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

{| class="wikitable" style="width: 25em;"
+ Constants
-!style="width:3em;"| "k_B"
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 systemrather 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

*$~\; U\; =\; N\; k\_B\; T^2\; left(frac\{partial\; ln\; Z\}\{partial\; T\}\; ight)\_V\; ~$
*$~\; S\; =\; frac\{U\}\{T\}\; +\; N\; k\_B\; ln\; Z\; ~$ Distinguishable Particles
*$~\; S\; =\; frac\{U\}\{T\}\; +\; N\; k\_B\; ln\; Z\; -\; N\; k\; ln\; N\; +\; Nk\; ~$ Indistinguishable Particles

*$~\; Z\_t\; =\; frac\{(2\; pi\; m\; k\_B\; T)^frac\{3\}\{2\}\; V\}\{h^3\}\; ~$
*$~\; Z\_v\; =\; frac\{1\}\{1\; -\; e^frac\{-h\; omega\}\{2\; pi\; k\_B\; T\; ~$
*$~\; Z\_r\; =\; frac\{2\; I\; k\_B\; T\}\{sigma\; (frac\{h\}\{2\; pi\})^2\}\; ~$
*$~\; sigma\; =\; 1\; ~$ 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

*$~\; C\_p=left\; (\; \{partial\; Uover\; partial\; T\}\; ight\; )\_p\; +\; p\; left\; (\; \{partial\; Vover\; partial\; T\}\; ight\; )\_p\; =\; left\; (\; \{partial\; Hover\; partial\; T\}\; ight\; )\_p=\; T\; left\; (\; \{partial\; Sover\; partial\; T\}\; ight\; )\_p\; ~$

Heat capacity at constant volume

*$~\; C\_V=left\; (\; \{partial\; Uover\; partial\; T\}\; ight\; )\_V=\; T\; left\; (\; \{partial\; Sover\; partial\; T\}\; ight\; )\_V\; ~$

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

* $~\; left\; (\; \{partial\; Tover\; partial\; V\}\; ight\; )\_\{S,N\}\; =\; -left\; (\; \{partial\; pover\; partial\; S\}\; ight\; )\_\{V,N\}\; ~$
* $~\; left\; (\; \{partial\; Tover\; partial\; p\}\; ight\; )\_\{S,N\}\; =\; left\; (\; \{partial\; Vover\; partial\; S\}\; ight\; )\_\{p,N\}\; ~$
* $~\; left\; (\; \{partial\; Tover\; partial\; V\}\; ight\; )\_\{p,N\}\; =\; -left\; (\; \{partial\; pover\; partial\; S\}\; ight\; )\_\{T,N\}\; ~$
* $~\; left\; (\; \{partial\; Tover\; partial\; p\}\; ight\; )\_\{V,N\}\; =\; left\; (\; \{partial\; Vover\; partial\; S\}\; ight\; )\_\{T,N\}\; ~$

Incremental processes

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

Compressibility at constant temperature

*$~\; K\_T\; =\; -\{\; 1over\; V\; \}\; left\; (\; \{partial\; Vover\; partial\; p\}\; ight\; )\_\{T,N\}\; ~$

More relations

* $~\; left\; (\; \{partial\; Sover\; partial\; U\}\; ight\; )\_\{V,N\}\; =\; \{\; 1over\; T\; \}\; ~$
* $~\; left\; (\; \{partial\; Sover\; partial\; V\}\; ight\; )\_\{N,U\}\; =\; \{\; pover\; T\; \}\; ~$
* $~\; left\; (\; \{partial\; Sover\; partial\; N\}\; ight\; )\_\{V,U\}\; =\; -\; \{\; mu\; over\; T\; \}\; ~$
* $~\; left\; (\; \{partial\; Tover\; partial\; S\}\; ight\; )\_V\; =\; \{\; T\; over\; C\_V\; \}\; ~$
* $~\; left\; (\; \{partial\; Tover\; partial\; S\}\; ight\; )\_p\; =\; \{\; T\; over\; C\_p\; \}\; ~$
* $~\; -left\; (\; \{partial\; pover\; partial\; V\}\; ight\; )\_T\; =\; \{\; 1\; over\; \{VK\_T\}\; \}\; ~$

