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.

Leave a Reply

Your email address will not be published. Required fields are marked *