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37 formulas

Thermodynamics

Heat, entropy, engines. Every formula below opens into a live, hands-on simulation.

gas laws
PV=nRTPV = nRT

Ideal Gas Law

Pressure times volume is proportional to temperature for a fixed amount of gas.

energy conservation
ΔU=Q−W\Delta U = Q - W

First Law of Thermodynamics

Energy in (heat) minus energy out (work) equals the change stored inside.

heat transfer
q=−kdTdxq = -k \frac{dT}{dx}

Fourier's Law of Heat Conduction

Heat flows from hot to cold, faster through better conductors and steeper gradients.

thermal properties
ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

Linear Thermal Expansion

Materials grow longer when heated — by an amount proportional to their length and temperature rise.

heat engines
η=1−TCTH\eta = 1 - \frac{T_C}{T_H}

Carnot Efficiency

No engine can beat the efficiency set by the ratio of its cold and hot reservoir temperatures.

radiation
P=σAT4P = \sigma A T^4

Stefan-Boltzmann Law

Hot objects radiate energy as light — and the power skyrockets with temperature (fourth power!).

entropy
ΔS=∫δQrevT\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}

Entropy Change

Entropy measures how much energy has spread out — it always increases in the universe overall.

statistical mechanics
f(v)=4πn(m2πkBT)3/2v2e−mv22kBTf(v) = 4\pi n \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}

Maxwell-Boltzmann Speed Distribution

Gas molecules have a spread of speeds — most cluster near a peak, with a long tail of fast outliers.

statistical mechanics
S=kBln⁡WS = k_B \ln W

Boltzmann Entropy

Entropy counts the number of microscopic ways a macroscopic state can be realized — more ways means higher entropy.

phase transitions
dPdT=LTΔv\frac{dP}{dT} = \frac{L}{T \Delta v}

Clausius-Clapeyron Equation

Steeper vapor-pressure curve where latent heat is high — phase boundary tilt is governed by entropy of vaporization.

calorimetry
Cv=(∂U∂T)VC_v = \left(\frac{\partial U}{\partial T}\right)_V

Heat Capacity at Constant Volume

Heat capacity at constant volume measures the energy needed to raise temperature when no work is done — pure internal energy storage.

calorimetry
Cp=(∂H∂T)PC_p = \left(\frac{\partial H}{\partial T}\right)_P

Heat Capacity at Constant Pressure

At constant P, some heat goes into work done expanding the gas — so C_p is always larger than C_v by exactly R (for ideal gases).

thermodynamic potentials
G=H−TSG = H - TS

Gibbs Free Energy

Gibbs free energy is the maximum non-PV work extractable from a system at constant T and P — and it must decrease for spontaneous processes.

thermodynamic potentials
F=U−TSF = U - TS

Helmholtz Free Energy

Helmholtz energy is the maximum work extractable at constant T and V. Like Gibbs but without the PV term — appropriate for sealed rigid containers.

blackbody radiation
λmax⁡T=b\lambda_{\max} T = b

Wien's Displacement Law

Hotter objects emit shorter-wavelength radiation; the peak wavelength is inversely proportional to temperature.

blackbody radiation
Bλ(T)=2hc2λ51ehc/λkBT−1B_\lambda(T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_B T} - 1}

Planck Radiation Law

Quantized photon energies cap the high-frequency emission of a blackbody, solving the ultraviolet catastrophe.

Statistical Mechanics
Z=∑ie−βEi,β=1kBTZ = \sum_i e^{-\beta E_i}, \quad \beta = \frac{1}{k_B T}

Canonical Partition Function

Z counts microstates weighted by their Boltzmann suppression; hotter systems explore more states.

Quantum Statistical Mechanics
f(E)=1e(E−μ)/(kBT)+1f(E) = \frac{1}{e^{(E-\mu)/(k_B T)} + 1}

Fermi-Dirac Distribution

Each quantum state can hold at most one fermion; the chemical potential μ acts as a 'cut-off' — states below are occupied, above are empty.

Quantum Statistical Mechanics
⟨ni⟩=1eβ(εi−μ)−1\langle n_i \rangle = \frac{1}{e^{\beta(\varepsilon_i - \mu)} - 1}

Bose-Einstein Distribution

Bosons actively prefer to share states — the more particles already in a state, the more likely the next one joins (stimulated emission).

Statistical Mechanics
S=NkB[ln⁡(VN(4πmE3Nh2)3/2)+52]S = Nk_B\left[\ln\left(\frac{V}{N}\left(\frac{4\pi m E}{3Nh^2}\right)^{3/2}\right) + \frac{5}{2}\right]

Sackur-Tetrode Equation

Entropy counts accessible microstates in units of h³ per degree of freedom; the 5/2 is the 3/2 kinetic + 1 from volume/particle indistinguishability.

Statistical Mechanics
Ω=−kBTln⁡Ξ,Ξ=∑N,ie−β(EN,i−μN)\Omega = -k_B T \ln \Xi, \quad \Xi = \sum_{N,i} e^{-\beta(E_{N,i} - \mu N)}

Grand Canonical Potential (Landau Free Energy)

Ω is the free energy cost of a system that can bleed particles; minimise it to find the equilibrium particle number at given T and μ.

