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Table of thermodynamic equations

From Wikipedia, the free encyclopedia

Thermodynamics
The classicalCarnot heat engine

Commonthermodynamic equations andquantities inthermodynamics, usingmathematical notation, are as follows:

Definitions

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Main articles:List of thermodynamic properties,Thermodynamic potential,Free entropy, andDefining equation (physical chemistry)

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

General basic quantities

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Quantity (common name/s)(Common) symbol/sSI unitDimension
Number of moleculesN11
Amount of substancenmolN
TemperatureTKΘ
Heat EnergyQ,qJML2T−2
Latent heatQLJML2T−2

General derived quantities

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Quantity (common name/s)(Common) symbol/sDefining equationSI unitDimension
Thermodynamic beta, inverse temperatureββ=1/kBT{\displaystyle \beta =1/k_{\text{B}}T}J−1T2M−1L−2
Thermodynamic temperatureττ=kBT{\displaystyle \tau =k_{\text{B}}T}

τ=kB(∂U/∂S)N{\displaystyle \tau =k_{\text{B}}\left(\partial U/\partial S\right)_{N}}1/τ=1/kB(∂S/∂U)N{\displaystyle 1/\tau =1/k_{\text{B}}\left(\partial S/\partial U\right)_{N}}

JML2T−2
EntropySS=−kB∑ipiln⁡pi{\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−1ML2T−2Θ−1
PressurePP=−(∂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}}

PaML−1T−2
Internal EnergyUU=∑ipiEi{\displaystyle U=\sum _{i}p_{i}E_{i}}JML2T−2
EnthalpyHH=U+pV{\displaystyle H=U+pV}JML2T−2
Partition FunctionZ11
Gibbs free energyGG=H−TS{\displaystyle G=H-TS}JML2T−2
Chemical potential (of componenti in a mixture)μiμi=(∂U/∂Ni)Nj≠i,S,V{\displaystyle \mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}}

μi=(∂F/∂Ni)T,V{\displaystyle \mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}}, whereF{\displaystyle F} is not proportional toN{\displaystyle N} becauseμi{\displaystyle \mu _{i}} depends on pressure.μi=(∂G/∂Ni)T,P{\displaystyle \mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}}, whereG{\displaystyle G} is proportional toN{\displaystyle N} (as long as the molar ratio composition of the system remains the same) becauseμi{\displaystyle \mu _{i}} depends only on temperature and pressure and composition.μi/τ=−1/kB(∂S/∂Ni)U,V{\displaystyle \mu _{i}/\tau =-1/k_{\text{B}}\left(\partial S/\partial N_{i}\right)_{U,V}}

JML2T−2
Helmholtz free energyA,FF=U−TS{\displaystyle F=U-TS}JML2T−2
Landau potential, Landau free energy,Grand potentialΩ,ΦGΩ=U−TS−μN {\displaystyle \Omega =U-TS-\mu N\ }JML2T−2
Massieu potential, Helmholtzfree entropyΦΦ=S−U/T{\displaystyle \Phi =S-U/T}J⋅K−1ML2T−2Θ−1
Planck potential, Gibbsfree entropyΞΞ=Φ−pV/T{\displaystyle \Xi =\Phi -pV/T}J⋅K−1ML2T−2Θ−1

