PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS
Quiz 1
SPRING 2002
Friday, February 22, 2002
NAME: ____________________________________________
To receive credit for a problem, you must show your work, or explain how you arrived at your
answer.
1. (25%) A small time gambler is running a game on a street corner. He tosses five coins at
once, and bettors can bet on how many heads will appear. Anyone who bets on zero or five,
and wins, is paid $30 for each dollar bet. Those who win with one or four are paid $6 for
each dollar bet, and they get $3 for each dollar bet if they win with two or three heads.
a) (10%) Find the probabilities that 0, 1, 2, 3, 4, and 5 heads will appear when the coins are
tossed.
b) (5%) Suppose someone played the game, and always bet that three heads would show up.
Based on the probability that you calculated, and the payoff offered for winning (3 to 1),
how would this person do financially, after playing for a long time, and making many
bets? Circle the correct answer, and support your choice with some numerical evidence.
WIN
LOSE
BREAK EVEN
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c) (5%) A street-smart character plans to improve his odds by switching the coins used in
the game. He has a set of five coins, each of which comes up heads 60% (0.60) of the
time. What is the probability that three heads will come up when these five loaded coins
are tossed?
d) (5%) Based on the probability that you just calculated, and the payoff offered for
winning, how would this character do after playing for a long time with the loaded coins,
and always betting on three heads? Circle the correct answer, and support your choice
with some numerical evidence.
WIN
LOSE
BREAK EVEN
2. (10%) Consider the expression, xy2 cos( 2 x 2 )dx  y sin( 2 x 2 )dy . Is it an exact differential?
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3. (25%) Energy produced in the center of the sun has a hard time finding its way out. On the
average, in the sun, a photon will go a distance of s0 = 1.0 cm (0.010 m) between collisions
with hydrogen atoms, and on the average, it is held about 10-8 s by the hydrogen atom before
being reemitted in a completely random direction. Hence, the photon takes about 108 “steps”
per second.
a) (10%) What is the average distance traveled in any one dimension per step. (You do not
have to do any calculations if you know the answer and can give a short explanation.)
b) (10%) The radius of the sun is about 7.0 ×108 m. How many “steps” must a photon take
before having a 32% chance of being beyond the edge of the sun in any one dimension?
(Hint: In a homework problem you found that the square of the standard deviation about
s2
the value given in part a) for any one step is given by (sz ) 2  0 ,where s0 is the average
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step size.)
c)
(5%) At the rate of 108 steps per second, how many years are needed to take the number
of steps calculated in part b)? (1 year = 3.16 ×107 s)
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4. (40%) One mole of an ideal gas is taken through the process shown on the graph. It starts at
point 1, with pressure p0 and volume V0. It expands, at constant pressure, to point 2, where its
volume is 2V0. Then it is compressed isothermally to point 3, where its pressure is 2p0 and its
volume is again V0. (For one mole of an ideal gas, pV = RT.) Express all answers in terms of
p0, V0, and R.
a) (5%) The internal energy of the gas at point 1 is E  32 p0V0 . What is the internal energy of
the gas at point 3?
b) (5%) How much work is done by the gas as it expands from 1 to 2?
c) (10%) How much work is done by the gas as it is compressed from 2 to 3?
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d) (10%) How much heat is added to the gas as it goes from 1 to 3?
e) (10%) Find the change in entropy of the gas as it goes from 1 to 3.
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