Table of thermodynamic equations explained

Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:

Definitions

See main article: article, List of thermodynamic properties, Thermodynamic potential, Free entropy and Defining equation (physical chemistry).

Many of the definitions below are also used in the thermodynamics of chemical reactions.

General basic quantities

Quantity (common name/s)	(Common) symbol/s	SI unit	Dimension
Number of molecules	N	1	1
Amount of substance	n	mol	N
Temperature	T	K	Θ
Heat Energy	Q, q	J	ML^2T^−2
Latent heat	QL	J	ML^2T^−2

General derived quantities

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Thermodynamic beta, inverse temperature	β	\beta = 1 /k _{B} T	J^−1	T^2M^−1L^−2
Thermodynamic temperature	τ	\tau = k _{B} T \tau = k _{B} \left (\partial U/\partial S \right ) _{N} 1 /\tau = 1 /k _{B} \left (\partial S/\partial U \right ) _{N}	J	ML^2T^−2
Entropy	S	S = - k _{B\sum} _{i} p _{iln} p _{i} S = - \left (\partial F/\partial T \right ) _{V} ,S = - \left (\partial G/\partial T \right ) _{N,P}	J⋅K^−1	ML^2T^−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	ML^2T^−2
Enthalpy	H	H = U + pV	J	ML^2T^−2
	Z		1	1
Gibbs free energy	G	G = H - TS	J	ML^2T^−2
Chemical potential (of component i in a mixture)	μi	\mu _{i} = \left (\partial U/\partial N _{i} \right ) _{N _{j} , S, V} \mu _{i} = \left (\partial F/\partial N _{i} \right ) _{T,} , where F is not proportional to N because \mu _{i} depends on pressure.\mu _{i} = \left (\partial G/\partial N _{i} \right ) _{T,} , 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 _{B} \left (\partial S/\partial N _{i} \right ) _{U,V}	J	ML^2T^−2
Helmholtz free energy	A, F	F = U - TS	J	ML^2T^−2
Landau potential, Landau free energy, Grand potential	Ω, ΦG	\Omega = U - TS - \mu N	J	ML^2T^−2
Massieu potential, Helmholtz free entropy	Φ	\Phi = S - U/T	J⋅K^−1	ML^2T^−2Θ^−1
Planck potential, Gibbs free entropy	Ξ	\Xi = \Phi - pV/T	J⋅K^−1	ML^2T^−2Θ^−1

Thermal properties of matter

See main article: articles, Heat capacity and Thermal expansion.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
General heat/thermal capacity	C	C = \partial Q/\partial T	J⋅K^−1	ML^2T^−2Θ^−1
Heat capacity (isobaric)	Cp	C _{p} = \partial H/\partial T	J⋅K^−1	ML^2T^−2Θ^−1
Specific heat capacity (isobaric)	Cmp	C _{mp} = \partial ^{2} Q/\partial m \partial T	J⋅kg^−1⋅K^−1	L^2T^−2Θ^−1
Molar specific heat capacity (isobaric)	Cnp	C _{np} = \partial ^{2} Q/\partial n \partial T	J⋅K^−1⋅mol^−1	ML^2T^−2Θ^−1N^−1
Heat capacity (isochoric/volumetric)	CV	C _{V} = \partial U/\partial T	J⋅K^−1	ML^2T^−2Θ^−1
Specific heat capacity (isochoric)	CmV	C _{mV} = \partial ^{2} Q/\partial m \partial T	J⋅kg^−1⋅K^−1	L^2T^−2Θ^−1
Molar specific heat capacity (isochoric)	CnV	C _{nV} = \partial ^{2} Q/\partial n \partial T	J⋅K⋅^−1 mol^−1	ML^2T^−2Θ^−1N^−1
Specific latent heat	L	L = \partial Q/ \partial m	J⋅kg^−1	L^2T^−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

Thermal transfer

See main article: Thermal conductivity.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	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 = d Q/ d t	W	ML^2T^−3
Thermal intensity	I	I = d P/ d A	W⋅m^−2	MT^−3
Thermal/heat flux density (vector analogue of thermal intensity above)	q	Q = \iint q ⋅ d S d t	W⋅m^−2	MT^−3

Equations

The equations in this article are classified by subject.

