23. The Second Law of Thermodynamics

The macrostate of a set of coins is given by the number of coins that are heads-up. If you have 100 coins, initially with 20 heads-up, what is ΔS when the system is changed to have 50 heads-up? Note that the multiplicity of k coins which are heads-up, out of N total coins, is $\Omega=\frac{N!}{k!\left(N-k\right)!}$. Does this change in macrostate satisfy the second law of thermodynamics?

A lonely party balloon with a volume of $2.40$ L and containing $0.100$ mol of air is left behind to drift in the temporarily uninhabited and depressurized International Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is $425$ m³. Calculate the entropy change of the air during the expansion.

A box is separated by a partition into two parts of equal volume. The left side of the box contains $500$ molecules of nitrogen gas; the right side contains $100$ molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature. On average, how many molecules of each type will there be in either half of the box?

Your calculator can't handle enormous exponents, but we can make sense of large powers of e by converting them to large powers of 10. If we write e = 10^α, then e^β = (10^α)^β = 10^αβ. What is the multiplicity of a macrostate with entropy S = 1.0 J/K? Give your answer as a power of 10.

Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining three heads and three tails?