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.

(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.

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?

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.