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%TCIDATA{Created=Tue Sep 23 17:59:11 1997}
%TCIDATA{LastRevised=Mon Nov 03 10:33:54 1997}

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\begin{document}

\SetTitle{Phy206 Exam II}
\SetAuthor{T. Curtright}
\Setdate{3 November 1997}
\TitlePage{This is a ``closed book'', multiple-choice exam. You are NOT allowed to use
any notes. \emph{Circle only one answer} out of the five choices for each
problem. Do ALL problems.\\
Sometimes, usually by accident, none of the listed answers is correct. If
you firmly believe that this is the case, then you should \emph{write-in}
what you believe to be \emph{the correct answer}.\\
You may ask the instructor questions, especially if you think a problem is
not clearly stated.\\
\bigskip \bigskip Good luck!\\
\textbf{\ \bigskip \bigskip ``On my honor, I have neither given nor received
any aid on this examination.''}\\
\bigskip
Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\
\bigskip ID \#:
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}

NOTE: This is the exam key. The answer (a) is correct for each question.

\smallskip \eject 

[1] A bullet of mass $m=0.040\;kg$, moving with a speed $v$ (in meters/sec),
lands in a thermally insulated bucket of water of mass $M=4.0\;kg.$ No water
splashes out of the bucket. After the bullet comes to rest, and the waves in
the water dissipate, it is observed that the water temperature has increased
by $\Delta T=0.19\;Kelvin$. Neglecting the heat capacity of the bullet and
the bucket, but not of the water, what was the initial speed $v$ of the
bullet?

(a) $4.0\times 10^{2}$meters/sec

(b) $2.0\times 10^{2}$meters/sec

(c) $4.0\times 10^{4}$meters/sec

(d) $16.\times 10^{4}$meters/sec

(e) $\pi \times 10^{3}$meters/sec

\ 

[2] Nitrogen gas consists of ``rigid'' diatomic molecules and has mass $%
M=28.0\times 10^{-3}\;kg/mole.$ What is the ratio $v_{sound}/v_{rms}$ at
temperature $T=300\;Kelvin$? Here $v_{sound}$ is the speed of (adiabatic)
sound waves in the nitrogen gas and $v_{rms}$ is the root-mean-square speed (%
$v_{rms}=\sqrt{\overline{v^{2}}}$) of the randomly moving nitrogen molecules.

(a) $\sqrt{7/15}=\allowbreak .\,68$

(b) $\sqrt{5/9}=\allowbreak .\,75$

(c) $\sqrt{3/3}=\allowbreak 1.0$

(d) $\sqrt{3/6}=\allowbreak .\,71$

(e) $\sqrt{6/3}=\allowbreak 1.\,4$

\ 

[3] A body of mass $m$ and specific heat $c$ initially ($t=0$) at a
temperature $T_{i}$ cools by convection and radiation in a room where the
temperature is $T_{0},$with $T_{i}>T_{0}$. The body obeys Newton's law of
cooling, given by $dQ/dt=-hA(T-T_{0})$, where $A$ is the surface area of the
body and $h$ is a constant (the ``surface coefficient of heat transfer'').
What is the temperature of the body at any later time?

(a) $T(t)=T_{0}+(T_{i}-T_{0})\exp (-\frac{hAt}{mc})$

(b) $T(t)=T_{0}+(T_{i}-T_{0})\exp (\frac{hAt}{mc})$

(c) $T(t)=T_{i}+T_{0}\frac{hAt}{mc}$

(d) $T(t)=T_{0}+(T_{i}-T_{0})\frac{mc}{hAt}$

(e) $T(t)=(T_{i}-T_{0})+T_{0}\exp (-\frac{hAt}{mc})$

\ 

[4] Suppose one mole of a not-quite-ideal gas is described by the Clausius
equation of state: $p(V-V_{0})=RT$. Here $V_{0}$ is a fixed volume
representing the constant size of all the molecules in the gas. What is the
work done by such a gas if it expands isothermally from volume $V_{1}$ to
volume $V_{2}$?

