Question

The entropy change for a particular process was measured to be -1.4 x 10-22 J/K. If the system initially had 4.4 x 1014 microstates at the beginning of the process, how many microstates are there at the end of the process?

Answer 1

From the Boltzmann hypothesis, we know,

S = k_B ln ( W_max) ... (1)

where, S = entropy , k_B = Boltzmann constant,

W_max = thermodynamic weight or number of microstates of the most dominating configuration.

So, let S' be the initial entropy with W_max= W' and

S" be the final entropy with W_max= W"

Therefore, from equation (1) we can write ,

dS = (S" - S')= k_b { ln W" - ln W' } = k_Bln(W"/W')

or, W"/W' = exp(dS / k_B )

so, W" = W' exp( dS / k_B)

Now, dS = -1.4 * 10^-22 J/K and W' = 4.4 * 10¹⁴ (given)

So, W" = (4.4 * 10¹⁴ )* exp [( - 1.4 * 10^-22)/ (1.38 * 10^-23)]

W" = 17.28 * 10⁹ (approx.)

So, finally after the end of the process there would be

17.28 * 10⁹ microstates.