Nash - Elements of Statistical Thermodynamics.

Plancks constant, h = 6.626 10 27 erg-sec

Boltzmann constant, k = 1.381 10 16 erg/K

Avogadros number, N = 6.023 1023 mole 1

Gas-law constant, R = 8.314 107 ergs/mole-K

= 1.987 cal/mole-K

The game of checkers is played on an array of 32 squares distinguished by their geometric placement. Consider a game that has reached a stage at which Black has 4 men and 4 kings while Red also has 4 men and 4 kings. Noting that the different kinds of counters (e.g., red man and red king, or red king and black king) are readily distinguishable from each other, but that counters of the same sort (e.g., the 4 red men) are not thus distinguishable, determine the number of recognizably different positions possible at this stage of the game.

a) Show that, in five-card stud poker, the number of possible five-card hands is

b) Consider the probability that, in a game of five-card stud, you will be dealt a flush (i.e., five cards of the same suit). By showing that the number of ways in which you may be dealt a flush is

demonstrate that your chance of being dealt a flush is of the order of .

a) In draw poker there is about a 49% chance that a randomly dealt five-card hand will contain at least two cards of the same denomination. This conclusion is perhaps most readily demonstrated by showing that the probability that the hand does not contain two cards of the same denomination is given by the expression

Justify this expression, and complete the indicated demonstration.

b) In a class of 50 randomly selected students, what is the probability that two will have the same birthday? This surprisingly large probability is easily obtained by adapting the development in part (a) to calculate the probability that two students will not have the same birthday.

[After Sussman, J. Chem. Educ. , 49 (1963).] a) In the neck of the sealed flask shown below, 5 white balls rest on top of 5 black balls. Suppose that the balls are tipped into the body of the flask, and that the flask is well shaken and then returned to the inverted position shown here. Demonstrate that the probability that the indicated arrangement will again turn up is

b) What is the physical significance of the function 10!/5! 5!?

c) Determine the number of distinguishable arrangements that would be possible if all ten balls were differently colored.

a) The total number of miscrostates () associated with all configurations possible when N distinguishable harmonic oscillators share Q energy quanta is:

To derive this formula, we symbolize each of the oscillators by an o , each of the quanta by a u . At the left end of a line we set down one o , and then consider in how many ways we can follow the one o , on the same line, with the remaining (N 1) o s and the Q u s. Conceiving each o as possessing all u s appearing to its right prior to the appearance of the next o , we see that the total number of permutations possible will also represent the total number of possible microstates (). Justify this statement, and derive the above formula.

b) Use the formula to confirm the statement made in the text that 2002 distinct microstates are possible when 10 oscillators share 5 energy quanta.

c) Use the formula, and Stirlings approximation, to confirm that 1000 oscillators sharing 1000 energy quanta can assume a total in the order of 10600 microstates.

a) Justify the claim that the formula W = N !/( H !)( T !) yields the number of ways ( W ) in which some particular numbers of heads ( H ) and tails ( T ) can be produced in a series of N tosses of a well-balanced coin.

b) For a series of 1000 tosses of a coin, we define A = W H T /W 500 500, where, W500500 is the number of microstates associated with the predominant configuration, and W H T is the number of microstates associated with the configuration characterized by some particular numbers of heads ( H ) and tails ( T ). Show that log A = 1000 log 500 H log H T log T .

c) Using the formula noted in (b), show that a 400-600 configuration possesses only about 10 9 as many microstates as the predominant 500-500 configuration.

d) Amending the formula in (b) as required for a series of 100 tosses, show that the 40-60 configuration has (as indicated in ) approximately