Table of thermodynamic equations

For more elaboration on these equations see: thermodynamic equations.

For list of mathematical notation used in these equations see: mathematical notation.

The following page is a concise list of common thermodynamic equations and quantities:

Variables

Main article: List of thermodynamic properties

Conjugate variables
p	Pressure
V	Volume
T	Temperature
S	Entropy
μ	Chemical potential
N	Particle number

Thermodynamic potentials
U	Internal energy
A	Helmholtz free energy
H	Enthalpy
G	Gibbs free energy

Material properties
ρ	Density
C_V	Heat capacity (constant volume)
C_P	Heat capacity (constant pressure)
β_T	Isothermal compressibility
β_S	Adiabatic compressibility
α	Coefficient of thermal expansion

Constants
k_B	Boltzmann constant
R	Ideal gas constant

Other conventional variables
$W$	Work done by the system on its surroundings
$Q$	Heat transferred from the surroundings into the system
δw	Infinitesimal amount of Work
δq	Infinitesimal amount of Heat

Equations

The equations in this article are classified by subject.

Entropy

$~ S = k_B (\ln \Omega) ~$ , where $k_B$ is the Boltzmann constant, and $\Omega$ denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
$~ dS = \frac{\delta Q}{T} ~$ , for reversible processes only

Quantum Properties

$~ U = N k_B T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V ~$

$~ S = \frac{U}{T} + N * ~ S = \frac{U}{T} + N k_B \ln Z - N k \ln N + Nk ~$ Indistinguishable Particles

where N is number of particles, Z is the partition function, h is Planck's constant, I is moment of inertia, Z_t is Z_translation, Z_v is Z_vibration, Z_r is Z_rotation

$~ Z_t = \frac{(2 \pi m k_B T)^\frac{3}{2} V}{h^3} ~$

$~ Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_B T}} ~$

$~ Z_r = \frac{2 I k_B T}{\sigma (\frac{h}{2 \pi})^2} ~$

where:

$~ \sigma = 1 ~$ , heteronuclear
$~ \sigma = 2 ~$ , homonuclear

Quasi-static and reversible processes

$~ dQ=C_p dT+l_v d_v =dU+PdV =TdS~$

Heat capacity at constant pressure

$~ C_p = \left ( {\partial Q_{rev} \over \partial T} \right )_p = \left ( {\partial U \over \partial T} \right )_p + p \left ( {\partial V \over \partial T} \right )_p = \left ( {\partial H \over \partial T} \right )_p = T \left ( {\partial S \over \partial T} \right )_p ~$

Heat capacity at constant volume

$~ C_V = \left ( {\partial Q_{rev} \over \partial T} \right )_V = \left ( {\partial U \over \partial T} \right )_V = T \left ( {\partial S \over \partial T} \right )_V ~$

Thermodynamic potentials and related concepts

Name	Symbol	Formula	Natural variables
Internal energy	$U$	$\int ( T dS - p dV + \sum_i \mu_i dN_i )$	$S, V, \{N_i\}$
Helmholtz free energy	$F$	$U-TS$	$T, V, \{N_i\}$
Enthalpy	$H$	$U+pV$	$S, p, \{N_i\}$
Gibbs free energy	$G$	$U+pV-TS$	$T, p, \{N_i\}$
Landau Potential (Grand potential)	$\Omega$ , $\Phi_{G}$	$U - T S -$ $\sum_i\,$ $\mu_i N_i$	$T, V, \{\mu_i\}$

Compressibility at constant temperature

$~ K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N} ~$

More relations

$~ \left ( {\partial S\over \partial U} \right )_{V,N} = { 1\over T } ~$
$~ \left ( {\partial S\over \partial V} \right )_{N,U} = { p\over T } ~$
$~ \left ( {\partial S\over \partial N} \right )_{V,U} = - { \mu \over T } ~$
$~ \left ( {\partial T\over \partial S} \right )_V = { T \over C_V } ~$
$~ \left ( {\partial T\over \partial S} \right )_p = { T \over C_p } ~$
$~ -\left ( {\partial p\over \partial V} \right )_T = { 1 \over {VK_T} } ~$