(a) $\Delta W=RT\,\ln \left( \frac{V_{2}-V_{0}}{V_{1}-V_{0}}\right) $

(b) $\Delta W=RT\,\ln \left( \frac{V_{2}}{V_{1}}\right) $

(c) $\Delta W=RT\,\left( V_{2}/V_{0}-V_{1}/V_{0}\right) $

(d) $\Delta W=RT\,(V_{2}/V_{1})$

(e) $0$

\eject 

[5] A crystal can be thought of as a collection of $N$ identical atoms
arranged in a cubical lattice with all the atoms connected with their
neighbors in the $x,\;y,$ and $z$ directions by ideal (Hooke's law) springs.
Using the equipartition theorem, what is the constant volume heat capacity
of the crystal?

(a) $3Nk_{Boltzmann}$

(b) $4Nk_{Boltzmann}$

(c) $6Nk_{Boltzmann}$

(d) $\frac{5}{2}Nk_{Boltzmann}$

(e) $\frac{3}{2}Nk_{Boltzmann}$\bigskip 

\ 

[6] Suppose one mole of an ideal monatomic gas undergoes reversible pressure
and temperature changes, as shown in Figure 1, such that $pV=RT$ always
holds. What is the change in entropy for the gas for the complete process $%
A\rightarrow B\rightarrow C\,$?

(a) $\Delta S=\frac{3}{2}\,R\,\ln 2$

(b) $\Delta S=\frac{5}{2}\,R\,\ln 2$

(c) $\Delta S=\frac{1}{2}\,R\,\ln 2$

(d) $\Delta S=R\,\ln 2$

(e) $\Delta S=\frac{5}{2}\,R\,$

\ 

[7] One mole of an ideal monatomic gas follows the cycle shown in Figure 2.
What is the net amount of heat exchanged between the gas and all external
heat reservoirs during each cycle?

(a) $(301\;Kelvin)\,R\,(-1+2\ln 2)$

(b) $(301\;Kelvin)\,R\,(\frac{3}{2}+2\ln 2)$

(c) $(301\;Kelvin)\,R\,(2\ln 2)$

(d) $(301\;Kelvin)\,\,(\frac{3R}{2})$

(e) $(602\;Kelvin)\,R\,(\frac{3}{2}+2\ln 2)$

\ 

[8] One mole of an ideal monatomic gas follows the cycle shown in Figure 2.
What is the amount of heat flowing \emph{into} the gas during each cycle?

(a) $7220\;Joules$

(b) $3750\;Joules$

(c) $3470\;Joules$

(d) $6250\;Joules$

(e) $970\;Joules$

\ 

\eject 

[9] One mole of an ideal monatomic gas, at an initial volume $%
V_{1}=25.0\;liters$, follows the cycle shown in Figure 3. All processes are
reversible, with the system always described by the ideal gas law. What is
the efficiency of an engine which uses this cycle?

(a) $15\%$

(b) $25\%$

(c) $35\%$

(d) $10\%$

(e) None of the other answers is correct.

\ 

[10] Assume the motion of the atmosphere is an adiabatic process of an ideal
gas, and that the variation of pressure with height is given by $%
dp/dh=-g\,\rho (h)\,$, where $\rho (h)$ is the mass density of the
atmosphere at height $h$, and where the gravitational acceleration is the
constant $g=9.8$ meters/sec$^{2}$. If the temperature at ground level is $%
330\;Kelvin$, and the temperature at a height of $100$ meters is $%
329\;Kelvin $, what is the temperature at a height of $800$ meters?

(a) $322\;Kelvin$

(b) $321\;Kelvin$

(c) $320\;Kelvin$

(d) $328\;Kelvin$

(e) There is not enough information given to determine the temperature.

