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?

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 six heads?

(II) Calculate the probabilities, when you throw two dice, of obtaining a 7.

(II) Calculate the probabilities, when you throw two dice, of obtaining an 11.

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. Construct a table showing the number of microstates that correspond to each macrostate.

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. Construct a table showing the number of microstates that correspond to each macrostate. Then, determine the probability of all 3 beans red.

Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

Use Eq. 20–14 to determine the entropy of each of the five macrostates listed in Table 20–1 on page 595.

Consider an isolated gas-like system consisting of a box that contains N = 10 distinguishable atoms, each moving at the same speed v. The number of unique ways that these atoms can be arranged so that N_L atoms are within the left-hand half of the box and N_R atoms are within the right-hand half of the box is given by N! / N_L!N_R!, where, for example, the factorial 4! = 4•3•2•1 (the only exception is that 0! = 1). Define each unique arrangement of atoms within the box to be a microstate of this system. Now imagine the following two possible macrostates: state A where all of the atoms are within the left-hand half of the box and none are within the right-hand half; and state B where the distribution is uniform (that is, there is the same number in each half). See Fig. 20–20. Assume the system is initially in state A and, at a later time, is found to be in state B. Determine the system’s change in entropy. Can this process occur naturally?