table of thermodynamic equations

Common thermodynamic equations and 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 reactions.

= General basic quantities =

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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 Energy \| Q, q \| J \| ML²T⁻²
Latent heat \| Q_L \| J \| ML²T⁻²

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scope="col" width="200" | Quantity (common name/s)

! scope="col" width="125" | (Common) symbol/s

! scope="col" width="125" | SI unit

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Number of molecules

| N

| 1

Amount of substance

| n

| mol

| N

Temperature

| T

| K

| Θ

Heat Energy

| Q, q

| J

| ML²T⁻²

Latent heat

| Q_L

| J

| ML²T⁻²

= General derived quantities =

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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 \| $\beta = 1/k_\text{B} T$ \| J⁻¹ \| T²M⁻¹L⁻²
Thermodynamic temperature \| τ \| $\tau = k_\text{B} T$ $\tau = k_\text{B} \left (\partial U/\partial S \right )_{N}$ $1/\tau = 1/k_\text{B} \left (\partial S/\partial U \right )_{N}$ \| J \| ML²T⁻²
Entropy \| S \| $S = -k_\text{B}\sum_i p_i\ln p_i$ $S = -\left (\partial F/\partial T \right )_{V,N}$ , $S = -\left (\partial G/\partial T \right )_{P,N}$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹
Pressure \| P \| $P = - \left (\partial F/\partial V \right )_{T,N}$ $P = - \left (\partial U/\partial V \right )_{S,N}$ \| Pa \| ML⁻¹T⁻²
Internal Energy \| U \| $U = \sum_i E_i$ \| J \| ML²T⁻²
Enthalpy \| H \| $H = U+pV$ \| J \| ML²T⁻²
Partition Function \| Z \| \| 1 \| 1
Gibbs free energy \| G \| $G = H - TS$ \| J \| ML²T⁻²
Chemical potential (of component i in a mixture) \| μ_i \| $\mu_i = \left (\partial U/\partial N_i \right )_{N_{j \neq i}, S, V }$ $\mu_i = \left (\partial F/\partial N_i \right )_{T, V }$ , where $F$ is not proportional to $N$ because $\mu_i$ depends on pressure. $\mu_i = \left (\partial G/\partial N_i \right )_{T, P }$ , where $G$ is proportional to $N$ (as long as the molar ratio composition of the system remains the same) because $\mu_i$ depends only on temperature and pressure and composition. $\mu_i/\tau = -1/k_\text{B} \left (\partial S/\partial N_i \right )_{U,V}$ \| J \| ML²T⁻²
Helmholtz free energy \| A, F \| $F = U - TS$ \| J \| ML²T⁻²
Landau potential, Landau free energy, Grand potential \| Ω, Φ_G \| $\Omega = U - TS - \mu N\$ \| J \| ML²T⁻²
Massieu potential, Helmholtz free entropy \| Φ \| $\Phi = S - U/T$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹
Planck potential, Gibbs free entropy \| Ξ \| $\Xi = \Phi - pV/T$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹

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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

| $\beta = 1/k_\text{B} T$

| J⁻¹

| T²M⁻¹L⁻²

Thermodynamic temperature

| τ

| $\tau = k_\text{B} T$

$\tau = k_\text{B} \left (\partial U/\partial S \right )_{N}$

$1/\tau = 1/k_\text{B} \left (\partial S/\partial U \right )_{N}$

| J

| ML²T⁻²

Entropy

| S

| $S = -k_\text{B}\sum_i p_i\ln p_i$

$S = -\left (\partial F/\partial T \right )_{V,N}$ ,

$S = -\left (\partial G/\partial T \right )_{P,N}$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

Pressure

| P

| $P = - \left (\partial F/\partial V \right )_{T,N}$

$P = - \left (\partial U/\partial V \right )_{S,N}$

| Pa

| ML⁻¹T⁻²

Internal Energy

| U

| $U = \sum_i E_i$

| J

| ML²T⁻²

Enthalpy

| H

| $H = U+pV$

| J

| ML²T⁻²

Partition Function

| Z

| 1

Gibbs free energy

| G

| $G = H - TS$

| J

| ML²T⁻²

Chemical potential (of component i in a mixture)

| μ_i

| $\mu_i = \left (\partial U/\partial N_i \right )_{N_{j \neq i}, S, V }$

$\mu_i = \left (\partial F/\partial N_i \right )_{T, V }$ , where $F$ is not proportional to $N$ because $\mu_i$ depends on pressure.

