Question

The connection between the number of microstates,W, and the entropy, S, of a system is expressed by the Boltzmann equation S=klnW where k is the Boltzmann constant per molecule (particle), 1.38×10−23 J/(K⋅particle). From this equation the entropy change, ΔS, for a system can be related to the change in the number of microstates as ΔS=klnWfinalWinitial where Wfinal in the final number of microstates and Winitial is the initial number of microstates. Part B A gaseous system undergoes a change in temperature and volume. What is the entropy change for a particle in this system if the final number of microstates is 0.505 times that of the initial number of microstates? Express your answer numerically in joules per kelvin per particle.

Answer 1

Boltzmann constant relates the entropy of a system to the number of microstates possible for the system.

S = k lnW

​where, k is the Boltzmann's constant, S is the entropy and W is the number of microstates.

The change in entropy is the difference between its final and initial values.

S = S_f - S_i = k lnW_f - k lnW_​​i = k ln(W_f/ W_i)

It is given that the final number of microstates is 0.505 times of the initial number of microstates.

Thus, W_f = 0.505*W_i

Hence, S = k ln(0.505*W_i / W_i) = k ln(0.505) = 1.38*10^-23 J/K.particle * ln(0.505) = - 9.43*10^-24 J/K.particle

For processes involving an decrease in the number of microstates, W_i >W_f , the entropy of the system decreases, S<0.