\ 

[11] An automobile fuel tank is filled to the brim with $50\;liters$ of
gasoline at $20{{}^\circ}C$, and the cap is left off of the tank opening. 
Immediately thereafter, the car is parked in the sun where the temperature
is $35{{}^\circ}C.$ Neglecting the expansion of the metal tank, but not of
the gasoline, how much fuel will overflow from the tank as a result of
thermal expansion? (Note the coefficient of thermal expansion for gasoline
is $\beta =9.6\times 10^{-4}/{{}^\circ}C$.)

(a) $\allowbreak 0.\,72\;liters$

(b)$\allowbreak $ $1.\,1\;liters$

(c) $\allowbreak 3.\,2\;liters$

(d) $\allowbreak 2.\,2\;liters$

(e) None of the other answers is correct.

\ 

[12] At a depth $h=25.0\;meters$ below the surface of the sea (water density 
$=\rho =1025\;kg/m^{3}$), where the temperature is $T_{C}=5.00{{}^\circ}C$,
a diver exhales an air bubble having a volume of $V_{0}=1.00\;cm^{3}$. If
the surface temperature of the sea is $T_{H}=20.0{{}^\circ}C$, what is the
volume of the air bubble right before it breaks the surface, where $%
P=1\;atm=1.01\times 10^{5}N/m^{2}$? (Neglect any absorption of the gas in
the bubble by the surrounding water.)

(a) $3.67\;cm^{3}$

(b) $\allowbreak 3.\,49\;cm^{3}$

(c) $\allowbreak 1.\,05\;cm^{3}$

(d) $13.\,9\;cm^{3}$

(e) None of the other answers is correct.

\ 

\eject 

[13] One mole of an ideal gas is enclosed in a cylinder that has a movable
piston on top. The piston has a mass $m$ and an area $A$ and is free to
slide up and down, keeping the pressure of the gas constant. How much work
is done by the gas as the temperature is slowly raised from $T_{1}$ to $T_{2}
$?

(a) $\Delta W=R\,(T_{2}-T_{1})$

(b) $\Delta W=R(T_{1}-T_{2})$

(c) $\Delta W=R\,T_{1}\left( 1+\ln (T_{2}/T_{1})\right) $

(d) $\Delta W=R\,T_{2}\left( 1+\ln (T_{1}/T_{2})\right) $

(e) $0$

\ 

[14] During a carefully controlled expansion, the pressure of a gas is given
by $P=P_{0}\exp (1-V/V_{0})$. How much work is done by the gas as it expands
from $V_{1}$ to $V_{2}$?

(a) $\Delta W=P_{0}V_{0}\left( \exp (1-V_{1}/V_{0})-\exp
(1-V_{2}/V_{0})\right) $

(b) $\Delta W=P_{0}V_{0}$

(c) $\Delta W=P_{0}V_{0}\exp (V_{2}/V_{0}-V_{1}/V_{0})$

(d) $\Delta W=P_{0}V_{0}\exp (V_{2}/V_{1})$

(e)$\Delta W=P_{0}V_{0}\exp (1-V_{1}/V_{2})$

\ 

[15] In a time $t$, $N$ hailstones strike a flat glass window of area $A$ at
an angle $\theta $ (measured from the plane of the glass so that vertically
falling hailstones which ``miss the glass'' would have $\theta =0$). Each
hailstone has a mass $m$ and a speed $v$. If the collisions are elastic
(i.e. no energy is lost), what is the average pressure on the window?

(a) $P=\frac{N}{At}\,2mv\sin \theta $

(b) $P=\frac{N}{At}\,mv\sin \theta $

(c) $P=\frac{N}{At}\,2mv\cos \theta $

(d) $P=\frac{N}{At}\,2mv\tan \theta $

(e) $P=\frac{N}{At}\,mv\tan \theta $

\ 

[16] Assuming the atmosphere is at constant temperature, $T$, and that the
acceleration of gravity is constant, $g$, what fraction of molecules, of
individual mass $m,$ in the gas of the atmosphere lie below a height $h$? ($%
k_{B}$ is Boltzmann's constant)

(a) $1-\exp (-\frac{mgh}{T\,k_{B}})$

(b) $\exp (-\frac{mgh}{T\,k_{B}})$

(c) $1-\exp (\frac{mgh}{T\,k_{B}})$

(d) $\exp (\frac{mgh}{T\,k_{B}})$

(e) $1-\frac{mgh}{T\,k_{B}}$

\ 

\eject 

[17] Two moles of an ideal monatomic gas (with $\gamma =1.67$) expands
slowly and adiabatically from a pressure of $P_{1}=5.00\;atm$ and a volume $%
V_{1}=10.0\;liters$, to a final volume of $V_{2}=20.0\;liters$. What is the
final pressure $P_{2}$ of the gas?

