Thermodynamics
Common thermodynamic equations and quantities in thermodynamics , using mathematical notation , are as follows:
Many of the definitions below are also used in the thermodynamics of chemical reactions .
General basic quantities [ edit ] 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 ML2 T−2 Latent heat QL J ML2 T−2
General derived quantities [ edit ] Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Thermodynamic beta , inverse temperature β β = 1 / k B T {\displaystyle \beta =1/k_{\text{B}}T} J−1 T2 M−1 L−2 Thermodynamic temperature τ τ = k B T {\displaystyle \tau =k_{\text{B}}T} τ = k B ( ∂ U / ∂ S ) N {\displaystyle \tau =k_{\text{B}}\left(\partial U/\partial S\right)_{N}} 1 / τ = 1 / k B ( ∂ S / ∂ U ) N {\displaystyle 1/\tau =1/k_{\text{B}}\left(\partial S/\partial U\right)_{N}}
J ML2 T−2 Entropy S S = − k B ∑ i p i ln p i {\displaystyle S=-k_{\text{B}}\sum _{i}p_{i}\ln p_{i}} S = − ( ∂ F / ∂ T ) V , N {\displaystyle S=-\left(\partial F/\partial T\right)_{V,N}} S = − ( ∂ G / ∂ T ) P , N {\displaystyle S=-\left(\partial G/\partial T\right)_{P,N}}
J⋅K−1 ML2 T−2 Θ−1 Pressure P P = − ( ∂ F / ∂ V ) T , N {\displaystyle P=-\left(\partial F/\partial V\right)_{T,N}} P = − ( ∂ U / ∂ V ) S , N {\displaystyle P=-\left(\partial U/\partial V\right)_{S,N}}
Pa ML−1 T−2 Internal Energy U U = ∑ i E i {\displaystyle U=\sum _{i}E_{i}} J ML2 T−2 Enthalpy H H = U + p V {\displaystyle H=U+pV} J ML2 T−2 Partition Function Z 1 1 Gibbs free energy G G = H − T S {\displaystyle G=H-TS} J ML2 T−2 Chemical potential (of component i in a mixture) μi μ i = ( ∂ U / ∂ N i ) N j ≠ i , S , V {\displaystyle \mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}} μ i = ( ∂ F / ∂ N i ) T , V {\displaystyle \mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}} F {\displaystyle F} N {\displaystyle N} μ i {\displaystyle \mu _{i}} μ i = ( ∂ G / ∂ N i ) T , P {\displaystyle \mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}} G {\displaystyle G} N {\displaystyle N} μ i {\displaystyle \mu _{i}} μ i / τ = − 1 / k B ( ∂ S / ∂ N i ) U , V {\displaystyle \mu _{i}/\tau =-1/k_{\text{B}}\left(\partial S/\partial N_{i}\right)_{U,V}}
J ML2 T−2 Helmholtz free energy A , F F = U − T S {\displaystyle F=U-TS} J ML2 T−2 Landau potential , Landau free energy, Grand potential Ω , ΦG Ω = U − T S − μ N {\displaystyle \Omega =U-TS-\mu N\ } J ML2 T−2 Massieu potential, Helmholtz free entropy Φ Φ = S − U / T {\displaystyle \Phi =S-U/T} J⋅K−1 ML2 T−2 Θ−1 Planck potential, Gibbs free entropy Ξ Ξ = Φ − p V / T {\displaystyle \Xi =\Phi -pV/T} J⋅K−1 ML2 T−2 Θ−1
Thermal properties of matter [ edit ] Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension General heat/thermal capacity C C = ∂ Q / ∂ T {\displaystyle C=\partial Q/\partial T} J⋅K−1 ML2 T−2 Θ−1 Heat capacity (isobaric) Cp C p = ∂ H / ∂ T {\displaystyle C_{p}=\partial H/\partial T} J⋅K−1 ML2 T−2 Θ−1 Specific heat capacity (isobaric) Cmp C m p = ∂ 2 Q / ∂ m ∂ T {\displaystyle C_{mp}=\partial ^{2}Q/\partial m\partial T} J⋅kg−1 ⋅K−1 L2 T−2 Θ−1 Molar specific heat capacity (isobaric) Cnp C n p = ∂ 2 Q / ∂ n ∂ T {\displaystyle