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Thermal Physics

Maxwell's Thermodynamic Relations

Maxwell's thermodynamic relations are fundamental equality statements involving partial derivatives of thermodynamic state variables. They are derived from exact differentials of thermodynamic potentials.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Direct application of the four primary Maxwell relations, cyclic rule, and simple partial differentiation in PVT systems.

PGT extension

Derivation of Maxwell relations from state functions, Tds equations, energy equations, and application to specific heat relations and Joule-Thomson effect.

Prerequisites

Concept of Temperature and internal energy
First and Second Laws of Thermodynamics
Exact and inexact differentials

Foundation of Maxwell Relations

Thermodynamic potentials (Internal Energy U, Helmholtz Free Energy F, Enthalpy H, Gibbs Free Energy G) are state functions with exact differentials.
For any exact differential d$z = M dx + N dy$, the condition for exactness is (dM/dy)_$x = (dN/dx)_y$.
Applying this mathematical theorem to the four thermodynamic potentials yields the four primary Maxwell relations.

First Maxwell Relation (from Internal Energy U)

Internal energy is expressed as d$U = T dS - P dV$.
The relation is obtained by equating the cross derivatives of temperature and pressure with respect to volume and entropy.
Equation: (dT/dV)_$S = - (dP/dS)_V$, linking thermal and mechanical variables adiabatically.

Second and Third Maxwell Relations (Helmholtz F and Enthalpy H)

Helmholtz free energy function $F = U - TS gives dF = - S dT - P dV$, leading to (dS/dV)_$T = (dP/dT)_V$.
Enthalpy function $H = U + PV gives dH = T dS + V dP$, leading to (dT/dP)_$S = (dV/dS)_P$.
These relations are extensively used to evaluate changes in entropy using measurable quantities like volume expansion coefficient.

Fourth Maxwell Relation (from Gibbs Free Energy G)

Gibbs free energy function $G = H - TS gives dG = - S dT + V dP$.
The resulting Maxwell relation is (dS/dP)_$T = - (dV/dT)_P$.
This relation is crucial in understanding the effect of pressure on phase transitions and melting points.

Derivations

Derivation of First Maxwell Relation from Internal Energy

Conditions: Single-phase, closed system in thermal and mechanical equilibrium

Write the combined First and Second Law of Thermodynamics: d$U = T dS - P dV$.
Treat U as a function of S and V, so that d$U = (\partial U/\partial S)_V dS + (\partial U/\partial V)_S dV$.
Identify coefficients: $T = (\partial U/\partial S)_V$ and -$P = (\partial U/\partial V)_S$.
Since dU is an exact differential, apply Euler's reciprocity condition: partial / partial V (partial U/partial S)_$V = \partial / \partial S (\partial U/\partial V)_S$.
Substitute the coefficients to obtain (partial T/partial V)_$S = - (\partial P/\partial S)_V$.

Result: \left(\frac{\partial T}{\partial V}\right)_$S = -\left(\frac{\partial P}{\partial S}\right)_V$

Derivation of Fourth Maxwell Relation from Gibbs Free Energy

Conditions: Constant composition, reversible processes

Define Gibbs free energy as $G = H - TS = U + PV - TS$.
Differentiate G: d$G = dU + P dV + V dP - T dS - S dT$.
Substitute d$U = T dS - P dV into the equation$, yielding d$G = -S dT + V dP$.
Treat G as a function of T and P: d$G = (\partial G/\partial T)_P dT + (\partial G/\partial P)_T dP$.
Apply the exactness condition to mixed partial derivatives: \partial / \partial P (\partial G/\partial T)_$P = \partial / \partial T (\partial G/\partial P)_T$.
Equate terms to arrive at -(\partial S/\partial P)_$T = (\partial V/\partial T)_P$.

Result: \left(\frac{\partial S}{\partial P}\right)_$T = -\left(\frac{\partial V}{\partial T}\right)_P$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
First Maxwell Relation	$$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$$	T = Temperature, V = Volume, S = Entropy, P = Pressure K m^{-3}	Reversible adiabatic process
Second Maxwell Relation	$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$$	S = Entropy, V = Volume, T = Temperature, P = Pressure J K^{-1} m^{-3}	Isothermal process
Third Maxwell Relation	$$\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$$	T = Temperature, P = Pressure, S = Entropy, V = Volume K Pa^{-1}	Isentropic (adiabatic) process
Fourth Maxwell Relation	$$\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$$	S = Entropy, P = Pressure, T = Temperature, V = Volume J K^{-1} Pa^{-1}	Isothermal process
First TdS Equation	$$T dS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV$$	$C_V = Heat capacity at constant volume$ J	General single-phase system
Second TdS Equation	$$T dS = C_P dT - T \left(\frac{\partial V}{\partial T}\right)_P dP$$	$C_P = Heat capacity at constant pressure$ J	General single-phase system
Energy Equation for Internal Energy	$$\left(\frac{\partial U}{\partial V}\right)_T = T \left(\frac{\partial P}{\partial T}\right)_V - P$$	U = Internal energy Pa	Valid for all substances
Difference in Heat Capacities	$$C_P - C_V = T \left(\frac{\partial P}{\partial T}\right)_V \left(\frac{\partial V}{\partial T}\right)_P$$	C_P, $C_V = Heat capacities$ J K^{-1}	General thermodynamic relation

Graphs and experiments

Isothermal P-V curves and Maxwell relations

Draw: A family of isotherms on a P-V plane showing slope (dP/dV)_T.