Equation Table for an Ideal Gas

Other useful identities

*$Delta\; U\; =\; q\_\{by\}\; +\; w\_\{on\}\; =\; q\_\{by\}\; -\; int\; p\_\{ext\}dV\; =\; q\_\{by\}\; -\; p\_\{ext\}Delta\; V$

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

*$G\; =\; H\; -\; TS\; =\; sum\_\{i\}\; mu\_\{i\}\; N\_\{i\}\; ,!$
*$dUleft(S,V,\{n\_\{i\; ight)\; =\; TdS\; -\; pdV\; +\; sum\_\{i\}\; mu\_\{i\}\; dN\_i$
*$dHleft(S,p,n\_\{i\}\; ight)\; =\; TdS\; +\; Vdp\; +\; sum\_\{i\}\; mu\_\{i\}\; dN\_\{i\}$
*$dAleft(T,V,n\_\{i\}\; ight)\; =\; -SdT\; -\; pdV\; +\; sum\_\{i\}\; mu\_\{i\}\; dN\_\{i\}$
*$dGleft(T,p,n\_\{i\}\; ight)\; =\; -SdT\; +\; Vdp\; +\; sum\_\{i\}\; mu\_\{i\}\; dN\_\{i\}$

*$mu\_\{JT\}\; =\; left(frac\{partial\; T\}\{partial\; p\}\; ight)\_H$
*$kappa\_\{T\}\; =\; -frac\{1\}\{V\}left(frac\{partial\; V\}\{partial\; p\}\; ight)\_T$
*$alpha\_\{p\}\; =\; frac\{1\}\{V\}left(frac\{partial\; V\}\{partial\; T\}\; ight)\_p$

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

Proof #1

An example using the above methods is:

:$left(frac\{partial\; T\}\{partial\; p\}\; ight)\_H\; =\; -frac\{1\}\{C\_p\}\; left(frac\{partial\; H\}\{partial\; p\}\; ight)\_T$

:$left(frac\{partial\; T\}\{partial\; p\}\; ight)\_Hleft(frac\{partial\; p\}\{partial\; H\}\; ight)\_Tleft(frac\{partial\; H\}\{partial\; T\}\; ight)\_p=\; -1$

:$left(frac\{partial\; T\}\{partial\; p\}\; ight)\_H\; =\; -left(frac\{partial\; H\}\{partial\; p\}\; ight)\_T\; left(frac\{partial\; T\}\{partial\; H\}\; ight)\_P$

::::$=\; frac\{-1\}\{left(frac\{partial\; H\}\{partial\; T\}\; ight)\_p\}\; left(frac\{partial\; H\}\{partial\; p\}\; ight)\_T$ ; $C\_p\; =\; left(frac\{partial\; H\}\{partial\; T\}\; ight)\_p$

:$Rightarrow\; left(frac\{partial\; T\}\{partial\; p\}\; ight)\_H=\; -frac\{1\}\{C\_p\}\; left(frac\{partial\; H\}\{partial\; p\}\; ight)\_T$

Proof #2

Another example:

:$C\_V\; =\; Tleft(frac\{partial\; S\}\{partial\; T\}\; ight)\_V$

:$U\; =\; q\; +\; w\; ,!$

:$dU\; =\; dq\_\{rev\}\; +\; w\_\{rev\}\; ;\; dS\; =\; frac\{dq\_\{rev\{T\},\; w\_\{rev\}\; =\; -pdV\; ,!$

::$=\; TdS-pdV\; ,!$

:$left(frac\{partial\; U\}\{partial\; T\}\; ight)\_V=\; Tleft(frac\{partial\; S\}\{partial\; T\}\; ight)\_V-\; pleft(frac\{partial\; V\}\{partial\; T\}\; ight)\_V\; ;\; C\_V\; =\; left(frac\{partial\; U\}\{partial\; T\}\; ight)\_V$

:$Rightarrow\; C\_V\; =\; Tleft(frac\{partial\; S\}\{partial\; T\}\; ight)\_V$

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.