C_{np}=\partial ^{2}Q/\partial n\partial T} J⋅K−1 ⋅mol−1 ML2 T−2 Θ−1 N−1 Heat capacity (isochoric/volumetric) CV C V = ∂ U / ∂ T {\displaystyle C_{V}=\partial U/\partial T} J⋅K−1 ML2 T−2 Θ−1 Specific heat capacity (isochoric) CmV C m V = ∂ 2 Q / ∂ m ∂ T {\displaystyle C_{mV}=\partial ^{2}Q/\partial m\partial T} J⋅kg−1 ⋅K−1 L2 T−2 Θ−1 Molar specific heat capacity (isochoric) CnV C n V = ∂ 2 Q / ∂ n ∂ T {\displaystyle C_{nV}=\partial ^{2}Q/\partial n\partial T} J⋅K⋅−1 mol−1 ML2 T−2 Θ−1 N−1 Specific latent heat L L = ∂ Q / ∂ m {\displaystyle L=\partial Q/\partial m} J⋅kg−1 L2 T−2 Ratio of isobaric to isochoric heat capacity, heat capacity ratio , adiabatic index, Laplace coefficient γ γ = C p / C V = c p / c V = C m p / C m V {\displaystyle \gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}} 1 1
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Temperature gradient No standard symbol ∇ T {\displaystyle \nabla T} K⋅m−1 ΘL−1 Thermal conduction rate, thermal current, thermal/heat flux , thermal power transfer P P = d Q / d t {\displaystyle P=\mathrm {d} Q/\mathrm {d} t} W ML2 T−3 Thermal intensity I I = d P / d A {\displaystyle I=\mathrm {d} P/\mathrm {d} A} W⋅m−2 MT−3 Thermal/heat flux density (vector analogue of thermal intensity above) q Q = ∬ q ⋅ d S d t {\displaystyle Q=\iint \mathbf {q} \cdot \mathrm {d} \mathbf {S} \mathrm {d} t} W⋅m−2 MT−3
The equations in this article are classified by subject.
Thermodynamic processes [ edit ] Physical situation Equations Isentropic process (adiabatic and reversible) Q = 0 , Δ U = − W {\displaystyle Q=0,\quad \Delta U=-W} For an ideal gas p 1 V 1 γ = p 2 V 2 γ {\displaystyle p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }} T 1 V 1 γ − 1 = T 2 V 2 γ − 1 {\displaystyle T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}} p 1 1 − γ T 1 γ = p 2 1 − γ T 2 γ {\displaystyle p_{1}^{1-\gamma }T_{1}^{\gamma }=p_{2}^{1-\gamma }T_{2}^{\gamma }}
Isothermal process Δ U = 0 , W = Q {\displaystyle \Delta U=0,\quad W=Q} For an ideal gas W = k T N ln ( V 2 / V 1 ) {\displaystyle W=kTN\ln(V_{2}/V_{1})} W = n R T ln ( V 2 / V 1 ) {\displaystyle W=nRT\ln(V_{2}/V_{1})}
Isobaric process p 1 = p 2 , p = constant W = p Δ V , Q = Δ U + p δ V {\displaystyle W=p\Delta V,\quad Q=\Delta U+p\delta V}
Isochoric process V 1 = V 2 , V = constant W = 0 , Q = Δ U {\displaystyle W=0,\quad Q=\Delta U}
Free expansion Δ U = 0 {\displaystyle \Delta U=0} Work done by an expanding gas Process W = ∫ V 1 V 2 p d V {\displaystyle W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V}
Net work done in cyclic processes W = ∮ c y c l e p d V = ∮ c y c l e Δ Q {\displaystyle W=\oint _{\mathrm {cycle} }p\mathrm {d} V=\oint _{\mathrm {cycle} }\Delta Q}
S = k B ln Ω {\displaystyle S=k_{\mathrm {B} }\ln \Omega } k B is the Boltzmann constant , and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability. d S = δ Q T {\displaystyle dS={\frac {\delta Q}{T}}} Statistical physics [ edit ] 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: θ = k B T / m c 2 {\displaystyle \theta =k_{\text{B}}T/mc^{2}} K 2 is the modified Bessel function of the second kind.
