Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Thermodynamics and Statistical Mechanics cheat sheet - grade college

Click image to open full size

Physics Grade college

Thermodynamics and Statistical Mechanics Cheat Sheet

A printable reference covering thermodynamic laws, entropy, free energies, partition functions, ensembles, and Maxwell relations for college physics.

TopicsThermodynamicsHeatEnergyAtomsMoleculesPhysics
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Thermodynamics studies heat, work, energy, entropy, and equilibrium behavior in macroscopic systems. Statistical mechanics explains those same ideas from microscopic states, probabilities, and particle energies. This cheat sheet connects the two views so students can move between state variables, thermodynamic potentials, and partition functions.

It is useful for solving problems involving engines, phase changes, ideal gases, ensembles, and equilibrium conditions.

The core ideas are the laws of thermodynamics, the fundamental relation dU=T dS−P dV+μ dNdU = T\,dS - P\,dV + \mu\,dN, and the statistical definition of entropy S=kBln⁡ΩS = k_B \ln \Omega. Free energies such as F=U−TSF = U - TS and G=H−TSG = H - TS identify natural variables and predict spontaneity under common constraints. In statistical mechanics, the partition function Z=∑ie−βEiZ = \sum_i e^{-\beta E_i} contains the main thermodynamic information, including F=−kBTln⁡ZF = -k_B T \ln Z and ⟨E⟩=−∂ln⁡Z∂β\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}.

Key Facts

  • The first law of thermodynamics is dU=δQ−δWdU = \delta Q - \delta W, and for pressure-volume work done by the system, \delta W = P\,dV.
  • For a reversible process, the entropy change is dS=δQrevTdS = \frac{\delta Q_{\mathrm{rev}}}{T}, and for an isolated system, ΔS≥0\Delta S \ge 0.
  • The fundamental thermodynamic identity for a simple compressible system is dU=T dS−P dV+μ dNdU = T\,dS - P\,dV + \mu\,dN.
  • The Helmholtz free energy is F=U−TSF = U - TS, with differential dF=−S dT−P dV+μ dNdF = -S\,dT - P\,dV + \mu\,dN.
  • The Gibbs free energy is G=H−TS=U+PV−TSG = H - TS = U + PV - TS, with differential dG=−S dT+V dP+μ dNdG = -S\,dT + V\,dP + \mu\,dN.
  • For the canonical ensemble, the partition function is Z=∑ie−βEiZ = \sum_i e^{-\beta E_i} where β=1kBT\beta = \frac{1}{k_B T}.
  • Canonical ensemble averages follow F=−kBTln⁡ZF = -k_B T \ln Z, ⟨E⟩=−∂ln⁡Z∂β\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}, and S=−(∂F∂T)V,NS = -\left(\frac{\partial F}{\partial T}\right)_{V,N}.
  • One Maxwell relation from dFdF is (∂S∂V)T,N=(∂P∂T)V,N\left(\frac{\partial S}{\partial V}\right)_{T,N} = \left(\frac{\partial P}{\partial T}\right)_{V,N}.

Vocabulary

Entropy
Entropy is a state function measuring energy dispersal or microscopic multiplicity, with statistical form S=kBln⁡ΩS = k_B \ln \Omega.
Temperature
Temperature is the thermodynamic variable defined by 1T=(∂S∂U)V,N\frac{1}{T} = \left(\frac{\partial S}{\partial U}\right)_{V,N}.
Partition function
The partition function Z=∑ie−βEiZ = \sum_i e^{-\beta E_i} is a weighted sum over accessible energy states that determines equilibrium thermodynamic properties.
Canonical ensemble
The canonical ensemble describes systems with fixed NN, VV, and TT that exchange energy with a heat bath.
Chemical potential
Chemical potential is the change in internal energy when particles are added, given by μ=(∂U∂N)S,V\mu = \left(\frac{\partial U}{\partial N}\right)_{S,V}.
Free energy
Free energy is a thermodynamic potential, such as F=U−TSF = U - TS or G=H−TSG = H - TS, used to predict equilibrium under specified constraints.

