Question

What is the change in entropy if you start with 10 coins in the 5 heads and 5 tails macrostate, toss them, and get 2 heads and 8 tails? (b) How much more likely is 5 heads and 5 tails than 2 heads and 8 tails? (Take the ratio of the number of microstates to find out.) (c) If you were betting on 2 heads and 8 tails, would you accept odds of 252 to 45? Explain why or why not.

a) 10 J/K, 5 times more likely, No
b) 20 J/K, 10 times more likely, Yes
c) 30 J/K, 15 times more likely, Yes
d) 40 J/K, 20 times more likely, No

Answer 1

The change in entropy can be calculated using the formula ΔS = kln(Wf / Wi) where Wf is the final number of microstates and Wi is the initial number of microstates. In this case, starting with 10 coins in the 5 heads and 5 tails macrostate and getting 2 heads and 8 tails, the change in entropy is approximately 20 J/K. The macrostate of 5 heads and 5 tails is approximately 5.6 times more likely than the macrostate of 2 heads and 8 tails. Whether to accept odds of 252 to 45 for betting on 2 heads and 8 tails depends on individual preference.

Step-by-step explanation:

The change in entropy can be calculated using the formula ΔS = kln(Wf / Wi), where ΔS is the change in entropy, k is the Boltzmann constant, Wf is the final number of microstates, and Wi is the initial number of microstates. In this case, we start with 10 coins in the macrostate of 5 heads and 5 tails, which corresponds to 252 microstates. After tossing the coins and getting 2 heads and 8 tails, there are 45 microstates. Plugging these values into the formula, we get:

ΔS = kln(45 / 252)

In part (b), we need to compare the likelihood of the macrostate with 5 heads and 5 tails to the macrostate with 2 heads and 8 tails. We can do this by taking the ratio of the number of microstates. The ratio is 252 / 45, which is approximately 5.6.

In part (c), we need to determine whether the odds of 252 to 45 are acceptable for betting on 2 heads and 8 tails. This can be subjective based on personal preference and risk tolerance. However, if the odds are favorable enough to yield a higher expected value, then it may be worth accepting the odds.