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Maxwell relations

In thermodynamics, Maxwell relations are relations between partial derivatives of quantities, which describe the properties of a system.

Topics in Calculus

Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Lists of integrals

Vector Calculus

Vector | Vector field | Matrix | Partial Derivative | Gradient | Flux | Divergence | Divergence Theorem | Del | Curl | Stokes' Theorem

Tensor Calculus

Tensor | Tensor field | Tensor product | Exterior power | Exterior Derivative | Covariant derivative | Manifold

Helmholtz free energy	F
Gibbs free energy	G
Enthalpy	H
Particle number	N
Pressure	P
Density	ρ
Entropy	S
Temperature	T
Internal energy	U
Volume	V

Equations

<math>\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V<math>	<math>\qquad \qquad \qquad (1)<math>
<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P<math>	<math>\qquad \qquad \qquad (2)<math>
<math>\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V<math>	<math>\qquad \qquad \qquad (3)<math>
<math>\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P<math>	<math>\qquad \qquad \qquad (4)<math>
<math>\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T<math>	<math>\qquad \qquad \qquad (5)<math>

Useful identities

<math>\Delta U = q_{by} + w_{on} = q_{by} - \int P_{ext}dV = q_{by} - P_{ext}\Delta V<math>

<math>H = U + PV \,\!<math>

<math>A = U - TS \,\!<math>

<math>G = H - TS = \sum_{i} \mu_{i} n_{i} \,\!<math>

<math>dU\left(S,V,{n_{i}}\right) = TdS - PdV + \sum_{i} \mu_{i} dn_i<math>

<math>dH\left(S,P,n_{i}\right) = TdS + VdP + \sum_{i} \mu_{i} dn_{i}<math>

<math>dA\left(T,V,n_{i}\right) = -SdT - PdV + \sum_{i} \mu_{i} dn_{i}<math>

<math>dG\left(T,P,n_{i}\right) = -SdT + VdP + \sum_{i} \mu_{i} dn_{i}<math>

<math>C_v = \left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V<math>

<math>C_p = \left(\frac{\partial H}{\partial T}\right)_P<math>

<math>\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H<math>

<math>\kappa_{T} = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T<math>

<math>\alpha_{P} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P<math>

<math>\left(\frac{\partial H}{\partial P}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_P<math>

<math>\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P<math>

<math>H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_P<math>

<math>U = -T^2\left(\frac{\partial \left(A/T\right)}{\partial T}\right)_V<math>

Pressure, temperature, and volume are easily measured properties.

Proofs

[Top]

Proof #2

<math>

\left(\frac{\partial T}{\partial P}\right)_H = -\frac{1}{C_p}

\left(\frac{\partial H}{\partial P}\right)_T

<math>

<math>

\left(\frac{\partial T}{\partial P}\right)_H \left(\frac{\partial P}{\partial H}\right)_T \left(\frac{\partial H}{\partial T}\right)_P = -1 <math>

<math>

\left(\frac{\partial T}{\partial P}\right)_H = -\left(\frac{\partial H}{\partial P}\right)_T

\left(\frac{\partial T}{\partial H}\right)_P

<math>

<math>

= \frac{-1}{\left(\frac{\partial T}{\partial H}\right)_P}

\left(\frac{\partial H}{\partial P}\right)_T

<math> ; <math>C_p = \left(\frac{\partial H}{\partial T}\right)_P<math>

<math>

\Rightarrow \left(\frac{\partial T}{\partial P}\right)_H = -\frac{1}{C_p}

\left(\frac{\partial H}{\partial P}\right)_T

<math>

[Top]

Proof #3

<math>

C_v = T\left(\frac{\partial S}{\partial T}\right)_V <math>

<math>

U = q + w \,\! <math>

<math>

dU = dq_{rev} + w_{rev} ; dS = \frac{dq_{rev}}{T}, w_{rev} = -PdV \,\! <math>

<math>

= TdS-PdV \,\! <math>

<math>

\left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V - P\left(\frac{\partial V}{\partial T}\right)_V ; C_v = \left(\frac{\partial U}{\partial T}\right)_V <math>

<math>

\Rightarrow C_v = T\left(\frac{\partial S}{\partial T}\right)_V <math>

References

• Peter Atkins and Julio de Paula, Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 [ISBN 0-7167-3539-3].
  • Chapters 1 - 10, Part 1: Equilibrium.

See Also

• Thermodynamics

General subfields within physics
Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism | General relativity | Particle physics | Quantum field theory | Quantum mechanics | Solid state physics | Special relativity | Statistical mechanics | Thermodynamics