Table of thermodynamic equations

For more elaboration on these equations see: thermodynamic equations.

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

Variables

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
CV	Heat capacity (constant volume)
CP	Heat capacity (constant pressure)
βT	Isothermal compressibility
βS	Adiabatic compressibility
α	Coefficient of thermal expansion

Other conventional variables

w	Work
q	Heat

Constants

kB	Boltzmann constant
R	Ideal gas constant

Equations

The equations in this article are classified by subject.

First law of thermodynamics

• $~dU=\delta q-\delta w~$

Note that the symbol $\delta$ represents the fact that because q and w are not state functions, $\delta q$ and $\delta w$ are inexact differentials.

In some fields such as physical chemistry, positive work is conventionally considered work done on the system rather than by the system, and the law is expressed as $dU=\delta q+\delta w$.

Entropy

• $~S=k_{B}(\ln \Omega )~$
• $~dS={\frac {\delta q}{T}}~$, for reversible processes only

Quantum Properties

• $~U=Nk_{B}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}~$
• $~S={\frac {U}{T}}+Nk_{B}\ln Z~$ Distinguishable Particles
• $~S={\frac {U}{T}}+Nk_{B}\ln Z-Nk\ln N+Nk~$ Indistinguishable Particles

• $~Z_{t}={\frac {(2\pi mk_{B}T)^{\frac {3}{2}}V}{h^{3}}}~$
• $~Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{B}T}}}}~$
• $~Z_{r}={\frac {2Ik_{B}T}{\sigma ({\frac {h}{2\pi }})^{2}}}~$
• $~\sigma =1~$ heteronuclear
• $~\sigma =2~$ homonuclear

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

Quasi-static and reversible process

• $~dQ=C_{p}dT+l_{v}d_{v}=dU+PdV=TdS~$

Heat capacity at constant pressure

• $~C_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}~$

Heat capacity at constant volume

• $~C_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}~$

Fundamental Equation of Thermodynamics

• $~U=TS-pV+\mu N$

Enthalpy

• $~H\equiv U+pV=\mu N+TS~$

Helmholtz free energy

• $~A\equiv U-TS=\mu N-pV~$

Gibbs free energy

• $~G\equiv U+pV-TS=H-TS=\mu N~$

Maxwell relations

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

Incremental processes

• $~dU=T\,dS-p\,dV+\mu \,dN~$
• $~dA=-S\,dT-p\,dV+\mu \,dN~$
• $~dG=-S\,dT+V\,dp+\mu \,dN=\mu \,dN+N\,d\mu ~$
• $~dH=T\,dS+V\,dp+\mu \,dN~$

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	Constant Pressure Δp = 0	Constant Volume ΔV = 0	Isothermal ΔT = 0	Adiabatic q = 0
Work ΔW	${\begin{matrix}w=-\int _{V_{1}}^{V_{2}}pdV\end{matrix}}$	$-p\left(V_{2}-V_{1}\right)\;$	$0\;$	$-nRT\ln {\frac {V_{2}}{V_{1}}}\;$	$C_{V}\left(T_{2}-T_{1}\right)\;$
Heat Capacity C		$C_{p}=(5/2)nR\;$	$C_{V}=(3/2)nR\;$	$C_{p}\;$ or $C_{V}\;$	$C_{p}\;$ or $C_{V}\;$
Internal Energy ΔU	$\Delta U=3/2*nR\Delta T\;$	$q+w\;$ $q_{p}+p\Delta V\;$	$q\;$ $C_{V}\left(T_{2}-T_{1}\right)\;$	$0\;$ $q=-w\;$	$w\;$ $C_{V}\left(T_{2}-T_{1}\right)\;$
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	${\begin{matrix}\Delta S=-\int _{T_{1}}^{T_{2}}{\frac {C}{T}}dT\end{matrix}}$	$C_{p}\ln {\frac {T_{2}}{T_{1}}}\;$	$C_{V}\ln {\frac {T_{2}}{T_{1}}}\;$	$nR\ln {\frac {V_{2}}{V_{1}}}\;$ ${\frac {q}{T}}\;$	$C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;$

Other useful identities

• $\Delta U=q_{by}+w_{on}=q_{by}-\int p_{ext}dV=q_{by}-p_{ext}\Delta V$

• $H=U+pV\,\!$
• $A=U-TS\,\!$

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

• $\mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}$
• $\kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T}$
• $\alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}$

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

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:

$C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}$

$U=q+w\,\!$

$dU=dq_{rev}+w_{rev};dS={\frac {dq_{rev}}{T}},w_{rev}=-pdV\,\!$

$=TdS-pdV\,\!$

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

$\Rightarrow C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}$

References

• 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.