Thermal properties of matter

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Main articles:Heat capacity andThermal expansion
Quantity (common name/s)(Common) symbol/sDefining equationSI unitDimension
General heat/thermal capacityCC=∂Q/∂T{\displaystyle C=\partial Q/\partial T}J⋅K−1ML2T−2Θ−1
Heat capacity (isobaric)CpCp=∂H/∂T{\displaystyle C_{p}=\partial H/\partial T}J⋅K−1ML2T−2Θ−1
Specific heat capacity (isobaric)CmpCmp=∂2Q/∂m∂T{\displaystyle C_{mp}=\partial ^{2}Q/\partial m\partial T}J⋅kg−1⋅K−1L2T−2Θ−1
Molar specific heat capacity (isobaric)CnpCnp=∂2Q/∂n∂T{\displaystyle C_{np}=\partial ^{2}Q/\partial n\partial T}J⋅K−1⋅mol−1ML2T−2Θ−1N−1
Heat capacity (isochoric/volumetric)CVCV=∂U/∂T{\displaystyle C_{V}=\partial U/\partial T}J⋅K−1ML2T−2Θ−1
Specific heat capacity (isochoric)CmVCmV=∂2Q/∂m∂T{\displaystyle C_{mV}=\partial ^{2}Q/\partial m\partial T}J⋅kg−1⋅K−1L2T−2Θ−1
Molar specific heat capacity (isochoric)CnVCnV=∂2Q/∂n∂T{\displaystyle C_{nV}=\partial ^{2}Q/\partial n\partial T}J⋅K⋅−1 mol−1ML2T−2Θ−1N−1
Specific latent heatLL=∂Q/∂m{\displaystyle L=\partial Q/\partial m}J⋅kg−1L2T−2
Ratio of isobaric to isochoric heat capacity,heat capacity ratio, adiabatic index,Laplace coefficientγγ=Cp/CV=cp/cV=Cmp/CmV{\displaystyle \gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}}11

Thermal transfer

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Main article:Thermal conductivity
Quantity (common name/s)(Common) symbol/sDefining equationSI unitDimension
Temperature gradientNo standard symbol∇T{\displaystyle \nabla T}K⋅m−1ΘL−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transferPP=dQ/dt{\displaystyle P=\mathrm {d} Q/\mathrm {d} t}WML2T−3
Thermal intensityII=dP/dA{\displaystyle I=\mathrm {d} P/\mathrm {d} A}W⋅m−2MT−3
Thermal/heat flux density (vector analogue of thermal intensity above)qQ=∬q⋅dSdt{\displaystyle Q=\iint \mathbf {q} \cdot \mathrm {d} \mathbf {S} \mathrm {d} t}W⋅m−2MT−3

Equations

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The equations in this article are classified by subject.

Thermodynamic processes

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Physical situationEquations
Isentropic process (adiabatic and reversible)Q=0,ΔU=−W{\displaystyle Q=0,\quad \Delta U=-W}

For an ideal gas
p1V1γ=p2V2γ{\displaystyle p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }}
T1V1γ−1=T2V2γ−1{\displaystyle T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}}
p11−γT1γ=p21−γT2γ{\displaystyle p_{1}^{1-\gamma }T_{1}^{\gamma }=p_{2}^{1-\gamma }T_{2}^{\gamma }}

Isothermal processΔU=0,W=Q{\displaystyle \Delta U=0,\quad W=Q}

For an ideal gas
W=kTNln⁡(V2/V1){\displaystyle W=kTN\ln(V_{2}/V_{1})}W=nRTln⁡(V2/V1){\displaystyle W=nRT\ln(V_{2}/V_{1})}

Isobaric processp1 =p2,p = constant

W=pΔV,Q=ΔU+pδV{\displaystyle W=p\Delta V,\quad Q=\Delta U+p\delta V}

Isochoric processV1 =V2,V = constant

W=0,Q=ΔU{\displaystyle W=0,\quad Q=\Delta U}

Free expansionΔU=0{\displaystyle \Delta U=0}
Work done by an expanding gasProcess

W=∫V1V2pdV{\displaystyle W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V}

Net work done in cyclic processes
W=∮cyclepdV=∮cycleΔQ{\displaystyle W=\oint _{\mathrm {cycle} }p\mathrm {d} V=\oint _{\mathrm {cycle} }\Delta Q}

Kinetic theory

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Ideal gas equations
Physical situationNomenclatureEquations
Ideal gas law
pV=nRT=kTN{\displaystyle pV=nRT=kTN}

p1V1p2V2=n1T1n2T2=N1T1N2T2{\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}}}}

Pressure of an ideal gas
  • m = mass ofone molecule
  • Mm = molar mass
p=Nm⟨v2⟩3V=nMm⟨v2⟩3V=13ρ⟨v2⟩{\displaystyle 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

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QuantityGeneral EquationIsobaric
Δp = 0		Isochoric
ΔV = 0		Isothermal
ΔT = 0		Adiabatic
Q=0{\displaystyle Q=0}
Work
W		δW=−pdV{\displaystyle \delta W=-pdV\;}−pΔV{\displaystyle -p\Delta V\;}0{\displaystyle 0\;}−nRTln⁡V2V1{\displaystyle -nRT\ln {\frac {V_{2}}{V_{1}}}\;}