$\mu_i = \left (\partial G/\partial N_i \right )_{T, P }$ , where $G$ is proportional to $N$ (as long as the molar ratio composition of the system remains the same) because $\mu_i$ depends only on temperature and pressure and composition.

$\mu_i/\tau = -1/k_\text{B} \left (\partial S/\partial N_i \right )_{U,V}$

| J

| ML²T⁻²

Helmholtz free energy

| A, F

| $F = U - TS$

| J

| ML²T⁻²

Landau potential, Landau free energy, Grand potential

| Ω, Φ_G

| $\Omega = U - TS - \mu N\$

| J

| ML²T⁻²

Massieu potential, Helmholtz free entropy

| Φ

| $\Phi = S - U/T$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

Planck potential, Gibbs free entropy

| Ξ

| $\Xi = \Phi - pV/T$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

= Thermal properties of matter =

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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 \| $C = \partial Q/\partial T$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹
Heat capacity (isobaric) \| C_p \| $C_{p} = \partial H/\partial T$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹
Specific heat capacity (isobaric) \| C_mp \| $C_{mp} = \partial^2 Q/\partial m \partial T$ \| J⋅kg⁻¹⋅K⁻¹ \| L²T⁻²Θ⁻¹
Molar specific heat capacity (isobaric) \| C_np \| $C_{np} = \partial^2 Q/\partial n \partial T$ \| J⋅K⁻¹⋅mol⁻¹ \| ML²T⁻²Θ⁻¹N⁻¹
Heat capacity (isochoric/volumetric) \| C_V \| $C_{V} = \partial U/\partial T$ \| J⋅K⁻¹ \| ML²T⁻²Θ⁻¹
Specific heat capacity (isochoric) \| C_mV \| $C_{mV} = \partial^2 Q/\partial m \partial T$ \| J⋅kg⁻¹⋅K⁻¹ \| L²T⁻²Θ⁻¹
Molar specific heat capacity (isochoric) \| C_nV \| $C_{nV} = \partial^2 Q/\partial n \partial T$ \| J⋅K⋅⁻¹ mol⁻¹ \| ML²T⁻²Θ⁻¹N⁻¹
Specific latent heat \| L \| $L = \partial Q/ \partial m$ \| J⋅kg⁻¹ \| L²T⁻²
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient \| γ \| $\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV}$ \| 1 \| 1

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scope="col" width="100" | Quantity (common name/s)

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General heat/thermal capacity

| C

| $C = \partial Q/\partial T$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

Heat capacity (isobaric)

| C_p

| $C_{p} = \partial H/\partial T$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

Specific heat capacity (isobaric)

| C_mp

| $C_{mp} = \partial^2 Q/\partial m \partial T$

| J⋅kg⁻¹⋅K⁻¹

| L²T⁻²Θ⁻¹

Molar specific heat capacity (isobaric)

| C_np

| $C_{np} = \partial^2 Q/\partial n \partial T$

| J⋅K⁻¹⋅mol⁻¹

| ML²T⁻²Θ⁻¹N⁻¹

Heat capacity (isochoric/volumetric)

| C_V

| $C_{V} = \partial U/\partial T$

| J⋅K⁻¹

| ML²T⁻²Θ⁻¹

Specific heat capacity (isochoric)

| C_mV

| $C_{mV} = \partial^2 Q/\partial m \partial T$

| J⋅kg⁻¹⋅K⁻¹

| L²T⁻²Θ⁻¹

Molar specific heat capacity (isochoric)

| C_nV

| $C_{nV} = \partial^2 Q/\partial n \partial T$

| J⋅K⋅⁻¹ mol⁻¹

| ML²T⁻²Θ⁻¹N⁻¹

Specific latent heat

| L

| $L = \partial Q/ \partial m$

| J⋅kg⁻¹

| L²T⁻²

Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient

| γ

| $\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV}$

| 1

= Thermal transfer =

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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 \| $\nabla T$ \| K⋅m⁻¹ \| ΘL⁻¹
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer \| P \| $P = \mathrm{d} Q/\mathrm{d} t$ \| W \| ML²T⁻³
Thermal intensity \| I \| $I = \mathrm{d} P/\mathrm{d} A$ \| W⋅m⁻² \| MT⁻³
Thermal/heat flux density (vector analogue of thermal intensity above) \| q \| $Q = \iint \mathbf{q} \cdot \mathrm{d}\mathbf{S}\mathrm{d} t$ \| W⋅m⁻² \| MT⁻³

