PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2001
Quiz 1 Tuesday, February 13, 2001
NAME: ____________________________________________
To receive credit for a problem, you must show your work, or explain how you arrived at your answer.
1. (15%) Suppose you get involved in a game in which six coins are tossed simultaneously. The object of the game is not to guess how many heads, or tails will show up in a toss. Instead, you are to guess what the distribution will be; i.e. will all six be the same, or will there be one of one type and five of the other, or two of one type and four of the other, or three of each. In such a game, what distribution has the greatest probability of occuring, and what is that probability?
Most probable distribution: __________________
Probability = __________________
2. (10%) Consider the expression
 yx e x

3

 x e dy . Is it an exact differential?
1

3. (30%) Consider a particle that is constrained to move in one dimension. Suppose this particle executes a random walk. Every 10
-6
s, it moves either exactly 1 nm to the right or exactly 1 nm to the left. a.) (10%) If the probability that the particle moves to the right is 0.6, what is the average distance the particle will have moved from its starting point after 10 s.
S = __________________ b.) (10%) If the probability that the particle moves to the right is 0.5 instead, what will the average displacement from the starting point be after 10 s.
S = __________________ c.) (10%) How long must you wait if you want the probability that the particle is more than 1 cm from its starting point to be greater than 32%. In this part, you should again assume that the probability that the particle moves to the right is 0.5. t = __________________
2

4. (45%) One mole of an ideal gas expands isothermally from point i to point f on the graph shown. At point i the volume of the gas is V
0
, and the pressure is p
0
. At point f the volume is
3 V
0
. (For one mole of an ideal gas, pV = RT .) Express all your answers in terms of p
0
, V
0
, and
R . a) (8%) How much work is done by the gas as it expands isothermally from V
0
to 3 V
0
?
W = ________________ b) (5%) What is the change in internal energy as the gas expands isothermally from V
0
to 3 V
0
?

E = ________________ c) (7%) How much heat is added to the gas as it expands isothermally from V
0
to 3 V
0
?
Q = ________________
3

d) (5%) Based on your answer to c), calculate the change in entropy of the gas as it expands isothermally from V
0
to 3 V
0
(Hint: Q = T

S )

S = ________________ e) (10%) The number of states available to the system increases as the gas expands isothermally from V
0 that available at V
0
to 3 V
0
? What is the ratio of the number of states available at 3 V
. (Hint:

= V
N
E

/2
)
0
to

3 V
0
/

V
0
= ________________ f) (10%) Based on your answer to e), calculate the change in entropy of the gas as it expands isothermally from V
0
to 3 V
0
?

S = ________________
4