Physics Formulae/Thermodynamics Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Thermodynamics.

Thermodynamics Laws

Zeroth Law of Thermodynamics	$(T_{A}=T_{B})\land (T_{B}=T_{C})\Rightarrow T_{A}=T_{C}\,\!$ (systems in thermal equilibrium)
First Law of Thermodynamics	$\Delta U=\Delta Q+\Delta W\,\!$ Internal energy increase $\Delta U>0\,\!$ , decrease $\Delta U<0\,\!$ Heat energy transferred to system $\Delta Q>0\,\!$ , from system $\Delta Q<0\,\!$ Work done transferred to system $\Delta W>0\,\!$ by system $\Delta W<0\,\!$
Second Law of Thermodynamics	$\Delta S\geq 0\,\!$
Third Law of Thermodynamics	$S=S_{\mathrm {structural} }+CT\,\!$

Thermodynamic Quantities

Quantity (Common Name/s)	(Common Symbol/s)	Defining Equation	SI Units	Dimension
Number of Molecules	$N\,\!$		dimensionless	dimensionless
Temperature	$T\,\!$		K	[Θ]
Heat Energy	$Q\,\!$		J	[M][L]²[T]^-2
Latent Heat	$Q_{L}\,\!$		J	[M][L]²[T]^-2
Entropy	$S\,\!$		J K^-1	[M][L]²[T]^-2 [Θ]^-1
Heat Capacity (isobaric)	$C_{p}\,\!$	$C_{p}={\frac {\partial Q}{\partial T}}\,\!$	J K ^-1	[M][L]²[T]^-2 [Θ]^-1
Specific Heat Capacity (isobaric)	$C_{mp}\,\!$	$C_{mp}={\frac {\partial ^{2}Q}{\partial m\partial T}}\,\!$	J kg^-1 K^-1	[L]²[T]^-2 [Θ]^-1
Molar Specific Heat Capacity (isobaric)	$C_{np}\,\!$	$C_{np}={\frac {\partial ^{2}Q}{\partial n\partial T}}\,\!$	J K ^-1 mol^-1	[M][L]²[T]^-2 [Θ]^-1 [N]^-1
Heat Capacity (isochoric)	$C_{V}\,\!$	$C_{V}={\frac {\partial Q}{\partial T}}\,\!$	J K ^-1	[M][L]²[T]^-2 [Θ]^-1
Specific Heat Capacity (isochoric)	$C_{mV}\,\!$	$C_{mV}={\frac {\partial ^{2}Q}{\partial m\partial T}}\,\!$	J kg^-1 K^-1	[L]²[T]^-2 [Θ]^-1
Molar Specific Heat Capacity (isochoric)	$C_{nV}\,\!$	$C_{nV}={\frac {\partial ^{2}Q}{\partial n\partial T}}\,\!$	J K ^-1 mol^-1	[M][L]²[T]^-2 [Θ]^-1 [N]^-1
Internal Energy Sum of all total energies which constitute the system	$U\,\!$	$U=\sum _{i}E_{i}\!$	J	[M][L]²[T]^-2
Enthalpy	$H\,\!$	$H=U+pV\,\!$	J	[M][L]²[T]^-2
Gibbs Free Energy	$\Delta G\,\!$	$\Delta G=\Delta H-T\Delta S\,\!$	J	[M][L]²[T]^-2
Helmholtz Free Energy	$A,F\,\!$	$A=U-TS\,\!$	J	[M][L]²[T]^-2
Specific Latent Heat	$L\,\!$	$L={\frac {Q}{m}}\,\!$	J kg^-1	[L]²[T]^-2
Ratio of Isobaric to Isochoric Heat Capacity, Adiabatic Index	$\gamma \,\!$	$\gamma ={\frac {C_{p}}{C_{V}}}={\frac {c_{p}}{c_{V}}}={\frac {C_{mp}}{C_{mV}}}\,\!$	dimensionless	dimensionless
Linear Coefficient of Thermal Expansion	$\alpha \,\!$	${\frac {\partial L}{\partial t}}=\alpha L\,\!$	K^-1	[Θ]^-1
Volume Coefficient of Thermal Expansion	$3\alpha \,\!$	${\frac {\partial V}{\partial t}}=3\alpha V\,\!$	K^-1	[Θ]^-1
Temperature Gradient	No standard symbol	$\nabla T\,\!$	K m^-1	[Θ][L]^-1
Thermal Conduction Rate/ Thermal Current	$P\,\!$	$P={\frac {\partial Q}{\partial t}}\,\!$	W = J s^-1	[M] [L]² [T]^-2
Thermal Intensity	$I\,\!$	$I={\frac {\partial P}{\partial A}}={\frac {\partial ^{2}P}{\partial A\partial t}}\,\!$	W m^-2	[M] [L]^-1 [T]^-2
Thermal Conductivity	$\kappa ,K,\lambda \,\!$	$\lambda =-{\frac {P}{\mathbf {A} \cdot \nabla T}}\,\!$	W m^-1 K^-1	[M] [L] [T]^-2 [Θ]^-1
Thermal Resistance	$R\,\!$	$R={\frac {\Delta x}{\lambda }}\,\!$	m² K W^-1	[L] [T]² [Θ]¹ [M]^-1
Emmisivity Coefficient	$\epsilon \,\!$	Can only be found from experiment $0\leqslant \epsilon \leqslant 1\,\!$ $\epsilon =0\,\!$ for perfect reflector $\epsilon =1\,\!$ for perfect absorber (true black body)	dimensionless	dimensionless