Interpret: The slope of the isotherm relates directly to (dS/dV)_T via the second Maxwell relation, illustrating how volume dependence of entropy relates to pressure variation with temperature.

Exam use: Tested conceptually in TGT to evaluate understanding of slopes of thermodynamic curves.

Phase equilibrium curves (Clapeyron latent heat relation)

Draw: Liquid-vapor coexistence curve on a P-T diagram.

Interpret: The slope dP/d$T = L / (T \Delta V)$ is derived using Maxwell relations and describes phase boundary conditions.

Exam use: Direct numerical and theoretical questions in PGT examinations regarding boiling point elevation with pressure.

Throttling Process (Joule-Thomson Expansion)

Draw: Enthalpy H as constant during porous plug experiment, relating temperature drop to pressure change.

Interpret: Joule-Thomson coefficient m$u = (dT/dP)_H$ is evaluated using Maxwell relations involving heat capacity C_P and expansion coefficient.

Exam use: Frequently featured in PGT multiple-choice and descriptive analytical problems.

Solved numericals

TGT core

Using the cyclic relation for thermodynamic variables P, V, and T, show that \left(\frac{\partial P}{\partial V}\right)_T \left(/left(\frac{\partial V}{\partial T}\right)_P \left(\frac{\partial T}{\partial P}\right)_$V = -1$.

Let a function f(P, V, T) = 0 define the state of a system.
Write the total differential of P as a function of V and T: d$P = (\partial P/\partial V)_T dV + (\partial P/\partial T)_V dT$.
Set dP = 0 for an isothermal process or substitute dT from d$T = (\partial T/\partial P)_V dP + (\partial T/\partial V)_P dV$.
Rearrange terms to obtain the standard cyclic chain rule identity: (\partial P/\partial V)_T (\partial V/\partial T)_P (\partial T/\partial P)_$V = -1$.

Answer: Proven analytically using partial derivative identities.

PGT extension

For a van der Waals gas with equation of state (P + a/V^2)(V - b) = RT, find the internal energy expression (\partial U/\partial V)_T using Maxwell thermodynamic relations.

Use the energy equation derived from Maxwell relations: (\partial U/\partial V)_$T = T (\partial P/\partial T)_V - P$.
From the van der Waals equation, express P as a function of T and V: $P = \frac{RT}{V-b} - \frac{a}{V^2}$.
Differentiate P with respect to T at constant V: (\partial P/\partial T)_$V = \frac{R}{V-b}$.
Substitute into the energy equation: (\partial U/\partial V)_$T = T \left(\frac{R}{V-b}\right) - \left(\frac{RT}{V-b} - \frac{a}{V^2}\right)$.
Simplify the expression to get a/V^2.

Answer: (\partial U/\partial V)_$T = \frac{a}{V^2}$

PGT extension

Prove that for any substance, C_P - $C_V = V \beta^2 T / \kappa_T$, where \beta is the volume expansivity and \kappa_T is the isothermal compressibility.

Start with the general relation C_P - $C_V = T (\partial P/\partial T)_V (\partial V/\partial T)_P$.
Define volume expansivity $\beta = \frac{1}{V} (\partial V/\partial T)_P$, so (\partial V/\partial T)_P = V \beta.
Define isothermal compressibility \kapp$a_T = -\frac{1}{V} (\partial V/\partial P)_T$.
Using the cyclic rule, (\partial P/\partial T)_$V = - (\partial P/\partial V)_T (\partial V/\partial T)_P = \frac{V \beta}{V \kappa_T} = \frac{\beta}{\kappa_T}$.
Substitute back into the difference equation to obtain C_P - $C_V = T (\beta / \kappa_T) (V \beta) = \frac{V \beta^2 T}{\kappa_T}$.

Answer: C_P - $C_V = \frac{V \beta^2 T}{\kappa_T}$

Assumptions and limitations

Applicable only to systems in thermodynamic equilibrium where state functions and exact differentials are well-defined.
Do not apply directly to irreversible or non-equilibrium processes unless evaluated across equilibrium states.

Common misconceptions

Common error: Maxwell relations are only valid for ideal gases.

