Thermal Physics Formulas
Remember that: \( \vec{a} = \mathbf{a} \) and \(\Delta a = a_{final} - a_{initial} = a_{f} - a_{i} = a_{2} - a_{1} \). The \(\propto\) symbol is read as "is proportional to".
Symbols
| name | symbol |
|---|---|
| area | $$A$$ |
| attraction between particles | $$a$$ |
| volume available for one mole of particles | $$b$$ |
| heat capacity | $$C$$ |
| specific heat capacity | $$C_{e},\ c$$ |
| Helmholtz free energy | $$F$$ |
| Gibbs free energy | $$G$$ |
| enthalpy | $$H$$ |
| reaction enthalpy | $$\Delta H_{r}$$ |
| reduced Planck constant | $$\hbar = 1.054572\ J \cdot s$$ |
| moment of inertia | $$I$$ |
| total angular momentum operator | $$J$$ |
| angular momentum quantum number | $$\hat{J}$$ |
| equilibrium constant | $$K$$ |
| Boltzmann constant | $$k = k_{B} = 1.380649 \times 10^{-23} \frac{J}{K}$$ |
| length | $$L$$ |
| mass | $$m$$ |
| amount of substance, mode of the system or degree of freedom | $$N,\ n$$ |
| quantum concentration | $$n_{Q}$$ |
| pressure | $$P,\ p$$ |
| heat | $$Q$$ |
| molar gas constant | $$R = 8.314462 \frac{J}{K \cdot mol}$$ |
| entropy | $$S$$ |
| Boltzmann entropy | $$S_{B}$$ |
| temperature | $$T$$ |
| internal energy | $$U$$ |
| volume | $$V$$ |
| molar volume | $$V_{m},\ v$$ |
| work, multiplicity or statistical weight | $$W$$ |
| partition function | $$Z$$ |
| thermal dilatation coefficient, quantity of the states of the system | $$\alpha$$ |
| volumetric expansion coefficient | $$\beta$$ |
| surface expansion coefficient | $$\gamma$$ |
| change in the value of a variable | $$\Delta$$ |
| total molar Gibbs function change in a reaction | $$\Delta_{r}G$$ |
| isothermal compressibility | $$\kappa$$ |
| thermal wavelength | $$\lambda_{th}$$ |
| chemical potential | $$\mu$$ |
| grand potential | $$\Phi,\ \Phi_{G}$$ |
| Stefan-Boltzmann constant | $$\sigma = 5.670374 \times 10^{-8} \frac{\textrm{W}}{\textrm{m}^{2} \cdot \textrm{K}^{4}}$$ |
| multiplicity or statistical weight, grand potential | $$\Omega$$ |
| thermodynamic quantity in the standard state | $$^{\sout{\circ}}$$ |
Thermodynamics
Thermology
| name | equation |
|---|---|
| thermal dilatation coefficient | $$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)$$ |
| linear dilatation | $$\Delta L = \alpha L_{0} \Delta T$$ |
| surface expansion coefficient | $$\gamma = \frac{1}{A} \frac{dA}{dT}$$ |
| surface expansion coefficient | $$\gamma = 2 \alpha$$ |
| surface dilatation | $$\Delta A = \gamma A_{i} \Delta T$$ |
| final area | $$ A_{f} = A_{i} \left[ 1 + 2 \alpha \left( T_{f} - T_{i} \right) \right]$$ |
| volumetric expansion coefficient | $$\beta = \frac{1}{V} \frac{dV}{dT}$$ |
| volumetric expansion coefficient | $$\beta = 3 \alpha$$ |
| volumetric dilatation | $$\Delta V = \beta V_{i} \Delta T$$ |
| final volume | $$V_{f} = V_{i} \left[ 1 + \beta \left( T_{f} - T_{i} \right) \right]$$ |
| isothermal compressibility | $$\kappa = -\frac{1}{V}\left( \frac{\partial V}{\partial P} \right)_{T}$$ |
| Stefan-Boltzmann law | $$E = \sigma T^{4}$$ |
Ideal gas
| name | equation |
|---|---|
| Boyle's law | $$P \propto \frac{1}{V}$$ |
| Charles' law | $$V \propto T$$ |
| Gay-Lussac's law | $$P \propto T$$ |
| ideal gas equation | $$PV = Nk_{B}T$$ |
Real gas
| name | equation |
|---|---|
| Van der Waals equation of state \( b = N_{A} V_{M} \) | $$\left( P + a \frac{N^{2}}{V^{2}} \right)\left( V - Nb \right) = NRT$$ |
| Redlich-Kwong equation of state | $$P = \frac{NRT}{V-b} - \frac{a}{\sqrt{T} }\frac{N^{2}}{V\left( V - Nb \right)}$$ |
State functions
| name | equation |
|---|---|
| internal energy | $$\Delta U = Q + W$$ |
| Van der Waals internal energy | $$U_{VW} = U_{ideal} - a\left( \frac{N}{V} \right)^{2} V$$ |
