Maxwell's Relations

A 19th-century physicist known as James Clark Maxwell derived Maxwell's relations. These said relations are basically a set of equations existing in thermodynamics. Mr Maxwell derived these relations using the theory of symmetry of second derivatives and Euler’s reciprocity relation. He also used the definitions provided by thermodynamic potentials to come up to these relations. Maxwell expresses these relations in partial differential form. The Maxwell relations consist of the various characteristic functions, these functions are enthalpy H, Helmholtz free energy F, internal energy U, and Gibbs free energy G. It also includes thermodynamic parameters such as Pressure P, entropy S, volume V, and temperature T.

The Maxwell equation in thermodynamics is very useful because these are the set of relations that allows the physicists to change certain unknown quantities, as these unknown quantities are hard to measure in the real world. So these quantities need to be replaced by some easily measured quantities.

Knowledge One Should Possess Before Starting With Maxwell Relations

Anyone going through Maxwell relations and equations must have deep knowledge in topics such as exact differentials, and partial differential relations. They must also be familiar with the basics of thermodynamics, the first and the second law of thermodynamics, entropy, etc.

The Maxwell relations is completely mathematics-based study and once the readers know about fundamental equations then everything else is a mathematical manipulation.

Thermodynamic Potentials

Before we continue with Maxwell's relations we will briefly explain all the four thermodynamic potentials which are also known as the characteristic functions that form the base of Maxwell's relations.

Some quantity that is used to represent some thermodynamic state in a system is known as thermodynamic potential. Each thermodynamic potential gives a different measure of the “type” of the energy system. Here we will discuss four types of potentials that help derive the Maxwell thermodynamic relation.

Internal energy- the energy contained in a system is the internal energy of a system. This energy excludes any outside energy that comes due to external forces. It also excludes the kinetic energy of a system as a whole. Internal energy includes only the energy of the system, which is due to the motion, and interactions of the particles that make up the system. 

Making use of the first law of thermodynamics, you can seek the differential form of the said internal energy:

dU = δQ+δW

dU = TdS - PdV

Enthalpy- the summation of internal energy and the product of volume and pressure gives enthalpy. The equation of enthalpy represents that the total heat content of a system is always the preferred potential to use when many chemical reactions are under study when such chemical reactions take place at a constant pressure. When the pressure here is constant, the change in the said internal energy is equal to the change in enthalpy of the system. The letter H represents the enthalpy.

H = U + PV

You can seek dH with the help of the above stated expression:

dH = dU + d (PV) = dU + PdV + VdP

dH = TdS - PdV + PdV + VdP 

-> dH = TdS + VdP

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Helmholtz free energy- Helmholtz free energy is the difference between the internal energy of the system and the product of entropy and temperature. This equation represents the amount of useful work that can be easily obtained from a close system when the temperature and the volume are constant. The letter F in this equation represents the said Helmholtz free energy.

F  = U -TS

From which you can find the differential form of the said equation above

dF = dU - d(TS) = dU - TdS - SdT

Substituting the differential form of the said internal energy (dU = TdS - PdV)

dF = TdS - PdV - TdS - SdT

 -> dF = -PdV - SdT

Gibbs Free Energy - This thermodynamic potential is the last potential that helps to calculate the quantity of work a system can do at constant pressure and temperature. It is a very useful concept while studying phase transitions that happen during such conditions. Gibbs can be defined as the said difference between the enthalpy of a system as well as the product of entropy and temperature of the system. The letter G here given in the equation represents the said Gibbs free energy.

Thus, G= H -TS


From which you can find the differential form of the said equation above:

dG = dH - d(TS) = dH -Tds - Sdt

Now, you need to substitute in the said differential form of the enthalpy (dH = TdS + VdP)

dG = TdS + VdP - TdS - SdT

 -> dG = VdP - SdT

This Table Summarizes the Differential Forms of the Four Types of Thermodynamic Potentials:

Thermodynamic Potentials

The Derived Derivational Form

The Natural Variables

Internal Energy depicted by U

dU = TdS − PdV

S and V

Enthalpy depicted by H

dH = TdS + VdP

S and P

Helmholtz Free Energy as depicted by F

dF = −PdV − SdT

V and T

Gibbs Free Energy as depicted by G

dG = VdP − SdT

P and T

FAQ (Frequently Asked Questions)

Why are Maxwell’s Relations Considered Imperative in Thermodynamics?

Maxwell’s relations are imperative in Thermodynamics because they allow you to relate the changes happening with one aspect of thermodynamic variables to other kinds of variables. Thus, if you want to know about the entropy change of a said system in relation to the pressure occurring at a constant enthalpy, you can make use of Maxwell's relations to do so. You can use Maxwell's relations to reduce the different aspects of something that you can measure.

Is It True that Maxwell’s Relations for Thermodynamics is Actually Applicable Only for Reversible Processes?

No, this is not true as the said equation dU = TdS - PdV can apply to the irreversible process as well in addition to the reversible processes. The reason being that these U, V, and S, are all basically state variables. Thus, a change in the said quantities during the said process cannot depend upon the process itself. However, they also depend upon the starting as well as the final stages.