pressure is shown on the graph below.

The Gibbs energy as the function of pressure of several phases

The variation of the Gibbs energy with the pressure is determined by the volume of the sample. Because the volume of the gaseous phase of a substance is greater than that of the same amount of liquid phase, and the entropy of the solid phase is smallest (for most substances), the Gibbs energy changes most steeply for the gas phase, followed by the liquid phase, and then the solid phase of the substance. Because the volumes of the solid and liquid phases of a substance are similar, their molar Gibbs energies vary by similar amounts as the pressure is changed. From the equation

we can integrate:

Then, if V does not depend on pressure such as in solid or liquid, the Gibbs energy is

where as in gas and we assume it as an ideal gas, the Gibbs energy is

Furthermore, the initial pressure is measured in standard pressure which the equation becomes

Now, let consider an equilibrium system of

The Gibbs energy for the reaction can be calculated as:

where

From our reversible process above, we can consider the general equilibrium and the Gibbs free energy as:

where

As mentioned early on the definition of Gibbs free energy, ΔG = 0 at the equilibrium and Q = K, thus

Furthermore, the composition of equilibrium mixture can be calculated prior to the experiment. From the equation above, when ΔG < 0, so K > 1 which means equilibrium towards product where as ΔG > 0 so K < 1 this means equilibrium towards reactants. Furthermore, when ΔG = 0 so K = 1 which means the equilibrium is between reactants and products. Then, the value of K does not depends on the pressure because ΔG is defined at constant pressure.

When we consider again our example

where

the value of K would not change, so when the pressure of the system is changed the equilibrium must counteract to keep K constant and this principle is know as Le Chatelier's principle.

A system at equilibrium when it is subjected to a disturbance, it responds in a way that tends to minimise the effect of the disturbance.

From the the relationship between G and K, we can derive the van't Hoff equation where it shows the dependence of K on temperature.

This equation shows that in equilibrium with exothermic reaction the increasing temperature would make the equilibrium shifts in the direction of reactants whereas the decreasing temperature would shift the equilibrium in the direction of products. In the other sides, in endothermic process, the effect would be the other way around, increasing in temperature shifts the equilibrium to the products and decreasing in temperature shifts the equilibrium to the reactants.