View source for Table of thermodynamic equations

{{Short description|Thermodynamics}}
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
Common [[thermodynamic equations]] and [[physical quantity|quantities]] in [[thermodynamics]], using [[mathematical notation]], are as follows:

== Definitions ==
{{Main article|List of thermodynamic properties|Thermodynamic potential|Free entropy|Defining equation (physical chemistry)}}

Many of the definitions below are also used in the thermodynamics of [[chemical reaction]]s.

=== General basic quantities ===

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s) 
! scope="col" width="125" | (Common) symbol/s 
! scope="col" width="125" | SI unit
! scope="col" width="100" | Dimension
|-
! Number of molecules
| ''N''
| 1
| 1
|-
! Amount of substance
| ''n''
| mol
| N
|-
! [[Temperature]]
| ''T''
| K
| Θ
|-
! [[Heat|Heat Energy]]
| ''Q'', ''q''
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Latent heat]]
| ''Q<sub>L</sub>''
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
|}

=== General derived quantities ===

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s) 
! scope="col" width="125" | (Common) symbol/s 
! scope="col" width="200" | Defining equation 
! scope="col" width="125" | SI unit
! scope="col" width="100" | Dimension
|-
! [[Thermodynamic beta]], inverse temperature 
| ''β''
| <math> \beta = 1/k_\text{B} T </math>
| J<sup>−1</sup>
| T<sup>2</sup>M<sup>−1</sup>L<sup>−2</sup>
|-
! [[Thermodynamic temperature]]
| ''τ''
| <math> \tau = k_\text{B} T </math>
<math> \tau = k_\text{B} \left (\partial U/\partial S \right )_{N} </math>
<math> 1/\tau = 1/k_\text{B} \left (\partial S/\partial U \right )_{N} </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Entropy]]
| ''S''
| <math>S = -k_\text{B}\sum_i p_i\ln p_i</math>
<math> S = -\left (\partial F/\partial T \right )_{V,N} </math>   ,
<math> S = -\left (\partial G/\partial T \right )_{P,N} </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! [[Pressure]]
| ''P'' 
| <math> P = - \left (\partial F/\partial V \right )_{T,N} </math>
<math> P = - \left (\partial U/\partial V \right )_{S,N} </math>
| Pa
| ML<sup>−1</sup>T<sup>−2</sup>
|-
! [[Internal Energy]]
| ''U''
| <math>U = \sum_i  E_i </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Enthalpy]]
| ''H''
| <math> H = U+pV </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Partition function (statistical mechanics)|Partition Function]] 
| ''Z''
|
| 1
| 1
|-
! [[Gibbs free energy]]
| ''G''
| <math> G = H - TS </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Chemical potential]] (of component ''i'' in a mixture)
| ''μ<sub>i</sub>''
| <math> \mu_i = \left (\partial U/\partial N_i \right )_{N_{j \neq i}, S, V } </math>
<math> \mu_i = \left (\partial F/\partial N_i \right )_{T, V } </math>, where <math>F</math> is not proportional to <math>N</math> because <math>\mu_i</math> depends on pressure.
<math> \mu_i = \left (\partial G/\partial N_i \right )_{T, P } </math>, where <math>G</math> is proportional to <math>N</math> (as long as the molar ratio composition of the system remains the same) because <math>\mu_i</math> depends only on temperature and pressure and composition.
<math> \mu_i/\tau = -1/k_\text{B} \left (\partial S/\partial N_i \right )_{U,V} </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Helmholtz free energy]]
| ''A'', ''F''
| <math> F = U - TS </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! [[Grand potential|Landau potential]], Landau free energy, [[Grand potential]]
| ''Ω'', ''Φ<sub>G</sub>''
| <math> \Omega = U - TS - \mu N\ </math>
| J
| ML<sup>2</sup>T<sup>−2</sup>
|-
! Massieu potential, Helmholtz [[free entropy]]
| ''Φ''
| <math> \Phi = S - U/T </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Planck potential, Gibbs [[free entropy]]
| ''Ξ''
| <math> \Xi = \Phi - pV/T </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
|}