Thermodynamic processes

Physical situation	Equations
Isentropic process (adiabatic and reversible)	Q = 0 , \Delta U = - W For an ideal gas p _{1} V ^{\gamma} _{1} = p _{2} V ^{\gamma} _{2} T _{1} V ^{\gamma -1} _{1} = T _{2} V ^{\gamma -1} _{2} p ^{1- \gamma} _{1} T ^{\gamma} _{1} = p ^{1- \gamma} _{2} T ^{\gamma} _{2}
Isothermal process	\Delta U = 0 , 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, Q = \Delta U + p \delta V
Isochoric process	V1 = V2, V = constant W = 0 , Q = \Delta U
Free expansion	\Delta U = 0
Work done by an expanding gas	Process W = \int ^{V _{2}} _{V _{1}} p d V Net work done in cyclic processes W = \oint _{cycle} p d V = \oint _{cycle\Delta} Q

Kinetic theory

Physical situation! scope="col" width="250"

Nomenclature	Equations
Ideal gas law	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 (p _{1} V _{1})/(p _{2} V _{2}) = (n _{1} T _{1})/(n _{2} T _{2}) = (N _{1} T _{1})/(N _{2} T _{2})
Pressure of an ideal gas	m = mass of one molecule Mm = molar mass	p = (Nm \langle v ^{2} \rangle)/(3 V) = (nM _{m} \langle v ^{2} \rangle)/(3 V) = (1)/(3) \rho \langle v ^{2} \rangle

Ideal gas

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic Q =0
Work W	\delta W = - p dV	- p\Delta V	0	- nRT ln (V _{2})/(V _{1}) - nRT ln (P _{1})/(P _{2})	(PV ^{\gamma} (V ^{1- \gamma} _{f} - V ^{1- \gamma} _{i} ))/(1- \gamma) = C _{V} \left(T _{2} - T _{1} \right)
Heat Capacity C	(as for real gas)	C _{p} = (5)/(2) nR (for monatomic ideal gas) C _{p} = (7)/(2) nR (for diatomic ideal gas)	C _{V} = (3)/(2) nR (for monatomic ideal gas) C _{V} = (5)/(2) nR (for diatomic ideal gas)
Internal Energy ΔU	\Delta U = C _{V} \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 ΔH	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 Δs	\Delta S = C _{V} ln {T _{2} \over T _{1}} + nR ln {V _{2} \over V _{1}} \Delta S = C _{p} ln {T _{2} \over T _{1}} - nR ln {p _{2} \over p _{1}} ^[1]	C _{p ln (T _{2})/(T _{1})}	C _{V ln (T _{2})/(T _{1})}	nR ln (V _{2})/(V _{1}) (Q)/(T)	C _{p ln (V _{2})/(V _{1})} + C _{V ln (p _{2})/(p _{1})} =0
Constant		(V)/(T)	(p)/(T)	p V	p V ^{\gamma}

Entropy

S = k _{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 = (\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.

Physical situation	Nomenclature	Equations
Maxwell–Boltzmann distribution	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 _{B} T/mc ^{2} K2 is the modified Bessel function of the second kind.	Non-relativistic speeds P\left ( v \right ) =4 \pi\left ( (m)/(2 \pi k _{B} T) \right ) ^{3/2} v ^{2} e ^{- mv ^{2/2} k _{B} T} Relativistic speeds (Maxwell–Jüttner distribution) f(p) = (1)/(4 \pi m ^{3} c ^{3} \theta K _{2(1/\theta)}) e ^{-\gamma(p)/\theta}
Entropy Logarithm of the density of states	Pi = probability of system in microstate i Ω = total number of microstates	S = - k _{B\sum} _{i} P _{i} ln P _{i} = k _{Bln} \Omega where: P _{i} = 1 /\Omega
Entropy change		\Delta S = \int ^{Q _{2}} _{Q _{1}} (d Q)/(T) \Delta S = k _{B} N ln (V _{2})/(V _{1}) + N C _{V} ln (T _{2})/(T _{1})
Entropic force		F _{S} = - T \nabla S
Equipartition theorem	df = degree of freedom	Average kinetic energy per degree of freedom\langle E _{k} \rangle = (1)/(2) kT Internal energy U = d _{f} \langle E _{k} \rangle = (d _{f})/(2) kT

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

Physical situation	Nomenclature	Equations
Mean speed		\langle v \rangle = \sqrt{ (8 k _{B} T)/(\pi m) }
Root mean square speed		v _{rms} = \sqrt{\langle v ^{2} \rangle} = \sqrt{ (3 k _{B} T)/(m) }
Modal speed		v _{mode} = \sqrt{ (2 k _{B} T)/(m) }
Mean free path	σ = effective cross-section n = volume density of number of target particles = 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

See main article: Thermodynamic potentials.

