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Thermodynamics and Statistical Mechanics cheat sheet - grade college

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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
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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.