(a) $1.\,57\;atm$

(b) $2.\,50\;atm$

(c) $\allowbreak 1.\,39\;atm$

(d) $15.\,9\;atm$

(e) $10.0\;atm$

\ \ 

[18] A mixture of two gases will ``diffuse'' through a filter at rates
proportional to their rms speeds. What is the ratio of diffusion rates for
two isotopes of chlorine, $\frac{rate(^{35}Cl)}{rate(^{37}Cl)}$ ? Assume the
two isotopes are in thermal equilibrium with each other.

(a) $\allowbreak 1.\,03$

(b) $\allowbreak 1.\,06$

(c) $0\allowbreak .\,973$

(d) $0\allowbreak .\,946$

(e) $1.00$

\ 

[19] A cylinder containing $n$ moles of an ideal gas (with constant volume
molar specific heat $C_{V}\,$) goes through a reversible adiabatic process,
starting with pressure, temperature, and volume $P_{i},\;T_{i},\;V_{i}$ and
ending with pressure, temperature, and volume $P_{f},\;T_{f},\;V_{f}$. What
is the net change in internal energy of the gas during the process?

(a)$\allowbreak $ $\Delta E=nC_{V}(T_{f}-T_{i})$

(b) $\Delta E=\frac{1}{\gamma -1}\,(P_{i}Vi-P_{f}V_{f})$

(c) $\Delta E=\gamma nC_{V}(T_{f}-T_{i})$

(d)$\allowbreak $ $\Delta E=(P_{i}V_{i}-P_{f}V_{f})$

(e) $\allowbreak \Delta E=\frac{\gamma }{\gamma -1}\,(P_{i}V_{i}-P_{f}V_{f})$

\ 

[20] A cylinder containing $n$ moles of an ideal gas (with constant volume
molar specific heat $C_{V}$, and constant pressure molar specific heat $%
C_{P}\,$) goes through an arbitrary reversible process, starting with
pressure, temperature, and volume $P_{i},\;T_{i},\;V_{i}$ and ending with
pressure, temperature, and volume $P_{f},\;T_{f},\;V_{f}$. What is the
change in entropy for the gas during the process?

(a) $\Delta S=nC_{V}\ln (T_{f}/T_{i})+nR\ln (V_{f}/V_{i})$

(b) $\Delta S=nC_{V}\ln (T_{f}/T_{i})$

(c) $\Delta S=nR\ln (V_{f}/V_{i})$

(d)$\;\Delta S=nC_{P}\ln (P_{f}/P_{i})+nR\ln (V_{f}/V_{i})$

(e) None of the other answers is correct.

\ 

\eject

[] An expandable cylinder has its top connected to a spring of constant $k$
as shown in Figure 4. The cylinder is filled with $V_{0}$ liters of an ideal
gas at atmospheric pressure $P_{0}$ and temperature $T_{0}$ (in $Kelvin$),
with the spring relaxed. If the lid has cross-sectional area $A$ and
negligible mass, how high does the lid rise when the temperature is slowly
raised to $T$?

(a) $PV=P(V_{0}+A\Delta x)=TP_{0}V_{0}/T_{0},\;\Delta x=(P-P_{0})A/k=\left( 
\frac{1}{(V_{0}+A\Delta x)}TP_{0}V_{0}/T_{0}-P_{0}\right) A/k$

(b)

(c)

(d)

(e) There is not enough information given to determine the answer.

\end{document}