Statistical Mechanics
⟨qi∂H∂qi⟩=kBT\left\langle q_i \frac{\partial H}{\partial q_i} \right\rangle = k_B T

Equipartition Theorem

Classical thermal fluctuations distribute energy democratically: every quadratic term in H gets the same share k_BT/2.

Non-equilibrium Statistical Mechanics
SFF(ω)=2kBT Re[χ(ω)]S_{FF}(\omega) = 2k_B T\, \text{Re}[\chi(\omega)]

Fluctuation-Dissipation Theorem

A system that dissipates energy (resistance) must also fluctuate spontaneously (noise) at the same rate — you can't have one without the other at finite temperature.

Statistical Mechanics
⟨Ekin⟩=−12⟨Epot⟩\langle E_{\text{kin}} \rangle = -\frac{1}{2}\langle E_{\text{pot}} \rangle

Virial Theorem (Statistical Mechanics)

In a bound system, twice the kinetic energy always equals the negative of the potential energy — energy is partitioned by the power law of the force.

Kinetic Theory
f(v)=4πn(m2πkBT)3/2v2exp⁡ ⁣(−mv22kBT)f(v) = 4\pi n\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2k_B T}\right)

Maxwell Speed Distribution (3D)

The v² factor (surface area of velocity sphere) competes with the Boltzmann suppression e^{−mv²/2k_BT} to create a peaked distribution; the peak shifts right as T rises.

Laws of Thermodynamics
ΔSuniverse=ΔSsys+ΔSsurr≥0\Delta S_{\text{universe}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0

Second Law of Thermodynamics

Isolated systems drift toward disorder; entropy only ever goes up.

Laws of Thermodynamics
(A∼C)∧(B∼C)  ⟹  (A∼B)(A \sim C) \wedge (B \sim C) \implies (A \sim B)

Zeroth Law of Thermodynamics

If two things each match a thermometer, they match each other.

Laws of Thermodynamics
lim⁡T→0S(T)=S0=kBln⁡g0\lim_{T \to 0} S(T) = S_0 = k_B \ln g_0

Third Law of Thermodynamics

Cool toward 0 K and entropy freezes to a constant you can never fully remove.

Thermodynamic Processes
PVγ=const,γ=CpCvP V^{\gamma} = \text{const}, \quad \gamma = \frac{C_p}{C_v}

Adiabatic Process

No heat in or out, so work done on the gas becomes its internal energy.

Equations of State
(P+an2V2)(V−nb)=nRT\left(P + \frac{a n^2}{V^2}\right)(V - n b) = n R T

Van der Waals Equation

Real molecules take up space (b) and attract each other (a), bending the ideal gas law.

Heat Engines & Cycles
COPref=QcW=QcQh−Qc≤TcTh−Tc\text{COP}_{\text{ref}} = \frac{Q_c}{W} = \frac{Q_c}{Q_h - Q_c} \leq \frac{T_c}{T_h - T_c}

How a Refrigerator Moves Heat Uphill

Spend work to pump heat from cold to hot; the colder the inside, the more work each joule costs.

Real Gases & Phase Changes
μJT=(∂T∂P)H=1Cp[T(∂V∂T)P−V]\mu_{\text{JT}} = \left(\frac{\partial T}{\partial P}\right)_H = \frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_P - V\right]

Joule-Thomson Effect

Throttle a real gas at constant enthalpy and it cools — below the inversion temperature.

Phase Transitions
dTmdP=T ΔVmeltLf≈−7.4×10−3 K/atm\frac{dT_m}{dP} = \frac{T\,\Delta V_{\text{melt}}}{L_f} \approx -7.4\times10^{-3}\ \text{K/atm}

Why Ice Skates Glide

A thin film of liquid water, not pressure alone, lets the blade slide.

Statistical Mechanics
Werase≥kBTln⁡2 per bitW_{\text{erase}} \geq k_B T \ln 2 \ \text{per bit}

Maxwell's Demon and Information

Sorting molecules looks free, but erasing the demon's memory pays the full entropy bill.

Statistical Mechanics
ni+1neni=2gi+1gi(2πmekBTh2)3/2e−χ/kBT\frac{n_{i+1} n_e}{n_i} = \frac{2 g_{i+1}}{g_i}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-\chi/k_B T}

Saha Ionization Equation

Hotter, thinner gas is more ionized; the Boltzmann factor e^(−χ/kT) sets the balance.

Heat Engines & Cycles
ηOtto=1−1rγ−1,r=V1V2\eta_{\text{Otto}} = 1 - \frac{1}{r^{\gamma - 1}}, \quad r = \frac{V_1}{V_2}

Otto Cycle Efficiency

Efficiency depends only on how much you compress the fuel-air mix before ignition.

Statistical Mechanics
ΔSmix=−NkB∑ixiln⁡xi  →  0 for identical species\Delta S_{\text{mix}} = -N k_B \sum_i x_i \ln x_i \;\to\; 0 \text{ for identical species}

Gibbs Paradox

Mixing distinct gases adds entropy; mixing identical ones adds none.