Physics 132-04 Test 1


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Questions (5 for 8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

  1. Consider a mass of ice that is being heated. What is the relationship between the temperature and the added heat after the ice has melted, but before the water begins to boil?







  2. You pour a mass $m_{p}$ of unknown pellets at a temperature $T_{p}$ into an aluminum calorimeter cup of mass $m_c$. The calorimeter cup contain a mass $m_w$ of water and a small mass $m_i$ of ice. The system comes to equilibrium at a final temperature $T_f$. You can look up the specific heats of the known components of the measurement. Write the full heat equation for the system. Is there enough information to determine the specific heat of the pellets? Explain.








  3. In our development of kinetic theory we claimed that on average $\langle {v_x^2} \rangle = \langle {v_y^2} \rangle = \langle {v_z^2}\rangle$ for a large number of particles in a box. Why?



    DO NOT WRITE BELOW THIS LINE.

  4. The figure shows the entropy of two solids $S_A$ and $S_B$ and their combined entropy $S_{AB}$ as a function of the internal energy in solid A $U_a$. The solids are in thermal contact. At the most probable macrostate we showed that

    \begin{displaymath} \frac{dS_A}{dU_A} = \frac{dS_B}{dU_B} \end{displaymath}

    and we require that $T_A = T_B$ at thermal equilibrium. Make a guess about the relationship between $dS/dU$ and the temperature $T$. Justify your choice.

  5. There are two types of electric charge - positive and negative. How did the behavior of the electroscope show this?

Problems (3). Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1. 12 pts. What is the magnitude and direction of the electric field that will balance the weight of a proton? The proton has a charge of $+e$.




2. 15 pts. A spherical balloon of volume $V$ contains helium at a pressure $P$. How many atoms of helium are in the balloon if the average kinetic energy of the helium atoms is $\langle E_{kin} \rangle$? Your answer should be in terms of $P$, $V$, and $\langle E_{kin} \rangle$.

3. 15 pts. Imagine that the entropy of a certain substance as a function of $N$ and $U$ is given by the formula $S = \alpha Nk_b U^3$. Using the definition of temperature, find an expression for the thermal energy $U$ of this substance in terms of its temperature $T$, the number of particle $N$, and any other necessary quantities. Is this result well behaved?

4. 18 pts. Suppose we wanted to terraform the Moon, i.e. change it's climate to an Earth-like one. We would have to release lots of oxygen $\rm O_2$ and nitrogen $\rm N_2$ gases (from the interior of the Moon maybe?). In sunlight temperatures on the Moon's surface can reach $T_{moon} = 500~K$. The escape velocity on the Moon (minimum speed needed for an object to escape the Moon's gravity) is $v_e = 2.4\times 10^3~m/s$. When we start releasing gases will the average $\rm O_2$ or $\rm N_2$ gas molecules in sunlight stay on the Moon or can they escape into outer space?


Physics 132-4 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 $6.37\times 10^6~m$
$e$ electronic charge $1.6\times 10^{-19}~C$ $k_e=1/4\pi\epsilon_0$ $8.99\times 10^9~N-m^2/C^2$

Physics 132-4 Equations and Constants



\begin{displaymath} \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 \end{displaymath}


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


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


\begin{displaymath} \vec I = \int \vec F dt = \langle \vec F \rangle \Delta t ... ...P = {\vert\vec F\vert \over A} \qquad PV = Nk_B T = nRT \qquad \end{displaymath}


\begin{displaymath} \langle KE\rangle = \langle E_{kin} \rangle = {1 \over 2}m... ... \qquad E_{int} = N \langle E_{kin} \rangle = {3\over 2} Nk_BT \end{displaymath}


\begin{displaymath} v_{rms} = \sqrt{\overline {v^2} } \qquad C_{V} = {f\over 2} ... ...\qquad E_f = { k_BT \over 2} \qquad E_{int} = {f\over 2} Nk_BT \end{displaymath}

f $\equiv$ number of degrees of freedom


\begin{displaymath} E_{atom} = (n_x + n_y + n_z + \frac{3}{2}) \epsilon \qquad ... ... q \epsilon \qquad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!} \end{displaymath}


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


\begin{displaymath} \vec F_C = k_e {q_1 q_2 \over r^2} \hat r \quad \vec E \equi... ... \sum_i {q_i \over r_i^2} \hat r_i = \int {dq \over r^2}\hat r \end{displaymath}


\begin{displaymath} \langle x\rangle = \frac{1}{N}\sum_i x_{i} \qquad \sigma = \... ...um_i \left( x_{i}-\left\langle x\right\rangle \right)^2}{N-1}} \end{displaymath}


\begin{displaymath} A = 4\pi r^2 \qquad V = Ah \qquad V = {4\over 3} \pi r^3\qquad \frac{dx^n}{dx} = nx^{n-1} \end{displaymath}


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


Image periodicChart



The Periodic Chart.