Final answer:
The entropy increase after mixing two different types of particles, known as the entropy of mixing, is quantified as 2N log 2 due to an increase in possible configurations. When identical particles are mixed, the number of configurations remains unchanged, so there is no increase in entropy, reflecting the Gibbs paradox.
Step-by-step explanation:
Entropy Increase Due to Mixing Different Particles
When two systems of N atoms of type A and B, respectively, are mixed, the total entropy increases due to the variety of possible configurations after mixing. The entropy increase is calculated as 2N log 2, known as the entropy of mixing. This occurs because each system goes from having one possible way to arrange its particles (before mixing) to having many possible arrangements (after mixing).
If the atoms A and B are identical, there is no change in the number of possible arrangements after mixing as the configurations before and after mixing remain the same. Therefore, there is no increase in entropy when identical atoms are mixed. This situation illustrates the concept known as the Gibbs paradox, which arises from the difference in behavior when mixing identical versus non-identical particles.
In the context of a dissolution process where a solid dissolves in a liquid, the increased entropy arises due to the greater freedom of motion and the additional interactions between the particles of the solid and the liquid. These result in a higher number of accessible microstates, thereby increasing the system's entropy.