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	$(T_A=T_B)\wedge (T_B=T_C)\Rightarrow T_A=T_C$ (systems in thermal equilibrium)
First Law of Thermodynamics	△Q = △U +△W Internal energy increase $\mathrm{\Delta }U>0$, decrease $\mathrm{\Delta }U<0$ Heat energy transferred to system $\mathrm{\Delta }Q>0$, from system $\mathrm{\Delta }Q<0$ Work done transferred to system $\mathrm{\Delta }W>0$ by system $\mathrm{\Delta }W<0$
Second Law of Thermodynamics	$\mathrm{\Delta }S\ge 0$
Third Law of Thermodynamics	$S=S_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}}+CT$

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	$\mathrm{\Delta }G$	$\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$	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	$\mathrm{\nabla }T$	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	$\kappa ,K,\lambda$	$\lambda =-\frac{P}{𝐀\cdot \mathrm{\nabla }T}$	W m-1 K-1	[M] [L] [T]-2 [Θ]-1
Thermal Resistance	$R$	$R=\frac{\mathrm{\Delta }x}{\lambda }$	m2 K W-1	[L] [T]2 [Θ]1 [M]-1
Emmisivity Coefficient	$ϵ$	Can only be found from experiment $0⩽ϵ⩽1$ $ϵ=0$ for perfect reflector $ϵ=1$ 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	$⟨E_{\mathrm{k}}⟩=\frac{f}{2}kT$
Internal Energy	$U=\frac{f}{2}NkT$

Thermal Transitions

Adiabatic	$\mathrm{\Delta }Q=0$ $\mathrm{\Delta }U=W$
Work by an Expanding Gas	Process $\mathrm{\Delta }W=\underset{V_1}{\overset{V_2}{\int }}p\mathrm{d}V$ Net Work Done in Cyclic Processes $\mathrm{\Delta }W={\oint }_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}p\mathrm{d}V$
Isobaric Transition	$\mathrm{\Delta }U=Q$
Cyclic Process	$Q+W=0$
Work, Isochoric	$W=0$
work, Isobaric	$W=p\mathrm{\Delta }V$
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	$\mathrm{\Delta }U=0$

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	$⟨v⟩=\sqrt{\frac{8kT}{\pi m}}$
Maxwell-Boltzmann Distribution Mode-Speed	$v_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}}=\sqrt{\frac{kT}{2m}}$
Root Mean Square Speed	$v_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{⟨v^2⟩}=\sqrt{\frac{kT}{3m}}$
Mean Free Path	$⟨x_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}⟩=\frac{⟨v⟩}{\sqrt{2}\pi d^{2}N}$?
Maxwell–Boltzmann Distribution	$P(v)=4\pi {\left(\frac{m}{2\pi kT}\right)}^{3/2}{v}^2{e}^{-m{v}^2/2kT}$
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	$\mathrm{\Delta }S=\underset{Q_1}{\overset{Q_2}{\int }}\frac{\mathrm{d}Q}{T}$ $\mathrm{\Delta }S=kN\ln \frac{V_2}{V_1}+N{C}_V\ln \frac{T_2}{T_1}$
Entropic Force	$F_S=-T\mathrm{\nabla }S$

Thermal Transfer

Stefan-Boltzmann Law	$I=\sigma ϵT^4$
Net Intensity Emmision/Absorbtion	$I=\sigma ϵ(T_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}}^4-T_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}^4)$
Internal Energy of a Substance	$\mathrm{\Delta }U=N{C}_V\mathrm{\Delta }T$
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	$ϵ=|W|/|Q_H|$
Carnot Engine Efficiency	${ϵ}_c=(|Q_H|-|Q_L|)/|Q_H|=(T_H-T_L)/T_H$
Refrigeration Performance	$K=|Q_L|/|W|$
Carnot Refrigeration Performance	$K_C=|Q_L|/(|Q_H|-|Q_L|)=T_L/(T_H-T_L)$