Quelltext der Seite Benutzer:Tefracky/Table of thermodynamic equations

{{Importartikel}}
{{For|list of mathematical notation used in these equations|mathematical notation}}
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
{{main|List of thermodynamic properties}}

This article is a summary of common [[equation]]s and [[physical quantity|quantities]] in [[thermodynamics]] (see [[thermodynamic equations]] for more elaboration). SI units are used for [[absolute temperature]], not Celsius or Fahrenheit.

==Definitions==

{{hatnote|Main articles: [[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 Units 
! scope="col" width="100" | Dimension
|-
!Number of molecules
| ''N''
| dimensionless
| dimensionless
|-
!Number of moles
|''n''
| mol
| [N]
|-
![[Temperature]]
| ''T''
| K
| [Θ]
|-
![[Heat|Heat Energy]]
| ''Q, q''
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Latent Heat]]
| ''Q<sub>L</sub>''
| J
| [M][L]<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 Units 
! scope="col" width="100" | Dimension
|-
![[Thermodynamic beta]], Inverse temperature 
|| ''β''
||<math> \beta = 1/k_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_B T \,\!</math>
<math> \tau = k_B \left (\partial U/\partial S \right )_{N} \,\!</math>
<math> 1/\tau = 1/k_B \left (\partial S/\partial U \right )_{N} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-
![[Entropy]]
| ''S''
| <math>S = -k_B\sum_i p_i\ln p_i</math>
<math> S = \left (\partial F/\partial T \right )_{V} \,\!</math>   ,
<math> S = \left (\partial G/\partial T \right )_{N,P} \,\!</math>
| J K<sup>−1</sup>
| [M][L]<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
| M  L<sup>−1</sup>T<sup>−2</sup>
|-
![[Internal Energy]]
| ''U''
|<math>U = \sum_i  E_i \!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Enthalpy]]
| ''H''
|<math> H = U+pV\,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Partition function (statistical mechanics)|Partition Function]] 
| ''Z''
|
| dimensionless
| dimensionless
|-
![[Gibbs free energy]]
| ''G''
|<math> G = H - TS \,\!</math>
| J
| [M][L]<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 F is not proportional to N because μ<sub>i</sub> depends on pressure.
<math> \mu_i = \left (\partial G/\partial N_i \right )_{T, P } \,\!</math>, where G is proportional to N (as long as the molar ratio composition of the system remains the same) because μ<sub>i</sub> depends only on temperature and pressure and composition.
<math> \mu_i/\tau = -1/k_B \left (\partial S/\partial N_i \right )_{U,V} \,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Helmholtz free energy]]
| ''A, F''
|<math> F = U - TS \,\!</math>
| J
| [M][L]<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
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
!Massieu Potential, Helmholtz [[free entropy]]
| ''Φ''
|<math> \Phi = S - U/T \,\!</math>
| J K<sup>−1</sup>
| [M][L]<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>
| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
|}

===Thermal properties of matter===

{{hatnote|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 units 
! scope="col" width="100" | Dimension
|-
!General heat/thermal capacity 
|| ''C''
||<math> C = \partial Q/\partial T\,\!</math>
|| J K <sup>−1</sup>
|| [M][L]<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>
|| [M][L]<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>
|| [M][L]<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 Q/\partial T \,\!</math>
|| J K <sup>−1</sup>
|| [M][L]<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>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup> [N]<sup>−1</sup>
|-
!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
|| ''γ''
||<math>\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} \,\!</math>
|| dimensionless
|| dimensionless
|-
|}

===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 units 
! scope="col" width="100" | Dimension
|-
![[Temperature gradient]]
|| No standard symbol 
||<math> \nabla T \,\!</math>
|| K m<sup>−1</sup>
|| [Θ][L]<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 = J s<sup>−1</sup>
|| [M] [L]<sup>2</sup> [T]<sup>−3</sup>
|-
!Thermal intensity
|| ''I''
||<math>I = \mathrm{d} P/\mathrm{d} A </math>
|| W m<sup>−2</sup>
|| [M] [T]<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>
|| [M] [T]<sup>−3</sup>
|-
|}

==Equations==

The equations in this article are classified by subject.

===Phase transitions===

:{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="10" | Equations
|-
!Adiabatic transition
|<math>\Delta Q = 0, \quad \Delta U = -\Delta W\,\!</math>
|-
!Isothermal transition
|<math>\Delta U = 0, \quad \Delta W =  \Delta Q \,\!</math>

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

Net Work Done in Cyclic Processes<br />
<math> \Delta W = \oint_\mathrm{cycle} p \mathrm{d}V \,\!</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
| <div class="plainlist">
*''p'' = pressure
*''V'' = volume of container
*''T'' = temperature 
*''n'' = number of moles
*''R'' = [[Gas constant]]
*''N'' = number of molecules
*''k'' = Boltzmann's constant
</div>
|<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
| <div class="plainlist">
*''m'' = mass of ''one'' molecule
*''M<sub>m</sub>'' = molar mass
</div>
| <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"
|+
|-
! 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_2}{P_1}\;</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><br><math>Q=-W\;</math>
| align="center" | <math>W\;</math><br><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_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
| <div class="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_B T/mc^2 \,\!</math> 
</div>