−nRTln⁡P1P2{\displaystyle -nRT\ln {\frac {P_{1}}{P_{2}}}\;}

PVγ(Vf1−γ−Vi1−γ)1−γ=CV(T2−T1){\displaystyle {\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}=C_{V}\left(T_{2}-T_{1}\right)}
Heat Capacity
C		(as for real gas)Cp=52nR{\displaystyle C_{p}={\frac {5}{2}}nR}
(for monatomic ideal gas)

Cp=72nR{\displaystyle C_{p}={\frac {7}{2}}nR}
(for diatomic ideal gas)

CV=32nR{\displaystyle C_{V}={\frac {3}{2}}nR}
(for monatomic ideal gas)

CV=52nR{\displaystyle C_{V}={\frac {5}{2}}nR\;}
(for diatomic ideal gas)

Internal Energy
ΔU		ΔU=CVΔT{\displaystyle \Delta U=C_{V}\Delta T\;}Q+W{\displaystyle Q+W\;}

Qp−pΔV{\displaystyle Q_{p}-p\Delta V}		Q{\displaystyle Q\;}

CV(T2−T1){\displaystyle C_{V}\left(T_{2}-T_{1}\right)}		0{\displaystyle 0\;}
Q=−W{\displaystyle Q=-W}		W{\displaystyle W\;}
CV(T2−T1){\displaystyle C_{V}\left(T_{2}-T_{1}\right)}
Enthalpy
ΔH		H=U+pV{\displaystyle H=U+pV\;}Cp(T2−T1){\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}QV+VΔp{\displaystyle Q_{V}+V\Delta p\;}0{\displaystyle 0\;}Cp(T2−T1){\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}
Entropy
Δs		ΔS=CVln⁡T2T1+nRln⁡V2V1{\displaystyle \Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}}
ΔS=Cpln⁡T2T1−nRln⁡p2p1{\displaystyle \Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}}[1]		Cpln⁡T2T1{\displaystyle C_{p}\ln {\frac {T_{2}}{T_{1}}}\;}CVln⁡T2T1{\displaystyle C_{V}\ln {\frac {T_{2}}{T_{1}}}\;}nRln⁡V2V1{\displaystyle nR\ln {\frac {V_{2}}{V_{1}}}\;}
QT{\displaystyle {\frac {Q}{T}}\;}		Cpln⁡V2V1+CVln⁡p2p1=0{\displaystyle C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;}
Constant{\displaystyle \;}VT{\displaystyle {\frac {V}{T}}\;}pT{\displaystyle {\frac {p}{T}}\;}pV{\displaystyle pV\;}pVγ{\displaystyle pV^{\gamma }\;}

Entropy

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Statistical physics

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Below are useful results from theMaxwell–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 situationNomenclatureEquations
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=kBT/mc2{\displaystyle \theta =k_{\text{B}}T/mc^{2}}

K2 is the modifiedBessel function of the second kind.

Non-relativistic speeds

P(v)=4π(m2πkBT)3/2v2e−mv2/2kBT{\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)=14πm3c3θK2(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 }}

EntropyLogarithm of thedensity of states
  • Pi = probability of system in microstatei
  • Ω = total number of microstates
S=−kB∑iPiln⁡Pi=kBln⁡Ω{\displaystyle S=-k_{\text{B}}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega }

where:
Pi=1/Ω{\displaystyle P_{i}=1/\Omega }

Entropy changeΔS=∫Q1Q2dQT{\displaystyle \Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}}

ΔS=kBNln⁡V2V1+NCVln⁡T2T1{\displaystyle \Delta S=k_{\text{B}}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}}

Entropic forceFS=−T∇S{\displaystyle \mathbf {F} _{\mathrm {S} }=-T\nabla S}
Equipartition theoremdf = degree of freedomAverage kinetic energy per degree of freedom

⟨Ek⟩=12kT{\displaystyle \langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT}

