Thermodynamic Potentials

Microstates and Macroscopic Engines

The entropy is a useful quantity to describe the second law of thermodynamics. It fundamentally describes the degree of disorder. In the 19th century, the analysis of thermodynamics relied only on measurable quantities like temperature, pressure, and volume. Boltzmann, supporting the view that matter is constructed by atoms, introduced the microscopic possibility (number of microstates) – a statistical quantity describing the number of ways the particles in a system can be arranged. This implies that the degree of disorder can be described by the number of ways particles can be arranged: nature favours large possibility.

He used an equation to relate the microstates with the entropy:

where is the Boltzmann constant.

The Central Equation

In the early era of thermodynamics, the first law was formulated as

The physicists found it troublesome that heat and work are both path-dependent functions.

When the second law was established, it allowed us to define entropy:

Thus giving for a reversible process.

In a reversible process, the work can be expressed as

Combining these three equations under the condition of a reversible process:

This is the Central (Fundamental) Equation of Thermodynamics.

All the variables in this equation are state functions. Just as when calculating entropy change, we can always choose a convenient reversible process to replace an irreversible one. The variables do not depend on whether the process is reversible. Therefore the central equation applies to both reversible and irreversible processes.

Thermodynamic Potentials

From the central equation we see that . We can easily measure and in the laboratory, but how can we measure the entropy ?

This is difficult. So physicists introduced the Legendre transformation: changing the independent variables to ones that are easier to obtain, thereby producing three additional ways to describe the total energy in a system.

Internal Energy,

This is the fundamental energy form, requiring the variables and which are hard to obtain in experiment and hard to hold constant.

In general form, we have

Differentiating again:

From Clairaut’s theorem, we obtain the first Maxwell relation:

Enthalpy,

We want to replace the term . From the product rule , we seek for isobaric conditions:

Thus

We define the enthalpy:

The enthalpy can be expressed as . It is useful at constant pressure (), so that .

Maxwell relation from :

Helmholtz Free Energy,

If the system is bounded by a diathermal boundary, the temperature of the system always equals that of the surroundings: . In this case the temperature is constant.

From the second law:

For the surroundings, if the temperature is constant and the process is reversible:

From the first law :

Thus

In the isothermal case we can write , giving:

The quantity is defined as the Helmholtz free energy :

Therefore, for a reversible process ():

In other words, under isothermal conditions, represents the energy available to do work after part of the internal energy is “used up” by entropy. This is why it is called free energy.

Maxwell relation from :

Gibbs Free Energy,

In a real laboratory, we often need both isothermal and isobaric conditions. In this case we replace both and with and .

Starting from the enthalpy, which is suited to the isobaric condition:

Using :

At constant temperature:

We define the Gibbs free energy :

Maxwell relation from :