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 with 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 a 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 + PVYou 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 - Sd

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

Maxwell relations thermodynamic potential is an extremely important concept in the study of physics. This concept is taught in brief in class 12 physics in the chapter thermodynamics. Students who want to study this discipline further should get a clear understanding of Maxwell relations in thermodynamics. Vedantu has curated all the study material related to Maxwell relations thermodynamics. This study material acts as a reference guide and helps students to achieve an excellent grades in the exam.

Maxwell’s relations are derived by James Clerk Maxwell who was a 19th-century physicist. These relations are a set of equations existing in thermodynamics and are derived from Euler's reciprocity relation. He used thermodynamic potentials to get to these relations. They are expressed in partial differential form. Maxwell relations have various characteristics, they are as follows-

Enthalpy H

Internal energy U

Helmholtz free energy F

Gibbs free energy - G

Thermodynamic parameters are also included-

Pressure P

Volume V

Entropy S

Temperature T

Laws of Thermodynamics

1. The first law of thermodynamics, also known as the law of conservation of energy, states- energy can’t be created or destroyed, it can only change from one form to another.

dQ = dU + PdV

2. According to the second law, - In a reversible heat transfer, the product of temperature and the sources of destination of heat is an element of heat transfer. δQ, with the increment (dS) of the system's conjugate variable, its entropy (S).

dQ = TdS

3. Third law of thermodynamics - the temperature approaches absolute zero as  the entropy of a system approaches a constant value. It can represented as- TdS = dU + PdV