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scope="col" width="100" | Quantity (common name/s)

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! scope="col" width="300" | Defining equation

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Temperature gradient

| No standard symbol

| $\nabla T$

| K⋅m⁻¹

| ΘL⁻¹

Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer

| P

| $P = \mathrm{d} Q/\mathrm{d} t$

| W

| ML²T⁻³

Thermal intensity

| I

| $I = \mathrm{d} P/\mathrm{d} A$

| W⋅m⁻²

| MT⁻³

Thermal/heat flux density (vector analogue of thermal intensity above)

| q

| $Q = \iint \mathbf{q} \cdot \mathrm{d}\mathbf{S}\mathrm{d} t$

| W⋅m⁻²

| MT⁻³

Equations

The equations in this article are classified by subject.

= Thermodynamic processes =

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scope="col" width="100" \| Physical situation ! scope="col" width="10" \| Equations
Isentropic process (adiabatic and reversible) \| $Q = 0, \quad \Delta U = -W$ For an ideal gas $p_1 V_1^{\gamma} = p_2 V_2^{\gamma}$ $T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}$ $p_1^{1-\gamma} T_1^{\gamma} = p_2^{1 - \gamma} T_2^{\gamma}$
Isothermal process \| $\Delta U = 0, \quad W = Q$ For an ideal gas $W=kTN \ln(V_2/V_1)$ $W=nRT \ln(V_2/V_1)$
Isobaric process \|p₁ = p₂, p = constant $W = p \Delta V, \quad Q = \Delta U + p \delta V$
Isochoric process \| V₁ = V₂, V = constant $W = 0, \quad Q = \Delta U$
Free expansion \| $\Delta U = 0$
Work done by an expanding gas \| Process $W = \int_{V_1}^{V_2} p \mathrm{d}V$ Net work done in cyclic processes $W = \oint_\mathrm{cycle} p \mathrm{d}V = \oint_\mathrm{cycle}\Delta Q$

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scope="col" width="100" | Physical situation

! scope="col" width="10" | Equations

Isentropic process (adiabatic and reversible)

| $Q = 0, \quad \Delta U = -W$

For an ideal gas

$p_1 V_1^{\gamma} = p_2 V_2^{\gamma}$

$T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}$

$p_1^{1-\gamma} T_1^{\gamma} = p_2^{1 - \gamma} T_2^{\gamma}$

Isothermal process

| $\Delta U = 0, \quad W = Q$

For an ideal gas

$W=kTN \ln(V_2/V_1)$

$W=nRT \ln(V_2/V_1)$

Isobaric process

|p₁ = p₂, p = constant

$W = p \Delta V, \quad Q = \Delta U + p \delta V$

Isochoric process

| V₁ = V₂, V = constant

$W = 0, \quad Q = \Delta U$

Free expansion

| $\Delta U = 0$

Work done by an expanding gas

| Process

$W = \int_{V_1}^{V_2} p \mathrm{d}V$

Net work done in cyclic processes

$W = \oint_\mathrm{cycle} p \mathrm{d}V = \oint_\mathrm{cycle}\Delta Q$

= 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}} \| $pV = nRT = kTN$ $\frac{p_1 V_1}{p_2 V_2} = \frac{n_1 T_1}{n_2 T_2} = \frac{N_1 T_1}{N_2 T_2}$
Pressure of an ideal gas \| {{plainlist}} m = mass of one molecule M_m = molar mass {{endplainlist}} \| $p = \frac{Nm \langle v^2 \rangle}{3V} = \frac{nM_m \langle v^2 \rangle}{3V} = \frac{1}{3}\rho \langle v^2 \rangle$