Kinetic Theory

Ideal Gas Law	$pV=nRT\,\!$ $pV=kTN\,\!$ ${\frac {p_{1}V_{1}}{n_{1}T_{1}}}={\frac {p_{2}V_{2}}{n_{2}T_{2}}}\,\!$ ${\frac {p_{1}V_{1}}{N_{1}T_{1}}}={\frac {p_{2}V_{2}}{N_{2}T_{2}}}\,\!$
Translational Energy	$\langle E_{\mathrm {k} }\rangle ={\frac {f}{2}}kT\,\!$
Internal Energy	$U={\frac {f}{2}}NkT\,\!$

Thermal Transitions

Adiabatic	$\Delta Q=0\,\!$ $\Delta U=W\,\!$
Work by an Expanding Gas	Process $\Delta W=\int _{V_{1}}^{V_{1}}p\mathrm {d} V\,\!$ Net Work Done in Cyclic Processes $\Delta W=\oint _{\mathrm {cycle} }p\mathrm {d} V\,\!$
Isobaric Transition	$\Delta U=Q\,\!$
Cyclic Process	$Q+W=0\,\!$
Work, Isochoric	$W=0\,\!$
work, Isobaric	$W=p\Delta V\,\!$
Work, Isothermal	$W=kTN\ln(V_{2}/V_{1})\,\!$
Adiabatic Expansion	$p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }\,\!$ $T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}\,\!$
Free Expansion	$\Delta U=0\,\!$

Statistical Physics

Below are useful results from the Maxell-Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity.

Degrees of Freedom	$f\,\!$
Maxwell-Boltzmann Distribution, Mean Speed	$\langle v\rangle ={\sqrt {\frac {8kT}{\pi m}}}\,\!$
Maxwell-Boltzmann Distribution Mode-Speed	$v_{\mathrm {mode} }={\sqrt {\frac {kT}{2m}}}\,\!$
Root Mean Square Speed	$v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {kT}{3m}}}\,\!$
Mean Free Path	$\langle x_{\mathrm {free} }\rangle ={\frac {\langle v\rangle }{{\sqrt {2}}\pi d^{2}N}}\,\!$ ?
Maxwell–Boltzmann Distribution	$P(v)=4\pi \left({\frac {m}{2\pi kT}}\right)^{3/2}v^{2}e^{-mv^{2}/2kT}\,\!$
Multiplicity of Configurations	$W={\frac {N!}{n_{1}!n_{2}!}}\,\!$
Microstate in one half of the box	$n_{1},n_{2}\,\!$
Boltzmann's Entropy Equation	$S=k\ln W\,\!$
Irreversibility	$\,\!$
Entropy	$S=-k\sum _{i}P_{i}\ln P_{i}\!\,\!$
Entropy Change	$\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}\,\!$ $\Delta S=kN\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}\,\!$
Entropic Force	$F_{S}=-T\nabla S\,\!$

Thermal Transfer

Stefan-Boltzmann Law	$I=\sigma \epsilon T^{4}\,\!$
Net Intensity Emmision/Absorbtion	$I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)\,\!$
Internal Energy of a Substance	$\Delta U=NC_{V}\Delta T\,\!$
Work done by an Expanding Ideal Gas	$\mathrm {d} W=p\mathrm {d} V=Nk\mathrm {d} T\,\!$
Meyer's Equation	$C_{p}-C_{V}=nR\,\!$

Thermal Efficiencies

Engine Efficiency	$\epsilon =\|W\|/\|Q_{H}\|\,\!$
Carnot Engine Efficiency	$\epsilon _{c}=(\|Q_{H}\|-\|Q_{L}\|)/\|Q_{H}\|=(T_{H}-T_{L})/T_{H}\,\!$
Refrigeration Performance	$K=\|Q_{L}\|/\|W\|\,\!$
Carnot Refrigeration Performance	$K_{C}=\|Q_{L}\|/(\|Q_{H}\|-\|Q_{L}\|)=T_{L}/(T_{H}-T_{L})\,\!$

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