Let k be the Boltzmann constant. If the thermodynamic state of gas at temperature T changes isothermally and reversibly to a state with three timesthe number of microstates as initially, the energy input to gas as heat is:
A) Q = 0
B) Q = 3kT
C) Q = -3kT
D) kTln3
E) -kTln3

Question

Quizplus · Accepted Answer

According to Boltzmann's formula, the entropy of a system is directly proportional to the logarithm of its number of microstates. So, if the number of microstates increases by a factor of 3, the entropy of the system also increases by a factor of ln3. As the process is isothermal and reversible, there is no change in temperature or internal energy of the gas. Therefore, the heat added to the system equals the work done on the system, which is given by ΔQ = -ΔW. As the process is reversible, the work done on the system is the maximum possible work, which equals the change in Helmholtz free energy (ΔA) of the gas. ΔA = -TΔS = -kTln3. Therefore, the heat added to the system is ΔQ = -ΔA = kTln3. Hence, the correct answer is D.

Let K Be the Boltzmann Constant