PHYSICS 313
Problem set 3 2001
Given September 24
Due Oct 1

Problem 1:
The total energy of two Einstein solids in thermal contact is such that $q=2000$ vibrational quanta are excited. Each solid contains 1000 atoms. What is the probability that each solid have exactly 1000 vibrational quanta?
Use Stirling's formula on the form

$$n!=n^ne^{-n}\sqrt{2\pi n}$$

Problem 2:
An Einstein solid contains 1 mole of atoms and $N_A=6.022\; 10^{23}$ quanta are excited. The energy of each quantum is such that $\hbar\omega/k_B=100$ K
a:
What is the entropy of the solid (in units of J K$^{-1}$)
b:
What is the temperature of the solid in K?
For this problem the simplified version of Stirling's formula

$$\ln n!\approx n\ln n -n$$

is adequate.
Problem 3:
One million monkeys have been typing away at the rate of 1 character per second hitting random keys since the beginning of the universe approximately $10^{17}$ s ago. Is it likely or unlikely that at least one of the monkeys would have succeeded in typing out this problem set? Assume that the keyboard of the monkey's typewriters have 40 characters (neglecting the difference between upper and lower case and formatting characters).