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|+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

| $pV = nRT = kTN$

$\frac{p_1 V_1}{p_2 V_2} = \frac{n_1 T_1}{n_2 T_2} = \frac{N_1 T_1}{N_2 T_2}$

Pressure of an ideal gas

| {{plainlist}}

m = mass of one molecule
M_m = molar mass

| $p = \frac{Nm \langle v^2 \rangle}{3V} = \frac{nM_m \langle v^2 \rangle}{3V} = \frac{1}{3}\rho \langle v^2 \rangle$

== Ideal gas ==

class="wikitable sortable mw-collapsible" \|+
Quantity ! General Equation ! Isobaric Δp = 0 ! Isochoric ΔV = 0 ! Isothermal ΔT = 0 ! Adiabatic $Q=0$
Work W \| align="center" \| $\delta W = -p dV\;$ \| align="center" \| $-p\Delta V\;$ \| align="center" \| $0\;$ \| align="center" \| $-nRT\ln\frac{V_2}{V_1}\;$ $-nRT\ln\frac{P_1}{P_2}\;$ \| align="center" \| $\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma} = C_V \left(T_2 - T_1 \right)$
Heat Capacity C \| align="center" \| (as for real gas) \| align="center" \| $C_p = \frac{5}{2}nR$ {{br}}(for monatomic ideal gas) $C_p = \frac{7}{2}nR$ {{br}}(for diatomic ideal gas) \| align="center" \| $C_V = \frac{3}{2}nR$ {{br}}(for monatomic ideal gas) $C_V = \frac{5}{2}nR \;$ {{br}}(for diatomic ideal gas) \| \|
Internal Energy ΔU \| align="center" \| $\Delta U = C_V \Delta T\;$ \| align="center" \| $Q + W\;$ {{br}}{{br}} $Q_p - p\Delta V$ \| align="center" \| $Q\;$ {{br}}{{br}} $C_V\left ( T_2-T_1 \right )$ \| align="center" \| $0\;$ {{br}} $Q=-W$ \| align="center" \| $W\;$ {{br}} $C_V\left ( T_2-T_1 \right )$
Enthalpy ΔH \| align="center" \| $H=U+pV\;$ \| align="center" \| $C_p\left ( T_2-T_1 \right )\;$ \| align="center" \| $Q_V+V\Delta p\;$ \| align="center" \| $0\;$ \| align="center" \| $C_p\left ( T_2-T_1 \right )\;$
Entropy Δs \| align="center" \| $\Delta S = C_V \ln{T_2 \over T_1} + nR \ln{V_2 \over V_1}$ {{br}} $\Delta S = C_p \ln{T_2 \over T_1} - nR \ln{p_2 \over p_1}$ Keenan, Thermodynamics, Wiley, New York, 1947 \| align="center" \| $C_p\ln\frac{T_2}{T_1}\;$ \| align="center" \| $C_V\ln\frac{T_2}{T_1}\;$ \| align="center" \| $nR\ln\frac{V_2}{V_1}\;$ {{br}} $\frac{Q}{T}\;$ \| align="center" \| $C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;$
Constant \| $\;$ \| align="center" \| $\frac{V}{T}\;$ \| align="center" \| $\frac{p}{T}\;$ \| align="center" \| $p V\;$ \| align="center" \| $p V^\gamma\;$

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Quantity

! General Equation

! Isobaric
Δp = 0

! Isochoric
ΔV = 0

! Isothermal
ΔT = 0

! Adiabatic
$Q=0$

Work
W

| align="center" | $\delta W = -p dV\;$

| align="center" | $-p\Delta V\;$

| align="center" | $0\;$

| align="center" | $-nRT\ln\frac{V_2}{V_1}\;$

$-nRT\ln\frac{P_1}{P_2}\;$

| align="center" | $\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma} = C_V \left(T_2 - T_1 \right)$

Heat Capacity
C

| align="center" | (as for real gas)

| align="center" | $C_p = \frac{5}{2}nR$ {{br}}(for monatomic ideal gas)

$C_p = \frac{7}{2}nR$ {{br}}(for diatomic ideal gas)

| align="center" | $C_V = \frac{3}{2}nR$ {{br}}(for monatomic ideal gas)