=== Thermal properties of matter ===

{{Main articles|Heat capacity|Thermal expansion}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI unit 
! scope="col" width="100" | Dimension
|-
! General heat/thermal capacity 
| ''C''
| <math> C = \partial Q/\partial T </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Heat capacity (isobaric)
| ''C<sub>p</sub>''
| <math> C_{p} = \partial H/\partial T </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Specific heat capacity (isobaric)
| ''C<sub>mp</sub>''
| <math> C_{mp} = \partial^2 Q/\partial m \partial T </math>
| J⋅kg<sup>−1</sup>⋅K<sup>−1</sup> 
| L<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Molar specific heat capacity (isobaric)
| ''C<sub>np</sub>''
| <math>C_{np} = \partial^2 Q/\partial n \partial T </math>
| J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>N<sup>−1</sup>
|-
! Heat capacity (isochoric/volumetric)
| ''C<sub>V</sub>''
| <math> C_{V} = \partial U/\partial T </math>
| J⋅K<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Specific heat capacity (isochoric)
| ''C<sub>mV</sub>''
| <math> C_{mV} = \partial^2 Q/\partial m \partial T </math>
| J⋅kg<sup>−1</sup>⋅K<sup>−1</sup>
| L<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>
|-
! Molar specific heat capacity (isochoric)
| ''C<sub>nV</sub>''
| <math> C_{nV} = \partial^2 Q/\partial n \partial T </math>
| J⋅K⋅<sup>−1</sup> mol<sup>−1</sup>
| ML<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>N<sup>−1</sup>
|-
! [[Latent heat|Specific latent heat]]
| ''L''
| <math>L = \partial Q/ \partial m </math>
| J⋅kg<sup>−1</sup>
| L<sup>2</sup>T<sup>−2</sup>
|-
! Ratio of isobaric to isochoric heat capacity, [[heat capacity ratio]], adiabatic index, [[Laplace]] coefficient
| ''γ''
| <math>\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} </math>
| 1
| 1
|-
|}

=== Thermal transfer ===

{{Main|Thermal conductivity}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI unit 
! scope="col" width="100" | Dimension
|-
! [[Temperature gradient]]
| No standard symbol 
| <math> \nabla T </math>
| K⋅m<sup>−1</sup>
| ΘL<sup>−1</sup>
|-
! Thermal conduction rate, thermal current, thermal/[[heat flux]], thermal power transfer
| ''P''
| <math> P = \mathrm{d} Q/\mathrm{d} t </math>
| W
| ML<sup>2</sup>T<sup>−3</sup>
|-
! Thermal intensity
| ''I''
| <math> I = \mathrm{d} P/\mathrm{d} A </math>
| W⋅m<sup>−2</sup>
| MT<sup>−3</sup>
|-
! Thermal/heat flux density (vector analogue of thermal intensity above)
| '''q'''
| <math> Q = \iint \mathbf{q} \cdot \mathrm{d}\mathbf{S}\mathrm{d} t </math>
| W⋅m<sup>−2</sup>
| MT<sup>−3</sup>
|-
|}

== Equations ==

The equations in this article are classified by subject.

=== Thermodynamic processes ===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="10" | Equations
|-
! [[Isentropic process]] (adiabatic and reversible)
| <math> Q = 0, \quad \Delta U = -W </math>

For an ideal gas<br />
<math>p_1 V_1^{\gamma} = p_2 V_2^{\gamma} </math><br />
<math>T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} </math><br />
<math>p_1^{1-\gamma} T_1^{\gamma} = p_2^{1 - \gamma} T_2^{\gamma} </math>
|-
![[Isothermal process]]
|<math>\Delta U = 0, \quad W =  Q </math>

For an ideal gas<br />
<math>W=kTN \ln(V_2/V_1) </math>
<math>W=nRT \ln(V_2/V_1) </math>
|-
![[Isobaric process]]
|''p''<sub>1</sub> = ''p''<sub>2</sub>, ''p'' = constant <br />
<math>W = p \Delta V, \quad Q = \Delta U + p \delta V </math>
|-
! [[Isochoric process]]
| ''V''<sub>1</sub> = ''V''<sub>2</sub>, ''V'' = constant <br />
<math>W = 0, \quad Q = \Delta U </math>
|-
! [[Joule expansion|Free expansion]]
| <math>\Delta U = 0 </math>
|-
! Work done by an expanding gas
| Process<br />
<math> W = \int_{V_1}^{V_2} p \mathrm{d}V </math>

