Why Are Maxwell’s Relations Important in Physics?
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
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
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
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
FAQs on Maxwell’s Relations: Thermodynamics Explained
1. What are Maxwell's relations in thermodynamics?
Maxwell's relations are a set of equations in thermodynamics that are derived from the four thermodynamic potentials. They are significant because they provide relationships between the partial derivatives of thermodynamic properties: Pressure (P), Volume (V), Temperature (T), and Entropy (S). These relations allow us to express quantities that are difficult to measure, like a change in entropy, in terms of quantities that are easily measurable, such as pressure and temperature.
2. What are the four thermodynamic potentials that form the basis of Maxwell's relations?
The four fundamental thermodynamic potentials are used to describe the energy of a system under different conditions. They are the foundation from which Maxwell's relations are derived. The potentials are:
- Internal Energy (U): The total energy contained within a system.
- Enthalpy (H): The total heat content of a system, defined as H = U + PV. It's particularly useful for processes at constant pressure.
- Helmholtz Free Energy (F): The 'useful' work obtainable from a closed system at constant temperature and volume, defined as F = U - TS.
- Gibbs Free Energy (G): The maximum amount of non-expansion work that can be extracted from a system at constant temperature and pressure, defined as G = H - TS.
3. Can you list the four main Maxwell's relations?
The four main Maxwell's relations are derived from the symmetry of the second derivatives (using Euler's reciprocity relation) of each of the four thermodynamic potentials. They are:
- From Internal Energy (U): (∂T/∂V)S = -(∂P/∂S)V
- From Enthalpy (H): (∂T/∂P)S = (∂V/∂S)P
- From Helmholtz Free Energy (F): (∂P/∂T)V = (∂S/∂V)T
- From Gibbs Free Energy (G): (∂V/∂T)P = -(∂S/∂P)T
4. How are Maxwell's relations derived from the thermodynamic potentials?
Maxwell's relations are derived using a mathematical property of state functions called Euler's reciprocity relation. The process involves these steps:
- Start with the differential form of a thermodynamic potential (e.g., for internal energy, dU = TdS - PdV).
- Since U, S, and V are state functions, the differential dU is an 'exact differential'.
- For an exact differential of the form dz = Mdx + Ndy, the reciprocity relation states that (∂M/∂y)x = (∂N/∂x)y.
- By applying this rule to the differential form of each of the four potentials, we can derive the four corresponding Maxwell's relations. For dU, T corresponds to M, -P to N, S to x, and V to y, yielding the first relation.
5. Why are Maxwell's relations considered so important in physics and chemistry?
The primary importance of Maxwell's relations is their ability to simplify thermodynamic calculations. They allow scientists and engineers to replace quantities that are difficult or impossible to measure directly in a lab (like the change in entropy with respect to pressure) with equivalent expressions involving quantities that are easy to measure (like changes in volume with respect to temperature). This makes it possible to calculate changes in thermodynamic properties like entropy and internal energy from experimental data of P, V, and T.
6. How do Maxwell's relations in thermodynamics differ from Maxwell's equations of electromagnetism?
This is a common point of confusion as both were developed by James Clerk Maxwell. The key difference is the field of physics they describe:
- Maxwell's Relations belong to thermodynamics. They describe relationships between macroscopic properties of a system like pressure, volume, temperature, and entropy.
- Maxwell's Equations belong to electromagnetism. They are a set of four fundamental equations that describe how electric and magnetic fields are generated and interact with each other and with charges and currents.
In short, one set is for heat and energy systems, while the other is for electricity and magnetism.
7. What is a practical example of applying a Maxwell relation?
A practical application is determining the change in temperature during an adiabatic compression or expansion, a process where no heat is exchanged (constant entropy). For example, the relation (∂T/∂P)S = (∂V/∂S)P can be used. By using another Maxwell relation and properties like heat capacity, this can be used to predict how much a gas will heat up when compressed adiabatically, a principle crucial for understanding engines and refrigerators.
8. Do Maxwell's relations apply to irreversible processes?
Yes, Maxwell's relations are valid for both reversible and irreversible processes. This is because the relations connect state functions (P, V, T, S, U, H, F, G). The change in a state function depends only on the initial and final states of the system, not on the path taken to get from one to the other. While the derivation often uses a reversible path (where dQ = TdS), the resulting equations relate the properties of the states themselves, making them universally applicable regardless of the process path.
