Chapter 20: The Micro/Macro Connection

6. Suppose you could suddenly increase the speed of every molecule in a gas by a factor of 2.

a. Would the RMS speed of the molecules increase by a factor of $2^{1/2},2,or{2}^2$? Explain.

b. Would the gas pressure increase by a factor of $2^{1/2},2$or $2^2$? Explain.

A lottery machine uses blowing air to keep 2000 Ping-Pong balls bouncing around inside a $1.0\mathrm{m}\times 1.0\mathrm{m}\times 1.0\mathrm{m}$box. The diameter of a Ping-Pong ball is $3.0\mathrm{cm}$. What is the mean free path between collisions? Give your answer in$\mathrm{cm}$.

The $2010$ Nobel Prize in Physics was awarded for the discovery of graphene, a two-dimensional form of carbon in which the atoms form a two-dimensional crystal-lattice sheet only one atom thick. Predict the molar specific heat of graphene. Give your answer as a multiple of$R$ .

The rms speed of the molecules in $1.0g$of hydrogen gas is$1800\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ .
a. What is the total translational kinetic energy of the gas molecules?
b. What is the thermal energy of the gas?
c. $500J$of work are done to compress the gas while, in the same process, $1200J$of heat energy are transferred from the gas to the environment. Afterward, what in the rms speed of the molecules?

At what temperature does the$rms$speed of $\left(a\right)$a nitrogen molecule and $\left(b\right)$a hydrogen molecule equal the escape speed from the earth's surface? $\left(c\right)$You'll find that these temperatures are very high, so you might think that the earth's gravity could easily contain both gases. But not all molecules move with$V_{rms}$. There is a distribution of speeds, and a small percentage of molecules have speeds several times $V_{rms}$ . Bit by bit, a gas can slowly leak out of the atmosphere as its fastest molecules escape. A reasonable rule of thumb is that the earth's gravity can contain a gas only if the average translational kinetic energy per molecule is less than $1\%$of the kinetic energy needed to escape. Use this rule to show why the earth's atmosphere contains nitrogen but not hydrogen, even though hydrogen is the most abundant element in the universe.

$n_1$moles of a monatomic gas and $n_2$moles of a diatomic gas are mixed together in a container.
a. Derive an expression for the molar specific heat at constant volume of the mixture.
b. Show that your expression has the expected behavior if $n_1\to 0$or$n_2\to 0$.

A $1.0kg$ball is at rest on the floor in a$2.0m\times 2.0m\times 2.0m$ room of air at $STP$. Air is$80\%$ nitrogen $\left(N_2\right)$and$20\%$ oxygen$\left(O_2\right)$ by volume.
a. What is the thermal energy of the air in the room?
b. What fraction of the thermal energy would have to be conveyed to the ball for it to be spontaneously launched to a height of$1.0m$ ?
c. By how much would the air temperature have to decrease to launch the ball?
d. Your answer to part $c$ is so small as to be unnoticeable, yet this event never happens. Why not?

An experiment you're designing needs a gas with $\gamma =1.50$. You recall from your physics class that no individual gas has this value, but it occurs to you that you could produce a gas with $\gamma =1.50$by mixing together a monatomic gas and a diatomic gas. What fraction of the molecules need to be monatomic?

III Consider a container like that shown in Figure \(20.12\), with \(n_{1}\) moles of a monatomic gas on one side and \(n_{2}\) moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1{\mathrm{li}}^{-}}$. The diatomic gas has initial temperature \(T_{2 \mathrm{i}}\).

a. Show that the equilibrium thermal energies are

Consider a container like that shown in Figure $20.12$, with $n_1$moles of a monatomic gas on one side and $n_2$moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1i}$. The diatomic gas has initial temperature$T_{2i}$ .
a. Show that the equilibrium thermal energies are

$\begin{array}{r}{E}_{1\mathrm{f}}=\frac{3n_1}{3n_1+5n_2}\left(E_{1\mathrm{i}}+E_{2\mathrm{i}}\right)\\ E_{2\mathrm{f}}=\frac{5n_2}{3n_1+5n_2}\left(E_{1\mathrm{i}}+E_{2\mathrm{i}}\right)\end{array}$

b. Show that the equilibrium temperature is

$T_{\mathrm{f}}=\frac{3n_1{T}_{1\mathrm{i}}+5n_2{T}_{2\mathrm{i}}}{3n_1+5n_2}$

c.$2.0g$ of helium at an initial temperature of role="math" localid="1648474536876" $300K$interacts thermally with $8.0g$of oxygen at an initial temperature of$600K$ . What is the final temperature? How much heat energy is transferred, and in which direction?