Boltzmann and microstates

Problem

A system has $W=10^{20}$ accessible microstates. Calculate $S$ in J/K using Boltzmann’s formula $S=k_B\ln W$.

Solution

We apply Boltzmann’s formula with $k_B=1{,}38\cdot 10^{-23}\;\text{J/K}$. Since $\ln(10^{20})=20\cdot\ln 10=20\cdot 2{,}303=46{,}05$: $S=k_B\ln W=1{,}38\cdot 10^{-23}\cdot\ln(10^{20})=1{,}38\cdot 10^{-23}\cdot 46{,}05=6{,}35\cdot 10^{-22}\;\text{J/K}$ $\ev{S\approx 6{,}35\cdot 10^{-22}\;\text{J/K}}$

Links

Topics: Entropy and the second law Concepts: Entropy Skills: Micro-macro interpretation