Internal energyU=df⟨Ek⟩=df2kT{\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 situationNomenclatureEquations
Mean speed⟨v⟩=8kBTπm{\displaystyle \langle v\rangle ={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}}
Root mean square speedvrms=⟨v2⟩=3kBTm{\displaystyle v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}}
Modal speedvmode=2kBTm{\displaystyle v_{\mathrm {mode} }={\sqrt {\frac {2k_{\text{B}}T}{m}}}}
Mean free path
  • σ = effective cross-section
  • n = volume density of number of target particles
  • ℓ = mean free path
ℓ=1/2nσ{\displaystyle \ell =1/{\sqrt {2}}n\sigma }

Quasi-static and reversible processes

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Forquasi-static andreversible processes, thefirst law of thermodynamics is:

dU=δQ−δW{\displaystyle dU=\delta Q-\delta W}

whereδQ is the heat suppliedto the system andδW is the work doneby the system.

Thermodynamic potentials

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Main article:Thermodynamic potentials
See also:Maxwell relations

The following energies are called thethermodynamic potentials,

NameSymbolFormulaNatural variables
Internal energyU{\displaystyle U}∫(TdS−pdV+∑iμidNi){\displaystyle \int \left(T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\right)}S,V,{Ni}{\displaystyle S,V,\{N_{i}\}}
Helmholtz free energyA{\displaystyle A}U−TS{\displaystyle U-TS}T,V,{Ni}{\displaystyle T,V,\{N_{i}\}}
EnthalpyH{\displaystyle H}U+pV{\displaystyle U+pV}S,p,{Ni}{\displaystyle S,p,\{N_{i}\}}
Gibbs free energyG{\displaystyle G}U+pV−TS{\displaystyle U+pV-TS}T,p,{Ni}{\displaystyle T,p,\{N_{i}\}}
Landau potential, or
grand potential		Ω{\displaystyle \Omega },ΦG{\displaystyle \Phi _{\text{G}}}U−TS−{\displaystyle U-TS-}∑i{\displaystyle \sum _{i}\,}μiNi{\displaystyle \mu _{i}N_{i}}T,V,{μi}{\displaystyle T,V,\{\mu _{i}\}}

and the correspondingfundamental thermodynamic relations or "master equations"[2] are:

PotentialDifferential
Internal energydU(S,V,Ni)=TdS−pdV+∑iμidNi{\displaystyle dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}}
EnthalpydH(S,p,Ni)=TdS+Vdp+∑iμidNi{\displaystyle dH\left(S,p,{N_{i}}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}}
Helmholtz free energydF(T,V,Ni)=−SdT−pdV+∑iμidNi{\displaystyle dF\left(T,V,{N_{i}}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}}
Gibbs free energydG(T,p,Ni)=−SdT+Vdp+∑iμidNi{\displaystyle dG\left(T,p,{N_{i}}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}}

Maxwell's relations

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The four most commonMaxwell's relations are:

Physical situationNomenclatureEquations
Thermodynamic potentials as functions of their natural variables(∂T∂V)S=−(∂p∂S)V=∂2U∂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=∂2H∂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=−∂2F∂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=∂2G∂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=1T{\displaystyle \left({\partial S \over \partial U}\right)_{V,N}={1 \over T}}(∂S∂V)N,U=pT{\displaystyle \left({\partial S \over \partial V}\right)_{N,U}={p \over T}}(∂S∂N)V,U=−μT{\displaystyle \left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}}
(∂T∂S)V=TCV{\displaystyle \left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}}(∂T∂S)p=TCp{\displaystyle \left({\partial T \over \partial S}\right)_{p}={T \over C_{p}}}
−(∂p∂V)T=1VKT{\displaystyle -\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}}

Other differential equations are:

NameHUG
Gibbs–Helmholtz equationH=−T2(∂(G/T)∂T)p{\displaystyle H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}}U=−T2(∂(F/T)∂T)V{\displaystyle U=-T^{2}\left({\frac {\partial \left(F/T\right)}{\partial T}}\right)_{V}}G=−V2(∂(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}

Quantum properties

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whereN is number of particles,h is thatPlanck constant,I ismoment of inertia, andZ is thepartition function, in various forms:

Degree of freedomPartition function
TranslationZt=(2πmkBT)32Vh3{\displaystyle Z_{t}={\frac {(2\pi mk_{\text{B}}T)^{\frac {3}{2}}V}{h^{3}}}}
VibrationZv=11−e−hω2πkBT{\displaystyle Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{\text{B}}T}}}}}
RotationZr=2IkBTσ(h2π)2{\displaystyle Z_{r}={\frac {2Ik_{\text{B}}T}{\sigma ({\frac {h}{2\pi }})^{2}}}}

Thermal properties of matter

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CoefficientsEquation
Joule-Thomson coefficientμJT=(∂T∂p)H{\displaystyle \mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}}
Compressibility (constant temperature)KT=−1V(∂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)αp=1V(∂V∂T)p{\displaystyle \alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}}
Heat capacity (constant pressure)Cp=(∂Qrev∂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)CV=(∂Qrev∂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}}
Derivation of heat capacity (constant pressure)

Since

(∂T∂p)H(∂p∂H)T(∂H∂T)p=−1{\displaystyle \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}
(∂T∂p)H=−(∂H∂p)T(∂T∂H)p=−1(∂H∂T)p(∂H∂p)T{\displaystyle {\begin{aligned}\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{aligned}}}
Cp=(∂H∂T)p{\displaystyle C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}}
⇒(∂T∂p)H=−1Cp(∂H∂p)T{\displaystyle \Rightarrow \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}
Derivation of heat capacity (constant volume)

Since

dU=δQrev−δWrev,{\displaystyle dU=\delta Q_{rev}-\delta W_{rev},}

(whereδWrev is the work done by the system),

δS=δQrevT,δWrev=pδV{\displaystyle \delta S={\frac {\delta Q_{rev}}{T}},\delta W_{rev}=p\delta V}
dU=TδS−pδV{\displaystyle dU=T\delta S-p\delta V}
(∂U∂T)V=T(∂S∂T)V−p(∂V∂T)V;CV=(∂U∂T)V{\displaystyle \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}}
⇒CV=T(∂S∂T)V{\displaystyle \Rightarrow C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}

Thermal transfer

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Physical situationNomenclatureEquations
Net intensity emission/absorption
  • Texternal = external temperature (outside of system)
  • Tsystem = internal temperature (inside system)
  • ε = emissivity
I=σϵ(Texternal4−Tsystem4){\displaystyle I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)}
Internal energy of a substance
  • CV = isovolumetric heat capacity of substance
  • ΔT = temperature change of substance
ΔU=NCVΔT{\displaystyle \Delta U=NC_{V}\Delta T}
Meyer's equation
  • Cp = isobaric heat capacity
  • CV = isovolumetric heat capacity
  • n = amount of substance
Cp−CV=nR{\displaystyle C_{p}-C_{V}=nR}
Effective thermal conductivities
  • λi = thermal conductivity of substancei
  • λnet = equivalent thermal conductivity
Series

λnet=∑jλj{\displaystyle \lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}}

Parallel1λnet=∑j(1λj){\displaystyle {\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)}

Thermal efficiencies

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Physical situationNomenclatureEquations
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=|WQH|{\displaystyle \eta =\left|{\frac {W}{Q_{\text{H}}}}\right|}

Carnot engine efficiency:
ηc=1−|QLQH|=1−TLTH{\displaystyle \eta _{\text{c}}=1-\left|{\frac {Q_{\text{L}}}{Q_{\text{H}}}}\right|=1-{\frac {T_{\text{L}}}{T_{\text{H}}}}}

RefrigerationK = coefficient of refrigeration performanceRefrigeration performance

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

Carnot refrigeration performanceKC=|QL||QH|−|QL|=TLTH−TL{\displaystyle K_{\text{C}}={\frac {|Q_{\text{L}}|}{|Q_{\text{H}}|-|Q_{\text{L}}|}}={\frac {T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}}}

See also

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References

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  1. ^Keenan,Thermodynamics, Wiley, New York, 1947
  2. ^Physical chemistry, P.W. Atkins, Oxford University Press, 1978,ISBN 0 19 855148 7
  • Atkins, Peter and de Paula, JulioPhysical Chemistry, 7th edition, W.H. Freeman and Company, 2002ISBN 0-7167-3539-3.
    • Chapters 1–10, Part 1: "Equilibrium".
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