Equation Table for an Ideal Gas

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic $Q=0$
Work $W$	$\delta W = p dV\;$	$p\Delta V\;$	$0\;$	$nRT\ln\frac{V_2}{V_1}\;$	$\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma}$ ^[1] = $C_V \left(T_1 - T_2 \right)$
Heat Capacity C	(as for real gas)	$C_p = \frac{5}{2}nR\;$ (for monatomic ideal gas)	$C_V = \frac{3}{2}nR \;$ (for monatomic ideal gas)
Internal Energy ΔU	$\Delta U = C_v \Delta T\;$	$Q - W\;$ $Q_p - p\Delta V\;$	$Q\;$ $C_V\left ( T_2-T_1 \right )\;$	$0\;$ $Q=-W\;$	$-W\;$ $C_V\left ( T_2-T_1 \right )\;$
Enthalpy ΔH	$H=U+pV\;$	$C_p\left ( T_2-T_1 \right )\;$	$Q_V+V\Delta p\;$	$0\;$	$C_p\left ( T_2-T_1 \right )\;$
Entropy ΔS	$\Delta S = C_v \ln{T_2 \over T_1} + R \ln{V_2 \over V_1}$ $\Delta S = C_p \ln{T_2 \over T_1} - R \ln{p_2 \over p_1}$ ^[2]	$C_p\ln\frac{T_2}{T_1}\;$	$C_V\ln\frac{T_2}{T_1}\;$	$nR\ln\frac{V_2}{V_1}\;$ $\frac{Q}{T}\;$	$C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;$
Constant	$\;$	$\frac{V}{T}\;$	$\frac{p}{T}\;$	$p V\;$	$p V^\gamma\;$

Other useful identities

$\Delta U = Q - W = Q - \int p_{ext} dV = Q - p_{ext}\Delta V$

$H = U + pV \,\!$
$A = U - TS \,\!$

$G = H - TS = \sum_{i} \mu_{i} N_{i} \,\!$
$dU\left(S,V,{n_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i$
$dH\left(S,p,n_{i}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}$
$dA\left(T,V,n_{i}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}$
$dG\left(T,p,n_{i}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}$

$\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H$
$\kappa_{T} = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T$
$\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$

$\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_p$
$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$
$H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p$
$U = -T^2\left(\frac{\partial \left(A/T\right)}{\partial T}\right)_V$

Proof #1

An example using the above methods is:

$\left(\frac{\partial T}{\partial p}\right)_H = -\frac{1}{C_p} \left(\frac{\partial H}{\partial p}\right)_T$

$\left(\frac{\partial T}{\partial p}\right)_H \left(\frac{\partial p}{\partial H}\right)_T \left(\frac{\partial H}{\partial T}\right)_p = -1$

$\left(\frac{\partial T}{\partial p}\right)_H = -\left(\frac{\partial H}{\partial p}\right)_T \left(\frac{\partial T}{\partial H}\right)_p$

$= \frac{-1}{\left(\frac{\partial H}{\partial T}\right)_p} \left(\frac{\partial H}{\partial p}\right)_T$ ; $C_p = \left(\frac{\partial H}{\partial T}\right)_p$

$\Rightarrow \left(\frac{\partial T}{\partial p}\right)_H = -\frac{1}{C_p} \left(\frac{\partial H}{\partial p}\right)_T$

Proof #2

Another example:

$C_V = T\left(\frac{\partial S}{\partial T}\right)_V$

$'''U = Q - W \,\!'''$

$dU = \delta Q_{rev} - \delta W_{rev} ; dS = \frac{\delta Q_{rev}}{T}, \delta W_{rev} = pdV \,\!$

$= TdS-pdV \,\!$

$\left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V - p\left(\frac{\partial V}{\partial T}\right)_V ; C_V = \left(\frac{\partial U}{\partial T}\right)_V$

$\Rightarrow C_V = T\left(\frac{\partial S}{\partial T}\right)_V$

References

^ http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html
^ Keenan, Thermodynamics, Wiley, New York, 1947

Atkins, Peter and de Paula, Julio Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 [ISBN 0-7167-3539-3].
- Chapters 1 - 10, Part 1: Equilibrium.
Bridgman, P.W., Phys. Rev., 3, 273 (1914).
Landsberg, Peter T. Thermodynamics and Statistical Mechanics. New York: Dover Publications, Inc., 1990. (reprinted from Oxford University Press, 1978).
Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.
Reichl, L.E., "A Modern Course in Statistical Physics", 2nd edition, New York: John Wiley & Sons, 1998.
Schroeder, Daniel V. Thermal Physics. San Francisco: Addison Wesley Longman, 2000 [ISBN 0-201-38027-7].
Silbey, Robert J., et al. Physical Chemistry. 4th ed. New Jersey: Wiley, 2004.
Callen, Herbert B. (1985). "Thermodynamics and an Introduction to Themostatistics", 2nd Ed., New York: John Wiley & Sons.

Branches of physics

Core

Divisions	Experimental physics Theoretical physics

Energy and motion	Classical mechanics Newtonian mechanics Lagrangian mechanics Hamiltonian mechanics Continuum mechanics Celestial mechanics Statistical mechanics Thermodynamics Fluid mechanics Biomechanics Quantum mechanics

Waves and fields	Gravitation Electromagnetism Quantum field theory Relativity Special relativity General relativity

Applied

Physical sciences and mathematics	Acoustics Astrophysics Atomic, molecular, and optical physics Atomic physics Chemical physics Computational physics Condensed matter physics High energy physics Mathematical physics Optics Plasma physics

Biology, geology, economics	Biophysics Medical physics Neurophysics Agrophysics Soil physics Atmospheric physics Econophysics Geophysics