Question

Starting from a symbolic expression of the Second Law of Thermodynamics, show how arguments based purely upon considerations of entropy lead to the formalism which we call Gibbs

Answer 1

Entropy has already been discussed in terms of disorder. The tendency of the universe to move towards disorder is a consequence of there being an infinite number of possible disordered states and very few ordered states. The probability of adoption of disorder is just too great for it not to happen.

For any process, physical or chemical, there is a before and after. We can consider a process to have two parts, a system and a surroundings (everything else that is not the system). The two parts together make up the whole universe.

If the process results in an increase in entropy in the universe then it is possible. There are two ways that universal entropy can increase.

1. The system produces a greater number of particles
2. The system releases energy that goes to increasing the disorder of the surroundings.

The entropy of the universe = the entropy of the system + the entropy of the surroundings, and any change in the universal entropy must be a consequence of either change in the entropy of the surroundings, the system or both.

?S(universe) = ?S(system) + ?S(surroundings)

Therefore:

?S(universe) - ?S(surroundings) = ?S(system)

Entropy change in the system can be gauged in terms of the number of particles with degrees of freedom (specifically gases). Change in the entropy of the surroundings is caused by release of energy.

From the above sub-section we se that there are two ways that the overall entropy of the universe can increase. Increasing the number of particles in the system, or releasing energy from the system that increases the entropy of the surroundings.

The effect of energy on the entropy change is dependent on the temperature of the surroundings. They are equated by the relationship:

?S(surroundings) = q/T

That is, the entropy change is the energy (q) released by the system (transferred to the surroundings) divided by the absolute temperature.

We call the energy released by chemicals in the course of a reaction, the enthalpy change (negative as the reaction is exothermic), so the relationship can also be written:

?S(surroundings) = -?H/T

If this is substituted into the universal entropy relationship:

?S(universe) = ?S(surroundings) + ?S(system)

we get:

?S(universe) = -?H/T + ?S(system)

Multiplying through by -T gives:

-T?S(universe) = ?H - T?S(system)

Gibbs recognised the importance of this relationship and defined a state symbol 'G' that represents the '-T?S(universe)', which he called ' Free Energy of the system'.

G = -T?S(universe)

It should be appreciated that when the universal entropy increases G must take a negative value.

Hence, Gibbs' free energy is related to the entropy of the universe. For any process to be possible, the change in Gibbs' free energy must be negative.

?G = ?H - T?S

When the entropy of the universe increases it means that the Gibbs Free Energy of the system decreases.