Non-relativistic speeds P ( v ) = 4 π ( m 2 π k B T ) 3 / 2 v 2 e − m v 2 / 2 k B T {\displaystyle P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{\text{B}}T}}
Relativistic speeds (Maxwell–Jüttner distribution) f ( p ) = 1 4 π m 3 c 3 θ K 2 ( 1 / θ ) e − γ ( p ) / θ {\displaystyle 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 Pi = probability of system in microstate i Ω = total number of microstates S = − k B ∑ i P i ln P i = k B ln Ω {\displaystyle S=-k_{\text{B}}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega } where: P i = 1 / Ω {\displaystyle P_{i}=1/\Omega }
Entropy change Δ S = ∫ Q 1 Q 2 d Q T {\displaystyle \Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}} Δ S = k B N ln V 2 V 1 + N C V ln T 2 T 1 {\displaystyle \Delta S=k_{\text{B}}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}}
Entropic force F S = − T ∇ S {\displaystyle \mathbf {F} _{\mathrm {S} }=-T\nabla S} Equipartition theorem d f = degree of freedom Average kinetic energy per degree of freedom ⟨ E k ⟩ = 1 2 k T {\displaystyle \langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT}
Internal energy U = d f ⟨ E k ⟩ = d f 2 k T {\displaystyle 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.
Physical situation Nomenclature Equations Mean speed ⟨ v ⟩ = 8 k B T π m {\displaystyle \langle v\rangle ={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}} Root mean square speed v r m s = ⟨ v 2 ⟩ = 3 k B T m {\displaystyle v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}} Modal speed v m o d e = 2 k B T m {\displaystyle v_{\mathrm {mode} }={\sqrt {\frac {2k_{\text{B}}T}{m}}}} Mean free path σ = effective cross-sectionn = volume density of number of target particlesℓ = mean free path ℓ = 1 / 2 n σ {\displaystyle \ell =1/{\sqrt {2}}n\sigma }
Quasi-static and reversible processes [ edit ] For quasi-static and reversible processes, the first law of thermodynamics is:
d U = δ Q − δ W {\displaystyle 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 [ edit ] The following energies are called the thermodynamic potentials ,
Name Symbol Formula Natural variables Internal energy U {\displaystyle U} ∫ ( T d S − p d V + ∑ i μ i d N i ) {\displaystyle \int \left(T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\right)} S , V , { N i } {\displaystyle S,V,\{N_{i}\}} Helmholtz free energy F {\displaystyle F} U − T S {\displaystyle U-TS} T , V , { N i } {\displaystyle T,V,\{N_{i}\}} Enthalpy H {\displaystyle H} U + p V {\displaystyle U+pV} S , p , { N i } {\displaystyle S,p,\{N_{i}\}} Gibbs free energy G {\displaystyle G} U + p V − T S {\displaystyle U+pV-TS} T , p , { N i } {\displaystyle T,p,\{N_{i}\}} Landau potential, or Ω {\displaystyle \Omega } Φ G {\displaystyle \Phi _{\text{G}}} U − T S − {\displaystyle U-TS-} ∑ i {\displaystyle \sum _{i}\,} μ i N i {\displaystyle \mu _{i}N_{i}} T , V , { μ i } {\displaystyle T,V,\{\mu _{i}\}}
and the corresponding fundamental thermodynamic relations or "master equations"[ 2]
Potential Differential Internal energy d U ( S , V , N i ) = T d S − p d V + ∑ i μ i d N i {\displaystyle dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}} Enthalpy d H ( S , p , N i ) = T d S + V d p + ∑ i μ i d N i {\displaystyle dH\left(S,p,{N_{i}}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}} Helmholtz free energy d F ( T , V , N i ) = − S d T − p d V + ∑ i μ i d N i {\displaystyle dF\left(T,V,{N_{i}}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}} Gibbs free energy d G ( T , p , N i ) = − S d T + V d p + ∑ i μ i d N i {\displaystyle dG\left(T,p,{N_{i}}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}}
[ edit ] The four most common Maxwell's relations are:
Physical situation Nomenclature Equations Thermodynamic potentials as functions of their natural variables U ( S , V ) {\displaystyle U(S,V)\,} Internal energy H ( S , P ) {\displaystyle H(S,P)\,} Enthalpy F ( T , V ) {\displaystyle F(T,V)\,} Helmholtz free energy G ( T , P ) {\displaystyle G(T,P)\,} Gibbs free energy ( ∂ T ∂ V ) S = − ( ∂ P ∂ S ) V = ∂ 2 U ∂ S ∂ V {\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}} ( ∂ T ∂ P ) S = + ( ∂ V ∂ S ) P = ∂ 2 H ∂ S ∂ P {\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}}
+ ( ∂ S ∂ V ) T = ( ∂ P ∂ T ) V = − ∂ 2 F ∂ T ∂ V {\displaystyle +\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}F}{\partial T\partial V}}}
− ( ∂ S ∂ P ) T = ( ∂ V ∂ T ) P = ∂ 2 G ∂ T ∂ P {\displaystyle -\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.
