(a) What is the difference between a state and a microstate of a system? (b) As a system goes from state A to state B, its entropy decreases. What can you say about the number of microstates corresponding to each state? (c) In a particular spontaneous process, the number of microstates available to the system decreases. What can you conclude about the sign of \(\Delta S\) surr?

A state of a system refers to the macroscopic properties of the system such as temperature, pressure, and volume. The macroscopic properties describe the overall behavior of the system. On the other hand, a microstate refers to a specific arrangement of particles in the system at a given energy level, which is compatible with a particular macroscopic state. Microstates represent all the possible arrangements of particles that can lead to the same macroscopic properties.

Entropy (S) is a measure of the number of microstates (W) that correspond to a particular macroscopic state. The relationship between entropy and the number of microstates is given by the Boltzmann's entropy formula: \(S = k_B \ln W\), where \(k_B\) is Boltzmann's constant. As the system goes from state A to state B, if its entropy decreases, it means that the number of microstates corresponding to state B is less than the number of microstates for state A. This can be seen from the formula as if W decreases, then S will also decrease.

For a spontaneous process, the total entropy change \(\Delta S_{tot}\) is positive. The total entropy change can be expressed as the sum of the entropy change of the system (\(\Delta S_{sys}\)) and the entropy change of the surroundings (\(\Delta S_{surr}\)): $$\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr}$$ In the given spontaneous process, if the number of microstates available to the system decreases, it means the system's entropy is decreasing, i.e., \(\Delta S_{sys} < 0\). Therefore, for the total entropy change to be positive, the entropy change of the surroundings must be positive, i.e., $$\Delta S_{surr} > 0$$ Hence, we can conclude that the sign of \(\Delta S_{surr}\) is positive when the number of microstates available to the system decreases in a spontaneous process.