$C_V = \frac{5}{2}nR \;$ {{br}}(for diatomic ideal gas)

Internal Energy
ΔU

| align="center" | $\Delta U = C_V \Delta T\;$

| align="center" | $Q + W\;$ {{br}}{{br}} $Q_p - p\Delta V$

| align="center" | $Q\;$ {{br}}{{br}} $C_V\left ( T_2-T_1 \right )$

| align="center" | $0\;$ {{br}} $Q=-W$

| align="center" | $W\;$ {{br}} $C_V\left ( T_2-T_1 \right )$

Enthalpy
ΔH

| align="center" | $H=U+pV\;$

| align="center" | $C_p\left ( T_2-T_1 \right )\;$

| align="center" | $Q_V+V\Delta p\;$

| align="center" | $0\;$

| align="center" | $C_p\left ( T_2-T_1 \right )\;$

Entropy
Δs

| align="center" | $\Delta S = C_V \ln{T_2 \over T_1} + nR \ln{V_2 \over V_1}$ {{br}} $\Delta S = C_p \ln{T_2 \over T_1} - nR \ln{p_2 \over p_1}$ Keenan, Thermodynamics, Wiley, New York, 1947

| align="center" | $C_p\ln\frac{T_2}{T_1}\;$

| align="center" | $C_V\ln\frac{T_2}{T_1}\;$

| align="center" | $nR\ln\frac{V_2}{V_1}\;$ {{br}} $\frac{Q}{T}\;$

| align="center" | $C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;$

Constant

| $\;$

| align="center" | $\frac{V}{T}\;$

| align="center" | $\frac{p}{T}\;$

| align="center" | $p V\;$

| align="center" | $p V^\gamma\;$

= Entropy =

$S = k_\mathrm{B} \ln \Omega$ , where k_B is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
$dS = \frac{\delta Q}{T}$ , 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.

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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: $\theta = k_\text{B} T/mc^2$ {{endplainlist}} K₂ is the modified Bessel function of the second kind. \| Non-relativistic speeds $P\left ( v \right )=4\pi\left ( \frac{m}{2\pi k_\text{B} T} \right )^{3/2} v^2 e^{-mv^2/2 k_\text{B} T}$ Relativistic speeds (Maxwell–Jüttner distribution) $f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} e^{-\gamma(p)/\theta}$
Entropy Logarithm of the density of states \| {{plainlist}} P_i = probability of system in microstate i Ω = total number of microstates {{endplainlist}} \| $S = - k_\text{B}\sum_i P_i \ln P_i = k_\mathrm{B}\ln \Omega$ where: $P_i = 1/\Omega$
Entropy change \| \| $\Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T}$ $\Delta S = k_\text{B} N \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1}$
Entropic force \| \| $\mathbf{F}_\mathrm{S} = -T \nabla S$
Equipartition theorem \| d_f = degree of freedom \| Average kinetic energy per degree of freedom $\langle E_\mathrm{k} \rangle = \frac{1}{2}kT$ Internal energy $U = d_\text{f} \langle E_\mathrm{k} \rangle = \frac{d_\text{f}}{2}kT$

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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: $\theta = k_\text{B} T/mc^2$

K₂ is the modified Bessel function of the second kind.

| Non-relativistic speeds

$P\left ( v \right )=4\pi\left ( \frac{m}{2\pi k_\text{B} T} \right )^{3/2} v^2 e^{-mv^2/2 k_\text{B} T}$

Relativistic speeds (Maxwell–Jüttner distribution)

$f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} e^{-\gamma(p)/\theta}$

Entropy Logarithm of the density of states

| {{plainlist}}

P_i = probability of system in microstate i
Ω = total number of microstates

| $S = - k_\text{B}\sum_i P_i \ln P_i = k_\mathrm{B}\ln \Omega$

where:

$P_i = 1/\Omega$

Entropy change

| $\Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T}$

$\Delta S = k_\text{B} N \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1}$

