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Unformatted text preview: Department of Physics, University of California, Berkeley
Physics 7B, Lecture 001, Spring 2010 (Xiaosheng Huang)
Midterm 1
Monday, February 22, 2010
6:00— 8:00 PM
Total Points: 100 + 10 Note: You are allowed one formula card (3.5 ” by 5 ”, double sided). N0 calculators or any other electronir
devices are permitted. 1. (20 pts) The volume coefﬁcient 5 is deﬁned as 5 = (l/V)(dV/dT). Calculate B as/a function 0:
temperature for the air in a hot balloon. Treat the air as an ideal gas. Assume that the air in the balloor
does not escape and the heating process is slow enough to be considered quasistatic. 2. (30 pts) In the summer an air conditioner keeps the temperature of a house at T,.. Assume the outside
temperature is T1 > T7. The temperature of the warm side of the air conditioner is Ta > T1. Assume the
air conditioner has perfrect Carnot efﬁciency. /\\ Air
Conditioner a) Suppose the air at a distance d from the air conditioner cools to the ambient temperature, T1 (see ﬁgure),
and the surface area of the warm side of the air conditioner is A. The conductivity of air is k. You may
assume heat conduction only takes place in the acdirection (In reality there will be heat conductin in other
directions. We are simply trying to make a reasonable estimate in this problem.) Calculate the amount oi
heat dumped outside per unit time, dQ/dt. b) Find the amount of power that needs to be supplied to the air conditioner. \ e) In the winter, if the temperature of the room is still at TT and the outside temperature is T 2 < TT ﬁnd
the heat 1033 rate an’ /IH fnr flu: Emma: 3. (15 pts) The osmium» Vi panoramas, GP, for a refridgerator is deﬁned as CP = QL/W. During
one cycle of the operation of a real refridgerator, IQ Ll amount of heat is absorbed from a low temperature
reservoir at TL and QH amount of heat is released to a high temperature reservoir at TH. Its CP is only a
fraction f (< l) of CPideal for a Carnot refridgerator. Find the total entropy change at the end of the cycle
in terms of Q E, TH/and J". You may assume C23 is positive and don’t have to write the absolute signs in
your calculation. (To) \\.,,.... \’ . 27‘; , . i q 1.:
l‘v’?" " " Av x
4. (25 pts) An nmole sample of an ideal diatomic gas at a pressure of P1 and temperature of T1 undergoes
a process in which its pressure increases linearly with temperature. The ﬁnal temperature and pressure are T2 and P2. (Hint: 15 volume necessarily a constant in this process?) ' ‘  l  S; n P. t i  1 j
a) Determine the change in internal energy. (Assume ﬁve actlve degrees of freedom.) 46M ' 7 “ ‘ ' b) Determine the work done by the gas. W‘ AEW * G 1 1 WV 3 I.“ M “ ““TL (1') i c) Determine the heat added to the gas. 5. (10 pts + 10 Bonus pts) Light can sometimes be treated as particles, called photons. Photons can form
a “gas”. The energy density of a photon gas is given by g = Ak4BT4 where A is a constant. This is closely related to the StefanBoltzmarm equation in the textbook. a) (10 pts) Find dS in terms of T and V for the photon gas by using the ﬁrst law of thermodynamics and
considering the isochoric process. b) (Bonus: 5 pts) Integrate the expression for d5 and ﬁnd 5' as a function of T. You may set the integration
constant to zero. c) (Bonus: 5 pts) Find the condition that V and T have to satisfy if a photon gas expands adiabaticaliy (or
isentropically). Suppose that the expansion of the universe is an adiabatic process and that we can ignore
all other contents of the universe except photons. Given that the temperature of the cosmic microwave
background radiation today is approximately 3 K, what was the size of the universe when the photon
temperature was at 3000 K, compared to its size today? If we further assume that the size of the universe,
a7 expands as a(t) = Bt, where B is a constant and t is time, at what fraction of the present age of the
universe was its temperature at 3000 K? The End ...
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 Spring '08
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