Net work done in cyclic processes<br />
<math> W = \oint_\mathrm{cycle} p \mathrm{d}V = \oint_\mathrm{cycle}\Delta Q </math>
|-
|}

=== Kinetic theory ===

{| class="wikitable"
|+Ideal gas equations
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! [[Ideal gas law]]
| {{plainlist}}
* ''p'' = pressure
* ''V'' = volume of container
* ''T'' = temperature 
* ''n'' = amount of substance
* ''R'' = [[gas constant]]
* ''N'' = number of molecules
* ''k'' = [[Boltzmann constant]]
{{endplainlist}}
| <math>pV = nRT = kTN </math><br />
<math>\frac{p_1 V_1}{p_2 V_2} = \frac{n_1 T_1}{n_2 T_2} = \frac{N_1 T_1}{N_2 T_2} </math>
|-
! Pressure of an ideal gas
| {{plainlist}}
* ''m'' = mass of ''one'' molecule
* ''M<sub>m</sub>'' = molar mass
{{endplainlist}}
| <math>p = \frac{Nm \langle v^2 \rangle}{3V} = \frac{nM_m \langle v^2 \rangle}{3V} = \frac{1}{3}\rho \langle v^2 \rangle </math>
|-
|}

==== Ideal gas ====

{| class="wikitable sortable mw-collapsible"
|+
|-
! Quantity
! General Equation
! Isobaric<br />Δ''p'' = 0
! Isochoric<br />Δ''V'' = 0
! Isothermal<br />Δ''T'' = 0
! Adiabatic<br /><math>Q=0</math>
|-
! Work <br /> ''W''
| align="center" | <math> \delta W = -p dV\;</math>
| align="center" | <math>-p\Delta V\;</math>
| align="center" | <math>0\;</math>
| align="center" | <math>-nRT\ln\frac{V_2}{V_1}\;</math>
<math>-nRT\ln\frac{P_1}{P_2}\;</math>
| align="center" | <math>\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma} = C_V \left(T_2 - T_1 \right)</math>
|-
! Heat Capacity <br /> ''C''
| align="center" | (as for real gas)
| align="center" | <math>C_p = \frac{5}{2}nR </math>{{br}}(for monatomic ideal gas) 
<math>C_p = \frac{7}{2}nR </math>{{br}}(for diatomic ideal gas)
| align="center" | <math>C_V = \frac{3}{2}nR </math>{{br}}(for monatomic ideal gas) 
<math>C_V = \frac{5}{2}nR \;</math>{{br}}(for diatomic ideal gas)
|
|
|-
! Internal Energy <br /> Δ''U'' 
| align="center" | <math>\Delta U = C_V \Delta T\;</math>
| align="center" | <math>Q + W\;</math>{{br}}{{br}}<math>Q_p - p\Delta V </math>
| align="center" | <math>Q\;</math>{{br}}{{br}}<math>C_V\left ( T_2-T_1 \right ) </math>
| align="center" | <math>0\;</math>{{br}}<math>Q=-W </math>
| align="center" | <math>W\;</math>{{br}}<math>C_V\left ( T_2-T_1 \right ) </math>
|-
! Enthalpy <br /> Δ''H''
| align="center" | <math>H=U+pV\;</math>
| align="center" | <math>C_p\left ( T_2-T_1 \right )\;</math>
| align="center" | <math>Q_V+V\Delta p\;</math>
| align="center" | <math>0\;</math>
| align="center" | <math>C_p\left ( T_2-T_1 \right )\;</math>
|-
! Entropy <br /> Δ''s''
| align="center" | <math>\Delta S = C_V \ln{T_2 \over T_1} + nR \ln{V_2 \over V_1}</math>{{br}}<math>\Delta S = C_p \ln{T_2 \over T_1} - nR \ln{p_2 \over p_1}</math><ref>Keenan, ''Thermodynamics'', Wiley, New York, 1947</ref>
| align="center" | <math>C_p\ln\frac{T_2}{T_1}\;</math>
| align="center" | <math>C_V\ln\frac{T_2}{T_1}\;</math>
| align="center" | <math>nR\ln\frac{V_2}{V_1}\;</math>{{br}}<math>\frac{Q}{T}\;</math>
| align="center" | <math>C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;</math>
|-
! Constant
| <math>\;</math>
| align="center" | <math>\frac{V}{T}\;</math>
| align="center" | <math>\frac{p}{T}\;</math>
| align="center" | <math>p V\;</math>
| align="center" | <math>p V^\gamma\;</math>
|}