Entropic force

| $\mathbf{F}_\mathrm{S} = -T \nabla S$

Equipartition theorem

| d_f = degree of freedom

| Average kinetic energy per degree of freedom

$\langle E_\mathrm{k} \rangle = \frac{1}{2}kT$

Internal energy

$U = d_\text{f} \langle E_\mathrm{k} \rangle = \frac{d_\text{f}}{2}kT$

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

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scope="col" width="100" \| Physical situation ! scope="col" width="250" \| Nomenclature ! scope="col" width="10" \| Equations
Mean speed \| \| $\langle v \rangle = \sqrt{\frac{8 k_\text{B} T}{\pi m}}$
Root mean square speed \| \| $v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k_\text{B} T}{m}}$
Modal speed \| \| $v_\mathrm{mode} = \sqrt{\frac{2 k_\text{B} T}{m}}$
Mean free path \| {{plainlist}} σ = effective cross-section n = volume density of number of target particles {{ell}} = mean free path {{endplainlist}} \| $\ell = 1/\sqrt{2} n \sigma$

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scope="col" width="100" | Physical situation

! scope="col" width="250" | Nomenclature

! scope="col" width="10" | Equations

Mean speed

| $\langle v \rangle = \sqrt{\frac{8 k_\text{B} T}{\pi m}}$

Root mean square speed

| $v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k_\text{B} T}{m}}$

Modal speed

| $v_\mathrm{mode} = \sqrt{\frac{2 k_\text{B} T}{m}}$

Mean free path

| {{plainlist}}

σ = effective cross-section
n = volume density of number of target particles
{{ell}} = mean free path

| $\ell = 1/\sqrt{2} n \sigma$

= Quasi-static and reversible processes =

For quasi-static and reversible processes, the first law of thermodynamics is:

: $dU=\delta Q - \delta W$

where δQ is the heat supplied to the system and δW is the work done by the system.

= Thermodynamic potentials =

The following energies are called the thermodynamic potentials,

: {{table of thermodynamic potentials}}

and the corresponding fundamental thermodynamic relations or "master equations"Physical chemistry, P.W. Atkins, Oxford University Press, 1978, {{ISBN|0 19 855148 7}} are:

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

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Potential

! Differential

Internal energy

| $dU\left(S,V,{N_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i$

Enthalpy

| $dH\left(S,p,{N_{i}}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}$

Helmholtz free energy

| $dF\left(T,V,{N_{i}}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}$

Gibbs free energy

| $dG\left(T,p,{N_{i}}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}$

= Maxwell's relations =

The four most common Maxwell's relations are:

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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}} $U(S,V)\,$ = Internal energy $H(S,p)\,$ = Enthalpy $F(T,V)\,$ = Helmholtz free energy $G(T,p)\,$ = Gibbs free energy {{endplainlist}} \| $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial p}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V}$ $\left(\frac{\partial T}{\partial p}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_p = \frac{\partial^2 H }{\partial S \partial p}$ $+\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V = - \frac{\partial^2 F }{\partial T \partial V}$ $-\left(\frac{\partial S}{\partial p}\right)_T = \left(\frac{\partial V}{\partial T}\right)_p = \frac{\partial^2 G }{\partial T \partial p}$

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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}}

$U(S,V)\,$ = Internal energy
$H(S,p)\,$ = Enthalpy
$F(T,V)\,$ = Helmholtz free energy
$G(T,p)\,$ = Gibbs free energy

| $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial p}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V}$

$\left(\frac{\partial T}{\partial p}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_p = \frac{\partial^2 H }{\partial S \partial p}$

$+\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V = - \frac{\partial^2 F }{\partial T \partial V}$

$-\left(\frac{\partial S}{\partial p}\right)_T = \left(\frac{\partial V}{\partial T}\right)_p = \frac{\partial^2 G }{\partial T \partial p}$

More relations include the following.

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\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} }

Other differential equations are:

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Name ! H ! U ! G
Gibbs–Helmholtz equation \| $H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p$ \| $U = -T^2\left(\frac{\partial \left(F/T\right)}{\partial T}\right)_V$ \| $G = -V^2\left(\frac{\partial \left(F/V\right)}{\partial V}\right)_T$
\| $\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_p$ \| $\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$ \|

