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 through a mixture of and to 100% pure . The entropy of mixing is given by
where is the fixed total number of molecules (100 here) and .
For a small system such as this, we can generate an array of values for each value of from 0 to 100. Plotting this as a bar chart, we get
Starting from where (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 to decreases. The changes in for the first few steps are:
add molecule number | change in |
The rate at which the entropy increases declines as we convert more molecules from to . If we add a slight impurity into an initially pure mixture of 100% , this generates a larger increase in entropy than if we add a bit more impurity to an already mixed system.