=== Entropy ===

* <math> S = k_\mathrm{B} \ln \Omega </math>, where ''k''<sub>B</sub> is the [[Boltzmann constant]], and Ω denotes the volume of [[macrostate]] in the [[phase space]] or otherwise called thermodynamic probability.
* <math> dS = \frac{\delta Q}{T} </math>, for reversible processes only

===Statistical physics===

Below are useful results from the [[Maxwell–Boltzmann distribution]] for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! [[Maxwell–Boltzmann distribution]]
| {{plainlist}}
* ''v'' = velocity of atom/molecule,
* ''m'' = mass of each molecule (all molecules are identical in kinetic theory),
* ''γ''(''p'') = Lorentz factor as function of momentum (see below)
* Ratio of thermal to rest mass-energy of each molecule: <math>\theta = k_\text{B} T/mc^2 </math> 
{{endplainlist}}

''K''<sub>2</sub> is the modified [[Bessel function]] of the second kind.
| Non-relativistic speeds<br />
<math>P\left ( v \right )=4\pi\left ( \frac{m}{2\pi k_\text{B} T} \right )^{3/2} v^2 e^{-mv^2/2 k_\text{B} T} </math>

Relativistic speeds (Maxwell–Jüttner distribution)<br />
<math> f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} e^{-\gamma(p)/\theta}</math>
|-
! Entropy [[logarithmic scale|Logarithm]] of the [[density of states]]
| {{plainlist}}
* ''P<sub>i</sub>'' = probability of system in microstate ''i''
* Ω = total number of microstates
{{endplainlist}}
| <math>S = - k_\text{B}\sum_i P_i \ln P_i = k_\mathrm{B}\ln \Omega </math>

where:<br />
<math>P_i = 1/\Omega </math>
|-
! Entropy change
|
| <math>\Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T} </math><br />
<math>\Delta S = k_\text{B} N \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1} </math>
|-
! Entropic force
|
| <math>\mathbf{F}_\mathrm{S} = -T \nabla S </math>
|-
! [[Equipartition theorem]]
| ''d''<sub>f</sub> = degree of freedom
| Average kinetic energy per degree of freedom
<math> \langle E_\mathrm{k} \rangle = \frac{1}{2}kT </math>

Internal energy
<math> U = d_\text{f} \langle E_\mathrm{k} \rangle = \frac{d_\text{f}}{2}kT </math>
|- 
|}

Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! Mean speed
|
| <math> \langle v \rangle = \sqrt{\frac{8 k_\text{B} T}{\pi m}} </math>
|-
! Root mean square speed
|
| <math> v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k_\text{B} T}{m}} </math>
|-
! Modal speed
|
| <math> v_\mathrm{mode} = \sqrt{\frac{2 k_\text{B} T}{m}} </math>
|-
! [[Mean free path]]
| {{plainlist}}
* ''σ'' = effective cross-section
* ''n'' = volume density of number of target particles
* ''{{ell}}'' = mean free path 
{{endplainlist}}
| <math>\ell = 1/\sqrt{2} n \sigma </math>
|-
|}

=== Quasi-static and reversible processes ===

For [[Quasistatic process|quasi-static]] and [[Reversible process (thermodynamics)|reversible]] processes, the [[first law of thermodynamics]] is:
: <math>dU=\delta Q - \delta W</math>
where ''δQ'' is the heat supplied ''to'' the system and ''δW'' is the work done ''by'' the system.