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Name

! H

! U

! G

Gibbs–Helmholtz equation

| $H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p$

| $U = -T^2\left(\frac{\partial \left(F/T\right)}{\partial T}\right)_V$

| $G = -V^2\left(\frac{\partial \left(F/V\right)}{\partial V}\right)_T$

\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_p

| $\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$

= Quantum properties =

$U = N k_\text{B} T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V$
$S = \frac{U}{T} + N k_\text{B} \ln Z - N k \ln N + Nk$ Indistinguishable Particles

where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

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Degree of freedom ! Partition function
Translation \| $Z_t = \frac{(2 \pi m k_\text{B} T)^\frac{3}{2} V}{h^3}$
Vibration \| $Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_\text{B} T}}$
Rotation \| $Z_r = \frac{2 I k_\text{B} T}{\sigma (\frac{h}{2 \pi})^2}$ {{plainlist}} where: σ = 1 (heteronuclear molecules) σ = 2 (homonuclear) {{endplainlist}}

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Degree of freedom

! Partition function

Translation

| $Z_t = \frac{(2 \pi m k_\text{B} T)^\frac{3}{2} V}{h^3}$

Vibration

| $Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_\text{B} T}}$

Rotation

| $Z_r = \frac{2 I k_\text{B} T}{\sigma (\frac{h}{2 \pi})^2}$

where:
σ = 1 (heteronuclear molecules)
σ = 2 (homonuclear)

Thermal properties of matter

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Coefficients ! Equation
Joule-Thomson coefficient \| $\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H$
Compressibility (constant temperature) \| $K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N}$
Coefficient of thermal expansion (constant pressure) \| $\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$
Heat capacity (constant pressure) \| $C_p = \left ( {\partial Q_{rev} \over \partial T} \right )_p = \left ( {\partial U \over \partial T} \right )_p + p \left ( {\partial V \over \partial T} \right )_p = \left ( {\partial H \over \partial T} \right )_p = T \left ( {\partial S \over \partial T} \right )_p$
Heat capacity (constant volume) \| $C_V = \left ( {\partial Q_{rev} \over \partial T} \right )_V = \left ( {\partial U \over \partial T} \right )_V = T \left ( {\partial S \over \partial T} \right )_V$

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Coefficients

! Equation

Joule-Thomson coefficient

| $\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H$

Compressibility (constant temperature)

| $K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N}$

Coefficient of thermal expansion (constant pressure)

| $\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$

Heat capacity (constant pressure)

| $C_p
= \left ( {\partial Q_{rev} \over \partial T} \right )_p
= \left ( {\partial U \over \partial T} \right )_p + p \left ( {\partial V \over \partial T} \right )_p
= \left ( {\partial H \over \partial T} \right )_p
= T \left ( {\partial S \over \partial T} \right )_p$

Heat capacity (constant volume)

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

class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Derivation of heat capacity (constant pressure)

Since

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

: $\begin{align}
\left(\frac{\partial T}{\partial p}\right)_H
& = -\left(\frac{\partial H}{\partial p}\right)_T
\left(\frac{\partial T}{\partial H}\right)_p
\\
& = \frac{-1}{\left(\frac{\partial H}{\partial T}\right)_p}
\left(\frac{\partial H}{\partial p}\right)_T
\end{align}$

: $C_p = \left(\frac{\partial H}{\partial T}\right)_p$

: $\Rightarrow \left(\frac{\partial T}{\partial p}\right)_H
= -\frac{1}{C_p}
\left(\frac{\partial H}{\partial p}\right)_T$

class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Derivation of heat capacity (constant volume)

Since

: $dU = \delta Q_{rev} - \delta W_{rev} ,$

(where δW_rev is the work done by the system),

: $\delta S = \frac{\delta Q_{rev}}{T}, \delta W_{rev}= p \delta V$

: $d U = T \delta S- p\delta V$

: $\left(\frac{\partial U}{\partial T}\right)_V
= T\left(\frac{\partial S}{\partial T}\right)_V
- p\left(\frac{\partial V}{\partial T}\right)_V ; C_V = \left(\frac{\partial U}{\partial T}\right)_V$

