# 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 | △Q = △U +△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 | $R\,\!$ | $R={\frac {\Delta x}{\lambda }}\,\!$ | m$^2$ 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 $\Delta W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V\,\!$ 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 | $K_{C}=|Q_{L}|/(|Q_{H}|-|Q_{L}|)=T_{L}/(T_{H}-T_{L})\,\!$ |