Question

(a) What is the change in entropy (in J/K) if you start with 100 coins in the 40 heads and 60 tails macrostate, toss them, and get 45 heads and 55 tails?

________J/K

(b) What if you get 5 heads and 95 tails?

________J/K

(c) How much more likely is 45 heads and 55 tails than 5 heads and 95 tails?

________times more likely

(d) Does either outcome violate the second law of thermodynamics? Explain.

Answer 1

(a)

Change in entropy,

S = k*ln(Wf / Wi)

Here, k = 1.38*10^(-23) J/K

Wi = number of microstates in having 40 heads and 60 tails = 1.4*10^28

Wf = number of microstates in having 45 heads and 55 tails = 6.1*10^28

S = 1.38*10^(-23) * ln (6.1*10^28 / 1.4*10^28)

S = 2.03*10^(-23) J/K

(b)

if you get 5 heads and 95 tails,

Wf = 7.5*10^7

S = 1.38*10^(-23) * ln (7.5*10^7 / 1.4*10^28)

S = -6.44*10^(-22) J/K

(c)

Ratio = Wf / Wf'

Ratio = 6.1*10^28 / 7.5*10^7

Ratio = 8.13*10^20

(d)

According to second law of thermodynamics,

total entropy of an isolated system can never decrease over time.

Here, in second case S is a negative number.

so,  outcome do not violate the second law of thermodynamics.