: $\Rightarrow C_V = T\left(\frac{\partial S}{\partial T}\right)_V$

= Thermal transfer =

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scope="col" width="100" \| Physical situation ! scope="col" width="250" \| Nomenclature ! scope="col" width="10" \| Equations
Net intensity emission/absorption \| {{plainlist}} T_external = external temperature (outside of system) T_system = internal temperature (inside system) ε = emissivity {{endplainlist}} \| $I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right )$
Internal energy of a substance \| {{plainlist}} C_V = isovolumetric heat capacity of substance ΔT = temperature change of substance {{endplainlist}} \| $\Delta U = N C_V \Delta T$
Meyer's equation \| {{plainlist}} C_p = isobaric heat capacity C_V = isovolumetric heat capacity n = amount of substance {{endplainlist}} \| $C_p - C_V = nR$
Effective thermal conductivities \| {{plainlist}} λ_i = thermal conductivity of substance i λ_net = equivalent thermal conductivity {{endplainlist}} \| Series $\lambda_\mathrm{net} = \sum_j \lambda_j$ Parallel $\frac{1}{\lambda}_\mathrm{net} = \sum_j \left ( \frac{1}{\lambda}_j \right )$

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scope="col" width="100" | Physical situation

! scope="col" width="250" | Nomenclature

! scope="col" width="10" | Equations

Net intensity emission/absorption

| {{plainlist}}

T_external = external temperature (outside of system)
T_system = internal temperature (inside system)
ε = emissivity

| $I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right )$

Internal energy of a substance

| {{plainlist}}

C_V = isovolumetric heat capacity of substance
ΔT = temperature change of substance

| $\Delta U = N C_V \Delta T$

Meyer's equation

| {{plainlist}}

C_p = isobaric heat capacity
C_V = isovolumetric heat capacity
n = amount of substance

| $C_p - C_V = nR$

Effective thermal conductivities

| {{plainlist}}

λ_i = thermal conductivity of substance i
λ_net = equivalent thermal conductivity

| Series

$\lambda_\mathrm{net} = \sum_j \lambda_j$

Parallel

$\frac{1}{\lambda}_\mathrm{net} = \sum_j \left ( \frac{1}{\lambda}_j \right )$

= Thermal efficiencies =

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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_H = heat energy in higher temperature reservoir Q_L = heat energy in lower temperature reservoir T_H = temperature of higher temp. reservoir T_L = temperature of lower temp. reservoir {{endplainlist}} \| Thermodynamic engine: $\eta = \left \|\frac{W}{Q_\text{H}} \right\|$ Carnot engine efficiency: $\eta_\text{c} = 1 - \left \| \frac{Q_\text{L}}{Q_\text{H}} \right \| = 1-\frac{T_\text{L}}{T_\text{H}}$
Refrigeration \| K = coefficient of refrigeration performance \| Refrigeration performance $K = \left \| \frac{Q_\text{L}}{W} \right \|$ Carnot refrigeration performance $K_\text{C} = \frac{\|Q_\text{L}$

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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_H = heat energy in higher temperature reservoir
Q_L = heat energy in lower temperature reservoir
T_H = temperature of higher temp. reservoir
T_L = temperature of lower temp. reservoir

| Thermodynamic engine:

$\eta = \left |\frac{W}{Q_\text{H}} \right|$

Carnot engine efficiency:

$\eta_\text{c} = 1 - \left | \frac{Q_\text{L}}{Q_\text{H}} \right | = 1-\frac{T_\text{L}}{T_\text{H}}$

Refrigeration

| K = coefficient of refrigeration performance

| Refrigeration performance

$K = \left | \frac{Q_\text{L}}{W} \right |$

Carnot refrigeration performance

$K_\text{C} = \frac{|Q_\text{L}$

Q_\text{H}

Q_\text{L}

= \frac{T_\text{L}}{T_\text{H}-T_\text{L}}

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".
{{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| bibcode=1914PhRv....3..273B }}
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 edition, New York: John Wiley & Sons.

External links

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

Category:Thermodynamic equations

Category:Thermodynamics

Category:Chemical engineering

table of thermodynamic equations

Definitions

= General basic quantities =

= General derived quantities =

= Thermal properties of matter =

= Thermal transfer =

Equations

= Thermodynamic processes =

= Kinetic theory =

== Ideal gas ==

= Entropy =

=Statistical physics=

= Quasi-static and reversible processes =

= Thermodynamic potentials =

= Maxwell's relations =

= Quantum properties =

Thermal properties of matter

= Thermal transfer =

= Thermal efficiencies =

See also

References

External links