# Entropy of mixing in a small system

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 5.57.

As an example of the entropy changes when two pure substances are mixed, consider a system of 100 molecules, which may vary in composition from 100% of species ${A}$ through a mixture of ${A}$ and ${B}$ to 100% pure ${B}$. The entropy of mixing is given by

$\displaystyle \Delta S_{mixing}=-Nk\left[x\ln x+\left(1-x\right)\ln\left(1-x\right)\right]$

where ${N=N_{A}+N_{B}}$ is the fixed total number of molecules (100 here) and ${x=N_{A}/N}$.

For a small system such as this, we can generate an array of ${\Delta S_{mixing}/k}$ values for each value of ${N_{A}}$ from 0 to 100. Plotting this as a bar chart, we get

Starting from ${N_{A}=0}$ where ${\Delta S/k=0}$ (since there is only one species at this point, there is no mixing), we see that the entropy increase per molecule as we convert successive molecules from ${A}$ to ${B}$ decreases. The changes in ${\Delta S/k}$ for the first few steps are:

 add molecule number change in ${\frac{\Delta S}{k}}$ ${1}$ ${5.60}$ ${2}$ ${4.20}$ ${3}$ ${3.67}$ ${4}$ ${3.32}$ ${5}$ ${3.06}$

The rate at which the entropy increases declines as we convert more molecules from ${A}$ to ${B}$. If we add a slight impurity into an initially pure mixture of 100% ${B}$, this generates a larger increase in entropy than if we add a bit more impurity to an already mixed system.