''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_B T} \right )^{3/2} v^2 e^{-mv^2/2 k_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]]
|<div class="plainlist">
* ''P<sub>i</sub>'' = probability of system in microstate ''i''
* Ω = total number of microstates
</div>
|<math>S = - k_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_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
| <div class="plainlist">
*''d<sub>f</sub>'' = degree of freedom
</div>
| Average kinetic energy per degree of freedom
<math> \langle E_\mathrm{k} \rangle = \frac{1}{2}kT\,\!</math>

Internal energy
<math> U = d_f \langle E_\mathrm{k} \rangle = \frac{d_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_B T}{\pi m}}\,\!</math>
|-
!Root mean square speed
|
| <math> v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3k_B T}{m}} \,\!</math>
|-
!Modal speed
|
|<math> v_\mathrm{mode} = \sqrt{\frac{2k_B T}{m}}\,\!</math>
|-
![[Mean free path]]
|<div class="plainlist">
*''σ'' = Effective cross-section
*''n'' = Volume density of number of target particles
*''{{ell}}'' = Mean free path 
</div>
|<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 δ''Q'' is the heat supplied ''to'' the system and δ''W'' 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
|<div class="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]] 
</div>
|<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_B T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V ~</math>
*<math> S =  \frac{U}{T} + N * ~ S =  \frac{U}{T} + N k_B \ln Z - N k \ln N + Nk ~</math>  Indistinguishable Particles

where ''N'' is number of particles, ''h'' is [[Planck's 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_B T)^\frac{3}{2} V}{h^3} </math>
|-
!Vibration
|<math> Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_B T}} </math>
|-
!Rotation
|<math> Z_r = \frac{2 I k_B T}{\sigma (\frac{h}{2 \pi})^2} </math>

<div class="plainlist">
*where:
*σ = 1 ([[heteronuclear molecule]]s)
*σ = 2 ([[homonuclear]])
</div>
|}

==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 δ''W''<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
| <div class="plainlist">
*''T''<sub>external</sub> = external temperature (outside of system)
*''T''<sub>system</sub> = internal temperature (inside system)
*''ε'' = emmisivity
</div>
|<math> I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) \,\!</math>
|-
!Internal energy of a substance
| <div class="plainlist">
*''C<sub>V</sub>'' = isovolumetric heat capacity of substance
*Δ''T'' = temperature change of substance
</div>
|<math>\Delta U = N C_V \Delta T\,\!</math>
|-
!Meyer's equation
|<div class="plainlist">
*''C<sub>p</sub>'' = isobaric heat capacity
*''C<sub>V</sub>'' = isovolumetric heat capacity
*''n'' = number of moles
</div>
|<math> C_p - C_V = nR \,\!</math>
|-
!Effective thermal conductivities
| <div class="plainlist">
*''λ<sub>i</sub>'' = thermal conductivity of substance ''i''
*''λ''<sub>net</sub> = equivalent thermal conductivity  
</div>
| 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
|<div class="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
</div>
|Thermodynamic engine:<br />
<math>\eta = \left |\frac{W}{Q_H} \right|\,\!</math>

Carnot engine efficiency:<br />
<math>\eta_c = 1 - \left | \frac{Q_L}{Q_H} \right | = 1-\frac{T_L}{T_H}\,\!</math>
|-
!Refrigeration
| <div class="plainlist">
*''K'' = coefficient of refrigeration performance
</div>
|Refrigeration performance
<math>K = \left | \frac{Q_L}{W} \right | \,\!</math>

Carnot refrigeration performance
<math>K_C = \frac{|Q_L|}{|Q_H|-|Q_L|} = \frac{T_L}{T_H-T_L}\,\!</math>
|-
|}

==See also==
{{multicol}}
*[[Antoine equation]]
*[[Bejan number]]
*[[Bowen ratio]]
*[[Bridgman's thermodynamic equations|Bridgman's equations]]
*[[Clausius–Clapeyron relation]]
*[[Departure function]]s
*[[Duhem–Margules equation]]
*[[Ehrenfest equations]]
{{multicol-break}}
*[[Gibbs–Helmholtz equation]]
*[[Gibbs' phase rule]]
*[[Kopp's law]]
*[[Kopp–Neumann law]]
*[[Noro–Frenkel law of corresponding states]]
*[[Onsager reciprocal relations]]
*[[Stefan number]]
*[[Triple product rule]]
{{multicol-end}}

==References==
<references/>
{{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''.
* Bridgman, P.W., ''Phys. Rev.'', 3, 273 (1914).
*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.
* Reichl, L.E., "A Modern Course in Statistical Physics", 2nd edition, New York: John Wiley & Sons, 1998.
*Schroeder, Daniel V. <u>Thermal Physics</u>.  San Francisco: Addison Wesley Longman, 2000 [ISBN 0-201-38027-7].
*Silbey, Robert J., et al. <u>Physical Chemistry</u>. 4th ed. New Jersey: Wiley, 2004.
*Callen, Herbert B. (1985). "Thermodynamics and an Introduction to Themostatistics", 2nd Ed., New York: John Wiley & Sons.
{{refend}}

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

{{Physics-footer}}

[[Category:Thermodynamic equations]]

</pre>