Physics 132-1 Test 1

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Questions (8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. We claim that matter consists of atoms and molecules. What evidence can you cite to support that claim?

2. A series of measurements of the pressure-temperature $P-T$ relationship of a gas at constant volume are made and found to lie along a line. Extrapolating to the point where $P=0$ yields the value of absolute zero. The average and standard deviation of the series is $-295 \pm 21 ^\circ \rm C$. Is this result consistent with the expected value of -273? Explain.

3. Consider the following results from the kinetic theory
   
   $$\langle E_{kin}\rangle = {3 \over 2}k_B T \qquad {\rm and} \qquad E_{int} = N \langle E_{kin}\rangle$$
   
   
   where $N$ is the number of particles in a gas, $T$ is the temperature, and $\langle E_{kin}\rangle$ is the average energy for a particle in the gas. Starting with these results show the following. Clearly show all your steps.
   
   $$\Delta E_{int} = {3 \over 2} N k_B \Delta T$$
   
   

4. Consider a gas in a container. Would it violate Newton's Laws or any other physical law if all the particles in the gas collided in such a way that all of the gas particles ended up in the bottom half of the container leaving the top half empty? Is such a scenario likely? Explain.

5. The plot below has the results of a calculation of the multiplicities of an Einstein solid with $N_A = 90$, $N_B=80$, and $q=25$. What is the width of the probability distribution? What is the most probable energy in terms of $\epsilon$, the energy quantum? Show your reasoning for full credit.

   

6. The force exerted by a gas on the walls of a container is constant in time (think about a balloon). Someone challenging the atomic picture of gases claims that you should see fluctuations in the pressure exerted on the walls like the fluctuations in the motion of small particles observed in Brownian motion. How would you respond?

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1.

15 pts.

A block of copper of mass $m = 2.0~kg$ at $T_0 = 20^\circ \rm C$ is dropped into a large vessel of liquid nitrogen at $T_1 = 77.3 \rm ~K$. How many kilograms of nitrogen boil away before the copper reaches the temperature $T_1$?

2.	17 pts.	In a period $\Delta t = 2.0~s$, $N = 4.0\times 10^{23}$ molecules of nitrogen gas strike a wall of area $A = 10.0~cm^2$. If the molecules move with a speed $v=350~m/s$ and strike the wall head on in elastic collisions, then what is the average pressure exerted on the wall? Get an equation for the pressure $P$ before you start inserting numbers. The mass of a nitrogen molecule is $m = 4.68\times 10^{-26}~kg$.
3.	20 pts.	A certain macropartition (call it microstate 1) of two Einstein solids has an entropy of $180k_b$. The next macropartition closer to the most probable one has an entropy of $205k_b$ (call it microstate 2). If the system is initially in microstate 1 and we check it again later, Will the system be more likely to be in microstate 1 or 2? By what factor? In other words, what is the ratio of the probabilities for being in microstates 1 and 2?

Physics 132-2 Equations and Constants

$T_{boiling}$ ($\rm N_2$)	$77~K$	$T_{freezing}$ ($\rm N_2$)	$63~K$
$T_{boiling}$ (water)	$373~K$ or $100^\circ\rm C$	$T_{freezing}$ (water)	$273~K$ or $0^\circ\rm C$
$L_v$(water)	$2.26\times 10^6~J/kg$	$L_f$ (water)	$3.33\times 10^5~J/kg$
$L_v$($\rm N_2$)	$2.01\times 10^5~J/kg$	$c$ (copper)	$3.87\times 10^2~J/kg-^\circ \rm C$
$c$ (water)	$4.19\times 10^3~J/kg-K$	$c$ (steam)	$0.69~J/kg-K$
$\rho$ (water)	$1.0\times 10^3 kg/m^3$	$P_{atm}$	$1.05\times 10^5 ~N/m^2$
$k_B$	$1.38\times 10^{-23}~J/K$	proton/neutron mass	$1.67\times 10^{-27}~kg$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
$0~K$	$\rm -273^\circ~C$	$1 ~ u$	$1.67\times 10^{-27}~kg$
Gravitation constant	$6.67 \times 10^{-11}~N-m^2/kg^2$	Earth's radius	$6.37\times 10^6~m$

$$\vec F = m \vec a = {d\vec p \over dt} \qquad KE = {1 \over ... ...E_1 \qquad \vec p = m \vec v \qquad \vec p_0 = \vec p_1 \qquad$$

$$Q = C\Delta T = cm\Delta T = n C_v \Delta T \qquad Q_{f,v} = m L_{f,v}$$

$$\Delta E_{int} = Q+W \qquad W = {\rm force} \times {\rm dist... ...\langle \vec F \rangle = {\Delta \vec p \over \Delta t} \qquad$$

$$P = {\vert\vec F\vert \over A} \qquad PV = Nk_B T = nRT \qquad$$

$$\vec I = \int \vec F dt = \langle \vec F \rangle \Delta t ... ...\quad E_{int} = N ~ \langle E_{kin} \rangle = {3\over 2} Nk_BT$$

$$v_{rms} = \sqrt{\overline {v^2} } \quad C_{V} = {f\over 2} N... ...2} Nk_BT \quad f \equiv {\rm number\ of\ degrees\ of\ freedom}$$

$$E_{atom} = (n_x + n_y + n_z + \frac{3}{2}) \hbar \omega_0 \q... ...ar \omega_0 \qquad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!}$$

$$S = k_B\ln \Omega \quad \frac{1}{T} = \frac{dS}{dE} \quad ... ...T} \quad E = 3Nk_B T \quad \frac{d \ln x}{dx} = \frac{1}{x}$$

$$\langle x\rangle = \frac{1}{N}\sum_i x_{i} \quad \sigma = \s... ...} \quad A = 4\pi r^2 \quad V = Ah \quad V = {4\over 3} \pi r^3$$

$$\frac{d f(x)}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x+... ...elta x} \quad \frac{df(y)}{dx} = \frac{df(y)}{dy}\frac{dy}{dx}$$