Common Mistakes to Avoid

  • Confusing heat with temperature is wrong because heat QQ is energy transferred by a temperature difference, while temperature TT is a state variable.
  • Using dS=δQTdS = \frac{\delta Q}{T} for irreversible processes is wrong because the equality requires a reversible path, so use dS=δQrevTdS = \frac{\delta Q_{\mathrm{rev}}}{T} for entropy changes.
  • Forgetting natural variables of thermodynamic potentials is wrong because derivatives like S=−(∂F∂T)V,NS = -\left(\frac{\partial F}{\partial T}\right)_{V,N} only hold when the correct variables are fixed.
  • Treating the partition function as just a normalization constant is wrong because ZZ also gives thermodynamic quantities through derivatives of ln⁡Z\ln Z.
  • Dropping the sign in work conventions is wrong because dU=δQ−P dVdU = \delta Q - P\,dV assumes work is done by the system during expansion.

Practice Questions

  1. 1 A monatomic ideal gas has n=2.00 moln = 2.00\,\mathrm{mol} and expands isothermally at T=300 KT = 300\,\mathrm{K} from Vi=1.00 LV_i = 1.00\,\mathrm{L} to Vf=4.00 LV_f = 4.00\,\mathrm{L}. Calculate W=nRTln⁡(VfVi)W = nRT\ln\left(\frac{V_f}{V_i}\right).
  2. 2 A two-level system has energies E0=0E_0 = 0 and E1=ϵE_1 = \epsilon. Write the canonical partition function ZZ and find the probability p1p_1 of occupying the excited state.
  3. 3 For a system with Helmholtz free energy F(T,V)=−aT4VF(T,V) = -aT^4V, where aa is a constant, find S=−(∂F∂T)VS = -\left(\frac{\partial F}{\partial T}\right)_V and P=−(∂F∂V)TP = -\left(\frac{\partial F}{\partial V}\right)_T.
  4. 4 Explain why minimizing FF is the correct equilibrium criterion for fixed TT, VV, and NN, while minimizing GG is used for fixed TT, PP, and NN.

Understanding Thermodynamics and Statistical Mechanics

A key skill is separating state functions from process quantities. Internal energy, entropy, pressure, volume, and temperature describe an equilibrium state. Their changes depend only on the starting and ending states.

Heat and work describe energy transferred during a particular process. They depend on the route taken. A gas can reach the same final volume by slow compression, rapid compression, heating, or cooling.

The work can differ in each case. Slow reversible paths are especially useful because the system stays close to equilibrium, so its pressure and temperature are well defined throughout.

Real processes usually create friction, turbulence, or finite temperature differences. These make entropy and often waste heat increase.

Thermodynamic potentials are tools for choosing the right energy account for the conditions held fixed. Helmholtz free energy is useful for a system in contact with a heat bath while volume stays fixed. This appears in models of gases in a rigid container, solids, and magnetic materials.

Gibbs free energy is useful when temperature and pressure stay fixed. Many chemistry and materials problems use these conditions because reactions happen in open containers at roughly atmospheric pressure.

At fixed temperature and pressure, a change that lowers Gibbs free energy is favored until equilibrium is reached. For phase changes, such as ice melting or water boiling, the two phases can coexist when their Gibbs free energies per particle are equal.

Statistical mechanics explains why temperature enters these rules. A macroscopic sample contains an enormous number of particles, so it can occupy a huge number of microscopic arrangements. In a system connected to a heat bath, low energy arrangements are more likely than high energy arrangements.

Higher temperature makes the energy preference less strict, allowing more excited states to contribute. The partition function adds the statistical weights of all allowed states. It works like a complete inventory of the accessible microscopic behavior.

Once it is known, energy, entropy, pressure, heat capacity, and other average properties can be found from it. Fluctuations still occur, but for a large sample their relative size is usually tiny. This is why pressure gauges and thermometers give stable readings.

The choice of ensemble matters because it states what the surroundings can exchange with the system. The microcanonical ensemble fits an isolated system with fixed energy, volume, and particle number. The canonical ensemble fits a sealed system that exchanges heat with a reservoir.

The grand canonical ensemble fits systems that exchange both heat and particles, such as gas adsorption on a surface or electrons moving between a small device and electrical contacts. Students should always write down the fixed variables before selecting a potential or partition function. This simple step prevents many sign and derivative mistakes.

Maxwell relations may look abstract, but they connect quantities that are easier to measure with quantities that are harder to measure directly. For example, a relation can connect how entropy changes when volume changes with how pressure changes when temperature changes. Pressure versus temperature data can therefore reveal entropy behavior.

These relations come from the fact that thermodynamic potentials are state functions with consistent mixed derivatives. When solving problems, pay close attention to subscripts on derivatives. They state which variables are held fixed.

Also check work sign conventions, since textbooks may define work done on a system instead of work done by it. A correct physical explanation should match the signs, units, and limiting behavior of the result.