Question

(e) Consider approximately 10²⁴ gas molecules in a box. A divider which can conduct heat, for example a rubber diaphragm, separates the box into two chambers of equal size. Each chamber contains half of the molecules. Assume that the internal energy of the gas is (approximately) fixed.

• Describe some possible macrostates of the gas in the two chambers of the box.
• Describe a low entropy macrostate.
• Describe a high-entropy macrostate.
• In which situation described below is the entropy of the gas greatest?
  1. When the energy is spread nearly evenly among all the molecules and all the molecules are clustered in small regions of each chamber.
  2. When the energy is concentrated in a small number of molecules that are spread nearly evenly throughout the box.
  3. When the energy is spread nearly evenly among all the molecules and all the molecules are spread nearly evenly throughout the box.
  4. When the energy is concentrated in a small number of molecules which are clustered in a small region of one chamber of the box.

Answer 1

Each partition contains 512 particles

The internal energy is fixed

the V of both the boxes is same,

as only energy exchange is possible, it is a canonical ensemble

Imagine all the gas molecules in the room you are sitting in. Now in your mind divide the room into two sides (left and right). Each molecule carrying some amount of energy could be anywhere in that room. Each molecule has a an equal chance of being either on the left side or the right side of the room. The odds that all of the molecules are on the right side of the room are very very jsmall. The odds that the molecules are essentially evenly distributed between both sides of the room are astronomically large. This is the essence of the microscopic view of entropy.

Quantitatively the entropy is given as

where, S is the entropy and is the number of microstates and k is just a constant

There are fewer ways to arrange the molecules and have them all on one side. Fewer microstates means low entropy. There are more ways to arrange the molecules and have them evenly distributed, therefore more microstate means more entropy.

so more evenly distributed equates to more entropy.

The entropy of the gas is highest when the energy is evenly spread among the molecules and the molecules are evenly spread in the volume.

The number of microstates in the two boxes is a product

and hence entropies of two independent systems is additive

this can be used to find possible macrostates of the system.

A low entropy macrostate would be the one where all the energy is concentrated and all the molecules are concentrated in a small volume.