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University of California, Berkeley
Physics 7B, Lecture 001, Spring 2009 (*Xiaosheng Huang*)

**Midterm 1
**

Monday, 2/23/2009 6:00-8:00 PM

Name:______________________

SID: ________________________

D/L Section:_______________

GSI: ________________________

**
Physical Constants:
**Avogadro's number, *NA*: 6.02×1023
Gas Constant, *R*: 8.315J/mol·K
Boltmann's Constant, *kB*: 1.38×10-23 J/K
Stefan-Boltzmann Constant, σ: 5.67×10-8 W/m2·K4
Specific heat for water: *c*=4.19×103 J/kg·oC
Heat of vaporization for water: *LV*=22.6×105 J/kg
Heat of fusion for water: *LF*= 3.33×105 J/kg
Standard Temperature and Pressure (STP): *T*=273K, *P*=1atm=1.01×105 Pa
Atomic mass unit (1u): 1.6605×10-27 kg
*Note: You are allowed one formula sheet (3½ by 5, double sided) and a calculator
(without wireless capabilities). Do NOT just write down an answer in the answer box;
show your steps. Formulaic answers may only involve the quantities given in a problem
and constants. Good Luck!
*

#1 #2 #3 #4

Total

1. (25 pts.) A spherical solid aluminum sphere with mass *M*, density ρ, specific heat *c*,
and emissivity *e*, floats in the vacuum of intergalactic space. The sphere is initially (*t*=0)
at temperature *T1*.
*a)* How long does it take for the aluminum sphere to cool to the cosmic microwave
background (CMB) temperature, *T0*, assuming *T1*>*T0*? (For this part, ignore the radiation
energy that the aluminum sphere will absorb from the (CMB)).
Formulaic Answer:
If
*T1*
=
300K
and
*T0*
=
2.7K
at
*t*=0
and
the
mass
of
the
aluminum
sphere
is
*M*=
100
kg,
find
the
numerical
value
for
the
cooling
time.
For
aluminum,
*e*
=
0.02,
ρ
= 2.7×103
kg/m3,
and
*c*=
0.90
×103
J/kg•K.
Numerical Answer:

For parts *b*) and *c*), take into consideration that the aluminum sphere will absorb radiation
energy from the CMB.
*b*) If the temperature of the aluminum sphere at time *t* is *T*, what is the net heat loss
(|*Qnet loss*|) of the aluminum sphere at this point?
Formulaic Answer:
|*Qnet loss*| =
*c*) Find a differential equation whose solution is *T*(*t*), the temperature of the aluminum
sphere as a function of time, *t.*
Formulaic Answer:

2. (25 pts.) The mint flavor comes from menthol, which has a molecular mass of
*M*=156.27, in atomic units. A bottle of mint oil at thermal equilibrium with the air in a
room at STP is opened at one end of the room at *t*=0. Assume the menthol molecule is
spherical in shape with a radius of *R*=0.5nm. Also assume the air in the room consists
only of *N*2 (ignore the *O*2 part) and treat the *N*2 as a spherical molecule with radius
*r*=0.3nm. Ignore all intermolecular interactions except elastic collisions.
*a*) Find the number density (number of molecules per unit volume, *N*/*V*) of the *N2* in the
room.
Numerical Answer:
*b*) Find the rms average speed, *vrms*, for menthol molecule.
Formulaic Answer:

*c*) Derive an expression for the mean free path *l *for the menthol molecule in terms of the
quantities given in the problem and *N*/*V*. (You don’t need to worry about factors of

€

2 .)
Formulaic Answer:
*
d*) Given *x*2=*sl*2, where *x* is the displacement, *l* the mean free path, and *s* the number of
collisions the molecule has suffered traveling through *x*, find the time, *t*, it takes for a
menthol molecule to reach a displacement of *x* on average.
Formulaic Answer:

*e*) Suppose the room is *x*=20m across, find the numerical value for the average time, *t*, it
takes for a menthol molecule to go from one end of the room to the other.
Numerical Answer:
*f*) (1 bonus pt.) Does your answer agree with what you know from experience? If not,
why not?

3. (25 pts.) At a steam power plant, steam engines work in pairs, the heat output of the
first one being the approximate heat input of the second. The operating temperatures of
the first are *T1*= 800 0C and *T2 *= 410 0C, and of the second *T3* = 400 0C and *T4* = 250 0C.
*a*) If the heat of combustion of coal is *b *= 3.0×107 J/kg, at what rate must coal be burned
if the plant is to put out *p*=1000 MW of power? Assume the efficiency of the engines is *f*
= 50% of the ideal (Carnot) efficiency.
Formulaic Answer
Numerical Answer

*b*) Water is used to cool the power plant. How much heat per second (Δ*Qw*/Δ*t*) does the
water have to absorb?
Formulaic Answer
Numerical Answer

*c*) If the water temperature is allowed to increase by no more than Δ*T* = 6 0C, estimate
how much water must pass through the plant per second. You may use Δ*Qw*/Δ*t* in your
formulaic answer.
Formulaic Answer
Numerical Answer:

4. (25 pts.) An ideal gas in a thermally insulated box is separated by a thermally
conducting partition into two parts. There are *n* moles of gas in each part. Initially, the
gas in part A has temperature *T1* and volume *V1* and in B temperature *T2* and volume *V2*.
The partition can slide without friction and the two parts have the same pressure, *P*.
*
a*) What is the final temperature when thermal equilibrium is reached?
Formulaic Answer
*b*) Calculate the entropy change for the gas in A and for the gas in B. (Please specify the
reversible path that you choose for this calculation.)
Formulaic Answer

*c)* Calculate the entropy change of the combined system of A and B, Δ*S*total.
Formulaic Answer
*d*) (4 bonus pts.) Show that Δ*S*total ≥ 0.