Correct: Maxwell relations are universal and apply to any pure substance in equilibrium, whether solid, liquid, or gas, ideal or real.

They originate purely from the mathematical property of exact differentials of state functions, independent of the equation of state.

Common error: The minus signs in Maxwell relations are arbitrary.

Correct: The negative signs are mathematically mandatory and arise directly from the cross-differentiation of potentials like U and G.

Swapping independent variables in differentials introduces negative signs through the cyclic rule and properties of partial derivatives.

Quick revision

The four potentials U, F, H, G generate the four Maxwell relations via exact differentials: dU (T,-P), dF (-S,-P), dH (T,V), dG (-S,V).
First Maxwell Relation: \left(\frac{\partial T}{\partial V}\right)_$S = -\left(\frac{\partial P}{\partial S}\right)_V$
Second Maxwell Relation: \left(\frac{\partial S}{\partial V}\right)_$T = \left(\frac{\partial P}{\partial T}\right)_V$

Must remember

Thermodynamic potentials ($U$, $F$, $H$, $G$) are state functions possessing exact differentials.
The first Maxwell relation is $(\frac{\partial T}{\partial V})_S = -(\frac{\partial P}{\partial S})_V$, derived from internal energy $U$.
The second Maxwell relation is $(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial_T})_V$, derived from Helmholtz free energy $F$.
The third Maxwell relation is $(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$, derived from Enthalpy $H$.
The fourth Maxwell relation is $(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial _T})_P$, derived from Gibbs free energy $G$.
Maxwell relations are universal and hold for any pure substance in thermodynamic equilibrium, not just ideal gases.
The difference in heat capacities is given by $C_P - C_V = T (\frac{\partial P}{\partial T})_V (\frac{\partial V}{\partial T})_P$.

Glossary

Internal Energy ($U$)The total energy contained within a thermodynamic system, with differential $dU = T dS - P dV$.

Helmholtz Free Energy ($F$)A thermodynamic potential measuring the useful work obtainable from a closed isothermal system, $F = U - TS$.

Enthalpy ($H$)A measure of total heat in a thermodynamic system, $H = U + PV$.

Gibbs Free Energy ($G$)Thermodynamic potential that measures maximum reversible work from a constant pressure and temperature system, $G = H - TS$.

Exact DifferentialA differential form $dz = M dx + N dy$ where $(\frac{\partial M}{\partial y})_x = (\frac{\partial N}{\partial x})_y$.

Comparison of the Four Thermodynamic Potentials and Maxwell Relations

Potential	Symbol & Formula	Differential Form	Associated Maxwell Relation
Internal Energy	$U$	$dU = T dS - P dV$	$(\frac{\partial T}{\partial V})_S = -(\frac{\partial P}{\partial S})_V$
Helmholtz Free Energy	$F = U - TS$	$dF = -S dT - P dV$	$(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$
Enthalpy	$H = U + PV$	$dH = T dS + V dP$	$(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$
Gibbs Free Energy	$G = H - TS$	$dG = -S dT + V dP$	$(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial T})_P$

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following thermodynamic potentials has the natural variables $S$ (entropy) and $V$ (volume)?

Explanation: Internal energy $U$ is expressed as $dU = T dS - P dV$, making its natural variables entropy ($S$) and volume ($V$).

2. The mathematical foundation of Maxwell's thermodynamic relations relies on which property of state functions?

Explanation: Thermodynamic potentials are state functions and possess exact differentials, allowing the equality of cross derivatives.

3. What is the expression for the first Maxwell relation derived from internal energy?

Explanation: From $dU = T dS - P dV$, exactness yields cross-derivative relation $(\frac{\partial T}{\partial V})_S = -(\frac{\partial P}{\partial S})_V$.

4. Helmholtz free energy $F$ is defined by the equation:

Explanation: Helmholtz free energy is defined as $F = U - TS$, leading to the differential $dF = -S dT - P dV$.

5. Which Maxwell relation is most directly used to evaluate the effect of pressure on melting points and phase transitions?

Explanation: The fourth Maxwell relation $(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial T})_P$ derived from Gibbs free energy $G$ is crucial for phase transitions.

6. Enthalpy $H$ is related to internal energy $U$ via the expression:

Explanation: Enthalpy is defined as $H = U + PV$.

7. The second Maxwell relation is given by:

Explanation: The second Maxwell relation, derived from Helmholtz free energy $F$, is $(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$.

8. Are Maxwell's thermodynamic relations valid exclusively for ideal gases?

Explanation: Maxwell relations are derived purely from mathematical properties of state functions and hold universally for any pure substance in equilibrium.

9. What is the third Maxwell relation associated with Enthalpy ($H$)?

Explanation: Enthalpy differential $dH = T dS + V dP$ yields the third Maxwell relation $(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$.

