Table of thermodynamic equations

This article is a summary of common equations and quantities in thermodynamics (see thermodynamic equations for more elaboration). SI units are used for absolute temperature, not Celsius or Fahrenheit. τ = k B ( ∂ U / ∂ S ) N {displaystyle au =k_{B}left(partial U/partial S ight)_{N},!} 1 / τ = 1 / k B ( ∂ S / ∂ U ) N {displaystyle 1/ au =1/k_{B}left(partial S/partial U ight)_{N},!} S = − ( ∂ F / ∂ T ) V {displaystyle S=-left(partial F/partial T ight)_{V},!} , S = − ( ∂ G / ∂ T ) N , P {displaystyle S=-left(partial G/partial T ight)_{N,P},!} P = − ( ∂ U / ∂ V ) S , N {displaystyle P=-left(partial U/partial V ight)_{S,N},!} component i in a mixture) μ i = ( ∂ F / ∂ N i ) T , V {displaystyle mu _{i}=left(partial F/partial N_{i} ight)_{T,V},!} , where F is not proportional to N because μi depends on pressure. μ i = ( ∂ G / ∂ N i ) T , P {displaystyle mu _{i}=left(partial G/partial N_{i} ight)_{T,P},!} , where G is proportional to N (as long as the molar ratio composition of the system remains the same) because μi depends only on temperature and pressure and composition. μ i / τ = − 1 / k B ( ∂ S / ∂ N i ) U , V {displaystyle mu _{i}/ au =-1/k_{B}left(partial S/partial N_{i} ight)_{U,V},!} 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 },!} For an ideal gas W = k T N ln ⁡ ( V 2 / V 1 ) {displaystyle W=kTNln(V_{2}/V_{1}),!} Δ W = p Δ V , Δ q = Δ U + p δ V {displaystyle Delta W=pDelta V,quad Delta q=Delta U+pdelta V,!} Δ W = 0 , Δ Q = Δ U {displaystyle Delta W=0,quad Delta Q=Delta U,!} Δ W = ∫ V 1 V 2 p d V {displaystyle Delta W=int _{V_{1}}^{V_{2}}pmathrm {d} V,!} p 1 V 1 p 2 V 2 = n 1 T 1 n 2 T 2 = N 1 T 1 N 2 T 2 {displaystyle {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}}},!} − n R T ln ⁡ P 1 P 2 {displaystyle -nRTln {frac {P_{1}}{P_{2}}};} C p = 7 2 n R {displaystyle C_{p}={frac {7}{2}}nR;} (for diatomic ideal gas) C V = 5 2 n R {displaystyle C_{V}={frac {5}{2}}nR;} (for diatomic ideal gas)K2 is the Modified Bessel function of the second kind. P ( v ) = 4 π ( m 2 π k B T ) 3 / 2 v 2 e − m v 2 / 2 k B T {displaystyle Pleft(v ight)=4pi left({frac {m}{2pi k_{B}T}} ight)^{3/2}v^{2}e^{-mv^{2}/2k_{B}T},!} where: P i = 1 / Ω {displaystyle P_{i}=1/Omega ,!} Δ S = k B N ln ⁡ V 2 V 1 + N C V ln ⁡ T 2 T 1 {displaystyle Delta S=k_{B}Nln {frac {V_{2}}{V_{1}}}+NC_{V}ln {frac {T_{2}}{T_{1}}},!} ⟨ E k ⟩ = 1 2 k T {displaystyle langle E_{mathrm {k} } angle ={frac {1}{2}}kT,!} ( ∂ T ∂ P ) S = + ( ∂ V ∂ S ) P = ∂ 2 H ∂ S ∂ P {displaystyle left({frac {partial T}{partial P}} ight)_{S}=+left({frac {partial V}{partial S}} ight)_{P}={frac {partial ^{2}H}{partial Spartial P}}} SinceSince λ n e t = ∑ j λ j {displaystyle lambda _{mathrm {net} }=sum _{j}lambda _{j},!} η = | W Q H | {displaystyle eta =left|{frac {W}{Q_{H}}} ight|,!} K = | Q L W | {displaystyle K=left|{frac {Q_{L}}{W}} ight|,!} This article is a summary of common equations and quantities in thermodynamics (see thermodynamic equations for more elaboration). SI units are used for absolute temperature, not Celsius or Fahrenheit. Many of the definitions below are also used in the thermodynamics of chemical reactions.