20-4 A Statistical View of Entropy#

Prompts

What is a microstate? A configuration? The multiplicity \(W\)?
For \(N\) molecules with \(n_1\) in one half and \(n_2\) in the other, write the multiplicity \(W\). Which configuration has the largest \(W\)?
State Boltzmann’s entropy equation. Why does entropy increase when a system moves to a more probable configuration?
For a free expansion (gas doubles its volume), show that the statistical definition \(S = k \ln W\) gives \(\Delta S = nR \ln 2\), matching thermodynamics.

Lecture Notes#

Entropy can be defined in terms of the number of ways a system can be arranged microscopically.
Microstate: a specific arrangement of molecules. Configuration: a group of microstates with the same macroscopic distribution (e.g., \(n_1\) in left half, \(n_2\) in right).
Multiplicity \(W\): number of microstates in a configuration.
Boltzmann: \(S = k \ln W\)—entropy is the logarithm of the number of ways to achieve a state.

For \(N\) identical molecules with \(n_1\) in one half of a box and \(n_2\) in the other (\(n_1 + n_2 = N\)):

(202)#\[ W = \frac{N!}{n_1!\,n_2!} \]

Example: 6 molecules, \(n_1 = 4\), \(n_2 = 2\) → \(W = 6!/(4!\,2!) = 15\).
The most probable configuration has \(n_1 \approx n_2\) (equal division)—it has the largest \(W\).

(203)#\[ S = k \ln W \]

\(k\): Boltzmann constant. \(W\): multiplicity.
Entropy is additive; probabilities multiply; \(\ln\) converts multiplication to addition.
Higher \(W\) → higher \(S\) → more probable configuration.

Assumption: All microstates are equally probable.

Configurations with larger \(W\) are more probable (more microstates).
For large \(N\) (e.g., \(10^{22}\)), almost all microstates have \(n_1 \approx n_2\)—the system is almost always in the central configuration.
The second law reflects the tendency to move toward more probable (higher \(W\), higher \(S\)) states.

For free expansion (gas doubles volume): \(N\) molecules initially all in left half → finally \(N/2\) in each half.

Initial: \(W_i = 1\) → \(S_i = 0\).
Final: \(W_f = N!/[(N/2)!]^2\) → using Stirling’s approximation, \(S_f = Nk \ln 2 = nR \ln 2\).
\(\Delta S = nR \ln 2\)—matches the thermodynamic result from section 20-1.

Stirling’s approximation

For large \(N\): \(\ln N! \approx N(\ln N) - N\). Used when factorials are too large to compute directly.

Poll: Multiplicity

For 4 molecules in a box, which configuration has the largest multiplicity?

\(W = N!/(n_1!\,n_2!)\)—multiplicity for two halves.
\(S = k \ln W\)—Boltzmann entropy.
Equal division (\(n_1 \approx n_2\)) has the largest \(W\) and \(S\) for large \(N\).
Second law: systems tend toward more probable (higher \(S\)) states.