10. What is the physical significance of the negative signs in Maxwell relations?

Explanation: Negative signs are mathematically mandatory and arise directly from the cross-differentiation of state potentials.

11. Which thermodynamic potential yields the differential $dG = -S dT + V dP$?

Explanation: Gibbs free energy is defined as $G = H - TS$, giving $dG = -S dT + V dP$.

12. The difference between heat capacities $C_P - C_V$ can be expressed using thermal expansion and pressure coefficients as:

Explanation: Using Maxwell relations and thermodynamic identities, $C_P - C_V = T (\frac{\partial P}{\partial T})_V (\frac{\partial V}{\partial T})_P$.

13. What condition applies to the first Maxwell relation $(\frac{\partial T}{\partial V})_S = -(\frac{\partial P}{\partial S})_V$?

Explanation: The derivative is taken at constant entropy $S$, which corresponds to an isentropic or reversible adiabatic process.

14. In the context of exact differentials, if $dz = M dx + N dy$, the condition for exactness is:

Explanation: A differential is exact if and only if $(\frac{\partial M}{\partial y})_x = (\frac{\partial N}{\partial x})_y$.

15. Which of the four potentials uses temperature ($T$) and pressure ($P$) as its natural variables?

Explanation: Gibbs free energy has the differential $dG = -S dT + V dP$, meaning its natural variables are $T$ and $P$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Which of the following represents the correct fourth Maxwell relation derived from Gibbs free energy?

Explanation: From $dG = -S dT + V dP$, applying the exactness condition yields $(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial T})_P$.

2. What is the thermodynamic relation for $(\frac{\partial U}{\partial V})_T$ in terms of pressure, temperature, and volume?

Explanation: The energy equation for internal energy is derived using the first TdS equation and Maxwell relations as $(\frac{\partial U}{\partial V})_T = T(\frac{\partial P}{\partial T})_V - P$.

3. For an ideal gas, what is the value of $(\frac{\partial U}{\partial V})_T$?

Explanation: For an ideal gas, $P = \frac{RT}{V}$, so $(\frac{\partial P}{\partial T})_V = \frac{R}{V} = \frac{P}{T}$. Substituting into $(\frac{\partial U}{\partial V})_T = T(\frac{\partial P}{\partial T})_V - P$ gives $T(\frac{P}{T}) - P = 0$.

4. Which Maxwell relation is used in the evaluation of entropy changes using volume expansion coefficient at constant volume?

Explanation: The second Maxwell relation $(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$ connects entropy change with volume under isothermal conditions.

5. Which of the following equations represents the First TdS equation?

Explanation: The First TdS equation is $T dS = C_V dT + T(\frac{\partial P}{\partial T})_V dV$.

6. What does the third Maxwell relation $(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$ apply to?

Explanation: The subscript $S$ indicates that entropy is held constant, which corresponds to an isentropic or reversible adiabatic process.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the differential form of Helmholtz free energy $F$?

Explanation: Since $F = U - TS$, $dF = dU - T dS - S dT = (T dS - P dV) - T dS - S dT = -S dT - P dV$.

2. Which Maxwell relation is derived directly from Enthalpy $H$?

Explanation: Enthalpy differential is $dH = T dS + V dP$, yielding cross-derivative $(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$.

3. The second TdS equation is given by:

Explanation: The second TdS equation is $T dS = C_P dT - T(\frac{\partial V}{\partial T})_P dP$.

4. If $dz = M dx + N dy$ is an exact differential, which condition must hold true?

Explanation: Clairaut's theorem for exact differentials requires equality of mixed partial derivatives: $(\frac{\partial M}{\partial y})_x = (\frac{\partial N}{\partial x})_y$.

5. Which thermodynamic variable is held constant in the fourth Maxwell relation?

Explanation: The fourth Maxwell relation is $(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial T})_P$, where temperature $T$ is held constant (isothermal).

6. What does the expression $(\frac{\partial P}{\partial T})_V$ represent physically?

Explanation: It represents the rate of change of pressure with temperature at constant volume.

7. For a van der Waals gas, the internal energy depends on:

Explanation: Unlike ideal gases, real gases described by van der Waals equation have internal energy dependent on both temperature and volume due to intermolecular forces.

8. Which of the following is correct regarding Gibbs free energy $G$?

Explanation: Gibbs free energy reaches a minimum at equilibrium for a closed system maintained at constant temperature and pressure.

9. In the derivation of Maxwell relations, how many primary thermodynamic potentials are commonly used?

Explanation: There are four primary potentials: Internal Energy ($U$), Helmholtz Free Energy ($F$), Enthalpy ($H$), and Gibbs Free Energy ($G$).

10. Which Maxwell relation equates $(\frac{\partial S}{\partial V})_T$ with $(\frac{\partial P}{\partial T})_V$?

Explanation: The second Maxwell relation, derived from Helmholtz free energy $F$, states that $(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$.