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# Table of thermodynamic equations

Thermodynamic equations
Laws of thermodynamics
Conjugate variables
Thermodynamic potential
Material properties
Maxwell relations
Bridgman's equations
Exact differential
Table of thermodynamic equations
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For more elaboration on these equations see: thermodynamic equations.

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

## List

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
α Coefficient of thermal expansion
Other conventional variables
w Work
q Heat
Constants
kB Boltzmann constant
R Ideal gas constant

The following equations are classified by subject.

### First law of thermodynamics

$~ dU=\delta q-\delta w ~$


Note that the symbol δ represents the fact that because q and w are not state functions, δq and δw 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 = δq + δw.

### Entropy

$~ S = k (\ln \Omega) ~$
$~ \Delta S = \frac{\Delta Q}{T} ~$, only for reversible processes


### Quantum Properties

$~ U = N k_B T^2 (\frac{\partial \ln Z}{\partial T})_V ~$
$~ S = \frac{U}{T} + N k_B \ln Z ~$  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, Zt is Ztranslation, Zv is Zvibration, Zr is Zrotation

### 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

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


### 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 (PVm = constant)

 Constant Pressure Constant Volume Isothermal Adiabatic Variable $\Delta p=0\;$ $\Delta V=0\;$ $\Delta T=0\;$ $q=0\;$ $m\;$ $0\;$ $\infty\;$ $1\;$ $\gamma=\frac {C_p}{C_V} \;$ Work$\begin{matrix}w=-\int_{V_1}^{V_2} pdV \end{matrix}$ $-p\left ( V_2-V_1 \right )\;$ $0\;$ $-nRT\ln\frac{V_2}{V_1}\;$ $C_V\left ( T_2-T_1 \right )\;$ 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, $\Delta U = 3/2 *nR\Delta T\;$ $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, $\Delta H\;$$H=U+pV\;$ $C_p\left ( T_2-T_1 \right )\;$ $q_V+V\Delta P\;$ $0\;$ $C_p\left ( T_2-T_1 \right )\;$ Entropy$\begin{matrix}\Delta S=-\int_{T_1}^{T_2} \frac {C}{T}dT \end{matrix}$ $C_p\ln\frac{T_2}{T_1}\;$ $C_V\ln\frac{T_2}{T_1}\;$ $nR\ln\frac{V_2}{V_1}\;$$\frac{q}{T}\;$ $0\;$

## Other useful identities

$\Delta U = q_{by} + w_{on} = q_{by} - \int p_{ext}dV = q_{by} - p_{ext}\Delta V$
$H = U + pV \,\!$
$A = U - TS \,\!$
$G = H - TS = \sum_{i} \mu_{i} N_{i} \,\!$
$dU\left(S,V,{n_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i$
$dH\left(S,p,n_{i}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}$
$dA\left(T,V,n_{i}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}$
$dG\left(T,p,n_{i}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}$
$C_V = \left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V$
$C_p = \left(\frac{\partial H}{\partial T}\right)_p$
$\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H$
$\kappa_{T} = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T$
$\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$
$\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_p$
$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$
$H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p$
$U = -T^2\left(\frac{\partial \left(A/T\right)}{\partial T}\right)_V$

### 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.
• 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.