( ∂ S ∂ U ) V , N = 1 T {\displaystyle \left({\partial S \over \partial U}\right)_{V,N}={1 \over T}} ( ∂ S ∂ V ) N , U = p T {\displaystyle \left({\partial S \over \partial V}\right)_{N,U}={p \over T}} ( ∂ S ∂ N ) V , U = − μ T {\displaystyle \left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}} ( ∂ T ∂ S ) V = T C V {\displaystyle \left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}} ( ∂ T ∂ S ) P = T C P {\displaystyle \left({\partial T \over \partial S}\right)_{P}={T \over C_{P}}} − ( ∂ p ∂ V ) T = 1 V K T {\displaystyle -\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 ( ∂ ( G / T ) ∂ T ) p {\displaystyle H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}} U = − T 2 ( ∂ ( F / T ) ∂ T ) V {\displaystyle U=-T^{2}\left({\frac {\partial \left(F/T\right)}{\partial T}}\right)_{V}} G = − V 2 ( ∂ ( F / V ) ∂ V ) T {\displaystyle G=-V^{2}\left({\frac {\partial \left(F/V\right)}{\partial V}}\right)_{T}} ( ∂ H ∂ p ) T = V − T ( ∂ V ∂ T ) P {\displaystyle \left({\frac {\partial H}{\partial p}}\right)_{T}=V-T\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ U ∂ V ) T = T ( ∂ P ∂ T ) V − P {\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial P}{\partial T}}\right)_{V}-P}
U = N k B T 2 ( ∂ ln Z ∂ T ) V {\displaystyle U=Nk_{\text{B}}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}} S = U T + N k B ln Z − N k ln N + N k {\displaystyle S={\frac {U}{T}}+Nk_{\text{B}}\ln Z-Nk\ln N+Nk} 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:
Degree of freedom Partition function Translation Z t = ( 2 π m k B T ) 3 2 V h 3 {\displaystyle Z_{t}={\frac {(2\pi mk_{\text{B}}T)^{\frac {3}{2}}V}{h^{3}}}} Vibration Z v = 1 1 − e − h ω 2 π k B T {\displaystyle Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{\text{B}}T}}}}} Rotation Z r = 2 I k B T σ ( h 2 π ) 2 {\displaystyle Z_{r}={\frac {2Ik_{\text{B}}T}{\sigma ({\frac {h}{2\pi }})^{2}}}}
Thermal properties of matter [ edit ] Coefficients Equation Joule-Thomson coefficient μ J T = ( ∂ T ∂ p ) H {\displaystyle \mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}} Compressibility (constant temperature) K T = − 1 V ( ∂ V ∂ p ) T , N {\displaystyle K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}} Coefficient of thermal expansion (constant pressure) α p = 1 V ( ∂ V ∂ T ) p {\displaystyle \alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}} Heat capacity (constant pressure) C p = ( ∂ Q r e v ∂ T ) p = ( ∂ U ∂ T ) p + p ( ∂ V ∂ T ) p = ( ∂ H ∂ T ) p = T ( ∂ S ∂ T ) p {\displaystyle 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 = ( ∂ Q r e v ∂ T ) V = ( ∂ U ∂ T ) V = T ( ∂ S ∂ T ) V {\displaystyle 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 efficiencies [ edit ] Physical situation Nomenclature Equations Thermodynamic engines η = efficiencyW = work done by engineQ H = heat energy in higher temperature reservoirQ L = heat energy in lower temperature reservoirT H = temperature of higher temp. reservoirT L = temperature of lower temp. reservoir Thermodynamic engine: η = | W Q H | {\displaystyle \eta =\left|{\frac {W}{Q_{\text{H}}}}\right|}
Carnot engine efficiency: η c = 1 − | Q L Q H | = 1 − T L T H {\displaystyle \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 = | Q L W | {\displaystyle K=\left|{\frac {Q_{\text{L}}}{W}}\right|}
Carnot refrigeration performance K C = | Q L | | Q H | − | Q L | = T L T H − T L {\displaystyle K_{\text{C}}={\frac {|Q_{\text{L}}|}{|Q_{\text{H}}|-|Q_{\text{L}}|}}={\frac {T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}}}
^ Keenan, Thermodynamics , Wiley, New York, 1947 ^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 7 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. (1 March 1914). "A Complete Collection of Thermodynamic Formulas" . Physical Review . 3 (4). American Physical Society (APS): 273–281. doi :10.1103/physrev.3.273 . ISSN 0031-899X . 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.
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