Theory — Boltzmann’s View of Entropy
In statistical thermodynamics, the entropy \(S\) of a macrostate is related to the number
of compatible microstates \(W\) (the multiplicity) by Boltzmann’s famous relation
\[
S \;=\; k_{\mathrm B}\,\ln W
\]
Here \(k_{\mathrm B}\) is the Boltzmann constant \((1.380\,649\times10^{-23}\ \mathrm{J\,K^{-1}})\).
The function \(\ln\) is the natural logarithm (base \(e\)).
A larger \(W\) means more ways to arrange the system at the same macroscopic conditions,
which corresponds to a larger \(S\).
Changes in entropy from changes in multiplicity
When a process takes the system from \(W_1\) microstates to \(W_2\) microstates,
\[
\Delta S \;=\; S_2 - S_1
\;=\; k_{\mathrm B}\,\ln\!\left(\frac{W_2}{W_1}\right)
\]
Doubling the number of microstates gives \(\Delta S = k_{\mathrm B}\ln 2\).
Numerically, \(\ln 2 \approx 0.693\), so
\(\Delta S \approx 9.57\times10^{-24}\ \mathrm{J\,K^{-1}}\) per system, and
per mole it is \(R\ln 2 \approx 5.76\ \mathrm{J\,mol^{-1}\,K^{-1}}\), where
\(R = N_{\mathrm A}k_{\mathrm B}\) is the gas constant.
Per-mole quantities
For a per-particle model, multiplying by Avogadro’s number converts to molar entropy:
\[
S_{\mathrm m} = N_{\mathrm A} S \;=\; R\,\ln W,
\qquad
\Delta S_{\mathrm m} = N_{\mathrm A}\Delta S \;=\; R\,\ln\!\left(\frac{W_2}{W_1}\right).
\]
Use this scaling only when the \(W\) you are using corresponds to a single representative entity;
for full many-particle macrostates \(W\) is already astronomically large and is defined for the entire system.
Forms used by the calculator
- Entropy from multiplicity:
\(S = k_{\mathrm B}\ln W\).
- Multiplicity from entropy:
\(W = \exp(S/k_{\mathrm B})\).
- Entropy change from multiplicities:
\(\Delta S = k_{\mathrm B}\ln(W_2/W_1)\).
- Final multiplicity from \(\Delta S\) and \(W_1\):
\(W_2 = W_1\,\exp(\Delta S/k_{\mathrm B})\).
Physical guidance
- At fixed total energy, increasing the accessible volume or loosening constraints
usually increases \(W\) and \(S\).
- At fixed volume, raising temperature typically makes more energy distributions accessible,
increasing \(W\) and \(S\).
Common pitfalls
- Use the natural log \(\ln\), not \(\log_{10}\).
- \(W\) must be dimensionless and \(W>0\).
- Keep units straight: \(S\) in \(\mathrm{J\,K^{-1}}\) per system; molar values in
\(\mathrm{J\,mol^{-1}\,K^{-1}}\).