See also: Maxwell relations.

The following energies are called the thermodynamic potentials,

and the corresponding fundamental thermodynamic relations or "master equations"^[2] are:

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

Maxwell's relations

The four most common Maxwell's relations are:

More relations include the following.

\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:

Name	H	U	G
Gibbs–Helmholtz equation	H = - T ^{2 \left( (\partial \left(G/T\right))/(\partial T)} \right) _{p}	U = - T ^{2 \left( (\partial \left(F/T\right))/(\partial T)} \right) _{V}	G = - V ^{2 \left( (\partial \left(F/V\right))/(\partial V)} \right) _{T}
	\left( (\partial H)/(\partial p) \right) _{T} = V - T\left( (\partial V)/(\partial T) \right) _{P}	\left( (\partial U)/(\partial V) \right) _{T} = T\left( (\partial P)/(\partial T) \right) _{V} - P

Quantum properties

U = N k _{B} T ^{2} \left( (\partial ln Z)/(\partial T) \right) _{V} S = (U)/(T) + N k _{B} ln Z - N k ln N + Nk Indistinguishable Particleswhere N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

Degree of freedom	Partition function
Translation	Z _{t} = (( 2 \pi m k _{B} T) ^{(3)/(2)} V)/(h ^{3})
Vibration	Z _{v} = (1)/(1 - e ^{(- h \omega)/(2 \pi k _{B} T)})
Rotation	Z _{r} = (2 I k _{B} T)/(\sigma ( (h)/(2 \pi) ) ^{2}) where: σ = 1 (heteronuclear molecules) σ = 2 (homonuclear)

Thermal properties of matter

Coefficients	Equation
Joule-Thomson coefficient	\mu _{JT} = \left( (\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} = (1)/(\left( V) (\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}

Thermal transfer

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	Texternal = external temperature (outside of system) Tsystem = internal temperature (inside system) ε = emissivity	I = \sigma \epsilon \left ( T ^{4} _{external} - T ^{4} _{system} \right )
Internal energy of a substance	CV = isovolumetric heat capacity of substance ΔT = temperature change of substance	\Delta U = N C _{V} \Delta T
Meyer's equation	Cp = isobaric heat capacity CV = isovolumetric heat capacity n = amount of substance	C _{p} - C _{V} = nR
Effective thermal conductivities	λi = thermal conductivity of substance i λnet = equivalent thermal conductivity	Seriesλ _{net} = \sum _{j} λ _{j} Parallel (1)/(λ) _{net} = \sum _{j} \left ( (1)/(λ) _{j} \right )

Thermal efficiencies

Physical situation	Nomenclature	Equations
Thermodynamic engines	η = efficiency W = work done by engine QH = heat energy in higher temperature reservoir QL = heat energy in lower temperature reservoir TH = temperature of higher temp. reservoir TL = temperature of lower temp. reservoir	Thermodynamic engine: η = \left	\frac \right	Carnot engine efficiency: η _{c} = 1 - \left	\frac \right	= 1-\frac
Refrigeration	K = coefficient of refrigeration performance	Refrigeration performanceK = \left	\frac \right	Carnot refrigeration performance K _{C} = (|Q _{L|})/(|Q _{H|-|Q} _{L|}) = (T _{L})/(T _{H-T} _{L})

See also

• List of thermodynamic properties
• Antoine equation
• Bejan number
• Bowen ratio
• Bridgman's equations
• Clausius–Clapeyron relation
• Departure functions
• 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

References

• Atkins, Peter and de Paula, Julio Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 .
  • Chapters 1–10, Part 1: "Equilibrium".
• Bridgman . P. W. . A Complete Collection of Thermodynamic Formulas . Physical Review . American Physical Society (APS) . 3 . 4 . 1 March 1914 . 0031-899X . 10.1103/physrev.3.273. 273–281.
• 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 .
• 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

• Thermodynamic equation calculator

Notes and References

1. Keenan, Thermodynamics, Wiley, New York, 1947
2. Physical chemistry, P.W. Atkins, Oxford University Press, 1978,