# Physics Formulae/Thermodynamics Formulae

Lead Article: Tables of Physics Formulae

## 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,

$\gamma \,\!$  $\gamma ={\frac {C_{p}}{C_{V}}}={\frac {c_{p}}{c_{V}}}={\frac {C_{mp}}{C_{mV}}}\,\!$  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 $\Delta W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V\,\!$ 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 $K_{C}=|Q_{L}|/(|Q_{H}|-|Q_{L}|)=T_{L}/(T_{H}-T_{L})\,\!$ 