=== Thermodynamic potentials ===
{{main|Thermodynamic potentials}}
{{See also|Maxwell relations}}

The following energies are called the [[thermodynamic potentials]],
: {{table of thermodynamic potentials}}
and the corresponding [[fundamental thermodynamic relation]]s or "master equations"<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, {{ISBN|0 19 855148 7}}</ref> are:

{| class="wikitable"
|-
! Potential 
! Differential
|-
! Internal energy 
| <math>dU\left(S,V,{N_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i</math>
|-
! Enthalpy
| <math>dH\left(S,p,{N_{i}}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}</math>
|-
! Helmholtz free energy
| <math>dF\left(T,V,{N_{i}}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}</math>
|-
! Gibbs free energy
| <math>dG\left(T,p,{N_{i}}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}</math>
|-
|}

=== Maxwell's relations ===

The four most common [[Maxwell's relations]] are:

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! Thermodynamic potentials as functions of their natural variables
| {{plainlist}}
* <math>U(S,V)\,</math> = [[Internal energy]]
* <math>H(S,P)\,</math> = [[Enthalpy]]
* <math>F(T,V)\,</math> = [[Helmholtz free energy]]
* <math>G(T,P)\,</math> = [[Gibbs free energy]] 
{{endplainlist}}
|<math> \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V} </math>

<math> \left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H }{\partial S \partial P}
</math>

<math> +\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V = - \frac{\partial^2 F }{\partial T \partial V} </math>

<math> -\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P = \frac{\partial^2 G }{\partial T \partial P} </math>
|-
|}

More relations include the following.

{| class="wikitable"
|-
| <math> \left ( {\partial S\over \partial U} \right )_{V,N} = { 1\over T } </math>
| <math> \left ( {\partial S\over \partial V} \right )_{N,U} = { p\over T } </math>
| <math> \left ( {\partial S\over \partial N} \right )_{V,U} = - { \mu \over T } </math>
|-
| <math> \left ( {\partial T\over \partial S} \right )_V = { T \over C_V } </math>
| <math> \left ( {\partial T\over \partial S} \right )_P = { T \over C_P } </math>
|
|-
| <math> -\left ( {\partial p\over \partial V} \right )_T = { 1 \over {VK_T} } </math>
|
|
|-
|}

Other differential equations are:

{| class="wikitable"
|-
! Name
! ''H''
! ''U''
! ''G''
|-
! [[Gibbs–Helmholtz equation]]
| <math>H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p</math>
| <math>U = -T^2\left(\frac{\partial \left(F/T\right)}{\partial T}\right)_V</math>
| <math>G = -V^2\left(\frac{\partial \left(F/V\right)}{\partial V}\right)_T</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>
|
|-
|}

=== Quantum properties ===

* <math> U = N k_\text{B} T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V </math>
* <math> S =  \frac{U}{T} + N k_\text{B} \ln Z - N k \ln N + Nk </math>  Indistinguishable Particles
where ''N'' is number of particles, ''h'' is that [[Planck constant]], ''I'' is [[moment of inertia]], and ''Z'' is the [[partition function (statistical mechanics)|partition function]], in various forms:

{| class="wikitable"
|-
! Degree of freedom
! Partition function
|-
! Translation
| <math> Z_t = \frac{(2 \pi m k_\text{B} T)^\frac{3}{2} V}{h^3} </math>
|-
! Vibration
| <math> Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_\text{B} T}} </math>
|-
! Rotation
| <math> Z_r = \frac{2 I k_\text{B} T}{\sigma (\frac{h}{2 \pi})^2} </math>

{{plainlist}}
* where:
* ''σ'' = 1 ([[heteronuclear molecule]]s)
* ''σ'' = 2 ([[homonuclear]])
{{endplainlist}}
|}

== Thermal properties of matter ==

{| class="wikitable"
|-
! Coefficients 
! Equation
|-
! [[Joule–Thomson effect|Joule-Thomson coefficient]]
| <math>\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H</math>
|-
! [[Compressibility]] (constant temperature)
| <math> K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N} </math>
|-
! [[Coefficient of thermal expansion]] (constant pressure)
| <math>\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p</math>
|-
! Heat capacity (constant pressure)
| <math>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 </math>
|-
! Heat capacity (constant volume)
| <math>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 </math>
|-
|}

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of heat capacity (constant pressure)
|-
|
Since
: <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>
\begin{align}
\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
\end{align}
</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>
|}

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of heat capacity (constant volume)
|-
|
Since
: <math>dU = \delta Q_{rev} - \delta W_{rev} ,</math>
(where ''δW''<sub>rev</sub> is the work done by the system),
: <math>\delta S = \frac{\delta Q_{rev}}{T}, \delta W_{rev}= p \delta V </math>
: <math> d U = T \delta S- p\delta V </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>
|}

