# Physics Formulae/Thermodynamics Formulae

Lead Article: Tables of Physics Formulae

## Thermodynamics Laws

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

## Thermodynamic Quantities

Quantity (Common Name/s) (Common Symbol/s) Defining Equation SI Units Dimension
Number of Molecules ${\displaystyle N\,\!}$  dimensionless dimensionless
Temperature ${\displaystyle T\,\!}$  K [Θ]
Heat Energy ${\displaystyle Q\,\!}$  J [M][L]2[T]-2
Latent Heat ${\displaystyle Q_{L}\,\!}$  J [M][L]2[T]-2
Entropy ${\displaystyle S\,\!}$  J K-1 [M][L]2[T]-2 [Θ]-1
Heat Capacity (isobaric) ${\displaystyle C_{p}\,\!}$  ${\displaystyle C_{p}={\frac {\partial Q}{\partial T}}\,\!}$  J K -1 [M][L]2[T]-2 [Θ]-1
Specific Heat Capacity (isobaric) ${\displaystyle C_{mp}\,\!}$  ${\displaystyle C_{mp}={\frac {\partial ^{2}Q}{\partial m\partial T}}\,\!}$  J kg-1 K-1 [L]2[T]-2 [Θ]-1
Molar Specific Heat

Capacity (isobaric)

${\displaystyle C_{np}\,\!}$  ${\displaystyle C_{np}={\frac {\partial ^{2}Q}{\partial n\partial T}}\,\!}$  J K -1 mol-1 [M][L]2[T]-2 [Θ]-1 [N]-1
Heat Capacity (isochoric) ${\displaystyle C_{V}\,\!}$  ${\displaystyle C_{V}={\frac {\partial Q}{\partial T}}\,\!}$  J K -1 [M][L]2[T]-2 [Θ]-1
Specific Heat Capacity (isochoric) ${\displaystyle C_{mV}\,\!}$  ${\displaystyle C_{mV}={\frac {\partial ^{2}Q}{\partial m\partial T}}\,\!}$  J kg-1 K-1 [L]2[T]-2 [Θ]-1
Molar Specific Heat

Capacity (isochoric)

${\displaystyle C_{nV}\,\!}$  ${\displaystyle C_{nV}={\frac {\partial ^{2}Q}{\partial n\partial T}}\,\!}$  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

${\displaystyle U\,\!}$  ${\displaystyle U=\sum _{i}E_{i}\!}$  J [M][L]2[T]-2
Enthalpy ${\displaystyle H\,\!}$  ${\displaystyle H=U+pV\,\!}$  J [M][L]2[T]-2
Gibbs Free Energy ${\displaystyle \Delta G\,\!}$  ${\displaystyle \Delta G=\Delta H-T\Delta S\,\!}$  J [M][L]2[T]-2
Helmholtz Free Energy ${\displaystyle A,F\,\!}$  ${\displaystyle A=U-TS\,\!}$  J [M][L]2[T]-2
Specific Latent Heat ${\displaystyle L\,\!}$  ${\displaystyle L={\frac {Q}{m}}\,\!}$  J kg-1 [L]2[T]-2
Ratio of Isobaric to

Isochoric Heat Capacity,

${\displaystyle \gamma \,\!}$  ${\displaystyle \gamma ={\frac {C_{p}}{C_{V}}}={\frac {c_{p}}{c_{V}}}={\frac {C_{mp}}{C_{mV}}}\,\!}$  dimensionless dimensionless
Linear Coefficient of Thermal Expansion ${\displaystyle \alpha \,\!}$  ${\displaystyle {\frac {\partial L}{\partial t}}=\alpha L\,\!}$  K-1 [Θ]-1
Volume Coefficient of Thermal Expansion ${\displaystyle 3\alpha \,\!}$  ${\displaystyle {\frac {\partial V}{\partial t}}=3\alpha V\,\!}$  K-1 [Θ]-1
Temperature Gradient No standard symbol ${\displaystyle \nabla T\,\!}$  K m-1 [Θ][L]-1
Thermal Conduction Rate/

Thermal Current

${\displaystyle P\,\!}$  ${\displaystyle P={\frac {\partial Q}{\partial t}}\,\!}$  W = J s-1 [M] [L]2 [T]-2
Thermal Intensity ${\displaystyle I\,\!}$  ${\displaystyle I={\frac {\partial P}{\partial A}}={\frac {\partial ^{2}P}{\partial A\partial t}}\,\!}$  W m-2 [M] [L]-1 [T]-2
Thermal Conductivity ${\displaystyle \kappa ,K,\lambda \,\!}$  ${\displaystyle \lambda =-{\frac {P}{\mathbf {A} \cdot \nabla T}}\,\!}$  W m-1 K-1 [M] [L] [T]-2 [Θ]-1
Thermal Resistance ${\displaystyle R\,\!}$  ${\displaystyle R={\frac {\Delta x}{\lambda }}\,\!}$  m2 K W-1 [L] [T]2 [Θ]1 [M]-1
Emmisivity Coefficient ${\displaystyle \epsilon \,\!}$  Can only be found from experiment

${\displaystyle 0\leqslant \epsilon \leqslant 1\,\!}$

${\displaystyle \epsilon =0\,\!}$  for perfect reflector

${\displaystyle \epsilon =1\,\!}$  for perfect absorber

(true black body)

dimensionless dimensionless

## Kinetic Theory

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

## Thermal Transitions

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

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

## Thermal Transfer

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

## Thermal Efficiencies

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