← Statistical Mechanics
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Topic 1 of 5

Microstates & Entropy

Statistical mechanics bridges microscopic physics and macroscopic thermodynamics. Entropy is defined as S = k_B ln Ω, where Ω is the number of microstates. The second law is simply the statement that macroscopic systems evolve toward states of higher multiplicity.

Key Concepts

  • Microstate: a specific configuration of all particles
  • Macrostate: defined by macroscopic variables (E, V, N)
  • Multiplicity Ω: number of microstates for a given macrostate
  • Boltzmann entropy: S = k_B ln Ω
  • Second law: isolated systems evolve to maximize Ω (maximize S)

Key Equations

Boltzmann entropy
S=kBln⁡ΩS = k_B \ln \Omega
Stirling approximation
ln⁡N!≈Nln⁡N−N\ln N! \approx N\ln N - N
Two-state multiplicity
Ω(N,n)=(Nn)=N!n!(N−n)!\Omega(N,n) = \binom{N}{n} = \frac{N!}{n!(N-n)!}
Entropy change
ΔS=kBln⁡(Ωf/Ωi)\Delta S = k_B \ln(\Omega_f/\Omega_i)
Worked Example

Example Problem

Problem

A system of 4 coins has 2 heads. Find the multiplicity and entropy.

Solution

Ω = C(4,2) = 6. S = k_B ln 6 = 1.38×10⁻²³ × 1.792 = 2.47×10⁻²³ J/K.

Practice

Exercises

7 problems
1 of 7

A system of 10 two-state particles has 5 in state "up". Find the multiplicity Ω.

2 of 7

For the same system (Ω=252), find S in units of k_B.

k_B
3 of 7

Two systems A and B are combined. Ω_A=100, Ω_B=50. Find the total entropy change ΔS when they are combined, in units of k_B (assume Ω_total = Ω_A × Ω_B).

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4 of 7

Use Stirling: ln(20!) ≈ 20 ln 20 - 20. Find ln(20!).

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5 of 7

A system can access 10⁶ microstates. Find S in J/K. (k_B = 1.38×10⁻²³ J/K)

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6 of 7

N=100 particles, n=50 in state up. Using Stirling, find S/k_B (= ln Ω ≈ N ln 2 at maximum).

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7 of 7

The entropy of mixing of 1 mole of ideal gas A with 1 mole of ideal gas B at constant T and V is ΔS = -nR ln(x_A)×... = 2nR ln 2 for equal moles. Find ΔS in J/K. (R=8.314 J/mol·K, n=1 mol each)

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Key Takeaways

  • Entropy S = k_B ln Ω counts the microscopic disorder of a macrostate
  • The second law of thermodynamics is a statistical statement about most-probable states
  • Stirling's approximation enables calculation of entropy for large N
  • Entropy is additive for independent subsystems since multiplicities multiply