=== Thermal transfer ===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! Net intensity emission/absorption
| {{plainlist}}
* ''T''<sub>external</sub> = external temperature (outside of system)
* ''T''<sub>system</sub> = internal temperature (inside system)
* ''ε'' = emissivity
{{endplainlist}}
| <math> I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) </math>
|-
! Internal energy of a substance
| {{plainlist}}
* ''C<sub>V</sub>'' = isovolumetric heat capacity of substance
* Δ''T'' = temperature change of substance
{{endplainlist}}
| <math>\Delta U = N C_V \Delta T </math>
|-
! Meyer's equation
| {{plainlist}}
* ''C<sub>p</sub>'' = isobaric heat capacity
* ''C<sub>V</sub>'' = isovolumetric heat capacity
* ''n'' = amount of substance
{{endplainlist}}
| <math> C_p - C_V = nR </math>
|-
! Effective thermal conductivities
| {{plainlist}}
* ''λ<sub>i</sub>'' = thermal conductivity of substance ''i''
* ''λ''<sub>net</sub> = equivalent thermal conductivity  
{{endplainlist}}
| Series
<math> \lambda_\mathrm{net} = \sum_j \lambda_j </math>

Parallel
<math> \frac{1}{\lambda}_\mathrm{net} = \sum_j \left ( \frac{1}{\lambda}_j \right ) </math>
|-
|}

=== Thermal efficiencies ===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! Thermodynamic engines
| {{plainlist}}
* ''η'' = efficiency
* ''W'' = work done by engine
* ''Q''<sub>H</sub> = heat energy in higher temperature reservoir
* ''Q''<sub>L</sub> = heat energy in lower temperature reservoir
* ''T''<sub>H</sub> = temperature of higher temp. reservoir
* ''T''<sub>L</sub> = temperature of lower temp. reservoir
{{endplainlist}}
| Thermodynamic engine:<br />
<math>\eta = \left |\frac{W}{Q_\text{H}} \right| </math>

Carnot engine efficiency:<br />
<math>\eta_\text{c} = 1 - \left | \frac{Q_\text{L}}{Q_\text{H}} \right | = 1-\frac{T_\text{L}}{T_\text{H}} </math>
|-
! Refrigeration
| ''K'' = coefficient of refrigeration performance
| Refrigeration performance
<math>K = \left | \frac{Q_\text{L}}{W} \right | </math>

Carnot refrigeration performance
<math>K_\text{C} = \frac{|Q_\text{L}|}{|Q_\text{H}|-|Q_\text{L}|} = \frac{T_\text{L}}{T_\text{H}-T_\text{L}} </math>
|-
|}

== See also ==
{{colbegin}}
* [[List of thermodynamic properties]]
* [[Antoine equation]]
* [[Bejan number]]
* [[Bowen ratio]]
* [[Bridgman's thermodynamic equations|Bridgman's equations]]
* [[Clausius–Clapeyron relation]]
* [[Departure function]]s
* [[Duhem–Margules equation]]
* [[Ehrenfest equations]]
* [[Gibbs–Helmholtz equation]]
* [[Phase rule]]
* [[Kopp's law]]
* [[Noro–Frenkel law of corresponding states]]
* [[Onsager reciprocal relations]]
* [[Stefan number]]
* [[Thermodynamics]]
* [[Timeline of thermodynamics]]
* [[Triple product rule]]
* [[Exact differential]]{{colend}}

== References ==
{{reflist}}
{{refbegin|2}}
* [[Peter Atkins|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".
* {{cite journal | last=Bridgman | first=P. W. | title=A Complete Collection of Thermodynamic Formulas | journal=Physical Review | publisher=American Physical Society (APS) | volume=3 | issue=4 | date=1 March 1914 | issn=0031-899X | doi=10.1103/physrev.3.273|url=https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=289 | pages=273–281}}
* Landsberg, Peter T.  <u>Thermodynamics and Statistical Mechanics</u>. 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.
* [[Linda Reichl|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 edition, New York: John Wiley & Sons.
{{refend}}

== External links ==
* [http://www.fxsolver.com/browse/?oc=3&cat=6&formulas=on Thermodynamic equation calculator]

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