PX284 - C1c - isothermal expansion

considering two systems of equal volume, $V_{0}$ , are connected with a pipe with a tap

in thermodynamics

\begin{aligned} Δ S & = \int_{i}^{f} d S \\ = \int_{V_{0}}^{2 V_{0}} \frac{p}{T} d V \\ = \int_{V_{0}}^{2 V_{0}} \frac{N k_{B}}{V} d V \\ ∴ Δ S & = N k_{B} \ln 2 \end{aligned}

in statistical mechanics

each molecule has two choices: it can either be in one system or the other

Ω_{t o t} = Ω_{0} \times 2^{N}

where, $Ω_{0}$ is the number of original microstates

using the boltzmann entropy formula:

Δ S = k_{B} \ln (2^{N} Ω_{0}) - k_{B} \ln Ω_{0} = N k_{B} \ln (2)