| entropy | $$dS = \frac{\delta Q}{T};\ \frac{dS}{dt}\geq 0$$ |
| heat capacity | $$C = \frac{dQ}{dT} = \frac{Q}{\Delta T} $$ |
| specific heat capacity | $$c = \frac{C}{m} = \frac{Q}{m \Delta T} $$ |
Thermodynamic potentials
| name | equation |
|---|---|
| internal energy | $$\Delta U = \int \left( TdS - PdV + \sum_{i} \mu_{i}dN_{i} \right)$$ |
| Helmholtz free energy | $$F = U - TS$$ |
| enthalpy | $$H = U + PV$$ |
| Gibbs free energy | $$\begin{align*}G &= U + PV - TS\\ &= H - TS\end{align*}$$ |
| grand potential | $$\Omega = \Phi_{G} = \sum_{i} U - TS - \mu_{i}N_{i}$$ |
Related formulas
| name | equation |
|---|---|
| reaction enthalpy | $$\Delta H_{r} = \sum H_{products} - \sum H_{reactants}$$ |
Statistical mechanics
| name | equation |
|---|---|
| Boltzmann entropy | $$S = k_{B}\ln\Omega$$ |
| average energy | $$\bar{E} = n \times \frac{1}{2}k_{B}T$$ |
| partition function | $$Z = \sum_\alpha e^{-\beta E_{\alpha}}$$ |
| Boltzmann factor | $$\beta = \frac{1}{k_{B} T}$$ |
|
partition function for two-level systems \(\left(-\Delta/2, \Delta/2\right)\) |
$$Z = 2 \cosh \left( \frac{\beta \Delta}{2} \right)$$ |
|
partition function for the \(N\)-level system \( \left(0, \hbar\omega, 2\hbar\omega, \ldots , \left( N-1 \right) \hbar \omega \right) \) |
$$Z = \frac{1 - e^{-N\beta\hbar\omega}}{1 - e^{-\beta\hbar\omega}}$$ |
|
partition function for the simple harmonic oscillator \( E = \left(n + \frac{1}{2} \hbar \omega\right),\ n = 0,1,2, \ldots \) |
$$Z = \frac{e^{-\frac{1}{2}\beta\hbar\omega}}{1 - e^{-\beta\hbar\omega}}$$ |
|
partition function for rotational energy levels \( E_{rotational} = \hat{J}^{2}/2I \), \(\hat{J}^{2} = \hbar^{2} J\left( J + 1 \right)\), \( J = 0,1,2, \ldots \) |
$$Z = \sum_{J = 0}^{\infty} \left( 2J + 1 \right) e^{ - \beta \hbar^{2} J\left( J + 1 \right)/ 2 I} $$ |
| ideal gas partition function | $$Z = \frac{V}{\lambda_{th}^{3}}$$ |
| thermal wavelength | $$\lambda_{th} = \frac{h}{\sqrt{2 \pi m k_{B} T}}$$ |
| quantum concentration | $$n_{Q} = \frac{1}{\lambda_{th}^{3}}$$ |
| grand partition function | $$\mathcal{Z} = \sum_{i}e^{\beta\left( \mu N_{i} - E_{i} \right)}$$ |
| grand potential | $$\begin{split}\Phi_{G} &= -k_{B} T \ln \mathcal{Z} \\ &= U - TS - \mu N \\ &=-PV \end{split}$$ |
| equilibrium constant | $$\ln K = -\frac{\Delta_{r} G^{\sout{\circ}}}{RT}$$ |
| temperature dependence of the equilibrium constant | $$\frac{\mathrm{d} \ln K}{\mathrm{d}T} = \frac{\Delta_{r}H^{\sout{\circ}}}{RT^{2}}$$ |
State functions
| name | function of state | statistical mechanical expression |
|---|---|---|
| internal energy | $$U$$ | $$U = -\frac{\mathrm{d} \ln Z}{\mathrm{d} \beta}$$ |
| Helmholtz free energy | $$F$$ | $$F = - k_{B} T \ln Z$$ |
| entropy | $$S = -\left( \frac{\partial F}{\partial T} \right)_{V} = \frac{U - F}{T}$$ | $$S = k_{B} \ln Z + k_{B} T \left( \frac{\partial \ln Z}{\partial T} \right)_{V}$$ |
| pressure | $$P = -\left( \frac{\partial F}{\partial V} \right)_{T} $$ | $$P = k_{B} T \left( \frac{\partial \ln Z}{\partial V} \right)_{T}$$ |
| enthalpy | $$H = U + PV$$ | $$H = k_{B} T \left[ T \left( \frac{\partial \ln Z}{\partial T} \right)_{V} + V \left( \frac{\partial \ln Z}{\partial V}\right)_{T} \right]$$ |
| Gibbs free energy | $$\begin{align*}G &= F + PV \\G &= H - TS\end{align*}$$ | $$G = k_{B} T \left[ -\ln Z + V \left( \frac{\partial \ln Z}{\partial V}\right)_{T} \right]$$ |
| heat capacity | $$C_{V} = \left( \frac{\partial U}{\partial T} \right)_{V}$$ | $$C_{V} = k_{B} T \left[ 2 \left( \frac{\partial \ln Z}{\partial T} \right)_{V} + T \left( \frac{\partial^{2} \ln Z}{\partial T^{2}}\right)_{V} \right]$$ |