|| The pressure inside a tank of neon is 150 atm. The temperature is $25\text{°}\text{C}$. On average, how many atomic diameters does a neon atom move between collisions?

|| Dust particles are $\approx 10\mu \text{m}$ in diameter. They are pulverized rock, with $\rho \approx 2500\text{kg}\text{/}{\text{m}}^3$. If you treat dust as an ideal gas, what is the rms speed of a dust particle at $20\text{°}\text{C}?$

|| A mad engineer builds a cube, 2.5 m on a side, in which 6.2-cm-diameter rubber balls are constantly sent flying in random directions by vibrating walls. He will award a prize to anyone who can figure out how many balls are in the cube without entering it or taking out any of the balls. You decide to shoot 6.2-cm-diameter plastic balls into the cube, through a small hole, to see how far they get before colliding with a rubber ball. After many shots, you find they travel an average distance of 1.8 m. How many rubber balls do you think are in the cube?

| Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not $\text{H}{}_2$). The number density is about $1\text{atom}\text{/}\text{c}{\text{m}}^3$ and the temperature is about 3 K.

1. Estimate the pressure in interstellar space. Give your answer in Pa and in atm.

2. What is the rms speed of the atoms?

3. What is the edge length L of an $L\times L\times L$ cube of gas with 1.0 J of thermal energy?

|| Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away.

1. How far would you be able to see if all the molecules around you suddenly doubled in volume?

2. How far would you be able to see if the temperature suddenly rose from 20°C to a blazing hot 1500°C with the pressure unchanged?

||| You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero’s molecule is about to crash head-on into the molecule on which a villain is riding. The villain’s molecule is initially 50 molecular radii away and, in the movie, it takes 3.5 s for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?

|||

1. CALC Find an expression for the $v_{\text{rms}}$ of gas molecules in terms of p, V, and the total mass of the gas M.

2. A gas cylinder has a piston at one end that is moving outward at speed $v_{\text{piston}}$ during an isobaric expansion of the gas. Find an expression for the rate at which $v_{\text{rms}}$ is changing in terms of $v_{\text{piston}}$, the instantaneous value of $v_{\text{rms}}$, and the instantaneous value L of the length of the cylinder.

3. A cylindrical sample chamber has a piston moving outward at 0.50 m/s during an isobaric expansion. The rms speed of the gas molecules is 450 m/s at the instant the chamber length is 1.5 m. At what rate is $v_{\text{rms}}$ changing?

|| Equation 20.3 is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius.

1. Find an expression for the mean free path of an electron through a gas.

2. Electrons travel 3.0 km through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least 50 km. What is the maximum possible pressure inside the accelerator tube, assuming $T=20\text{°}\text{C}?$ Give your answer in both Pa and atm.

|| Uranium has two naturally occurring isotopes. ${}^{238}\text{U}$ has a natural abundance of 99.3% and ${}^{235}\text{U}$ has an abundance of 0.7%. It is the rarer ${}^{235}\text{U}$ that is needed for nuclear reactors. The isotopes are separated by forming uranium hexafluoride, $\text{UF}{}_6$, which is a gas, then allowing it to diffuse through a series of porous membranes. ${}^{235}\text{UF}{}_6$ has a slightly larger rms speed than ${}^{238}\text{UF}{}_6$ and diffuses slightly faster. Many repetitions of this procedure gradually separate the two isotopes. What is the ratio of the rms speed of ${}^{235}\text{UF}{}_6$ to that of ${}^{238}\text{UF}{}_6?$

| On earth, STP is based on the average atmospheric pressure at the surface and on a phase change of water that occurs at an easily produced temperature, being only slightly cooler than the average air temperature. The atmosphere of Venus is almost entirely carbon dioxide $(\text{CO}{}_2)$, the pressure at the surface is a staggering 93 atm, and the average temperature is $470\text{°}\text{C}$. Venusian scientists, if they existed, would certainly use the surface pressure as part of their definition of STP. To complete the definition, they would seek a phase change that occurs near the average temperature. Conveniently, the melting point of the element tellurium is $450\text{°}\text{C}$. What are (a) the rms speed and (b) the mean free path of carbon dioxide molecules at Venusian STP based on this phase change in tellurium? The radius of a $\text{CO}{}_2$ molecule is $1.5\times {10}^{-10}\text{m}$.

||| $5.0\times {10}^{23}$ nitrogen molecules collide with a $10{\text{cm}}^2$ wall each second. Assume that the molecules all travel with a speed of $400\text{m}\text{/}\text{s}$ and strike the wall head-on. What is the pressure on the wall?

||| A $10\text{cm}\times 10\text{cm}\times 10\text{cm}$ box contains 0.010 mol of nitrogen at $20\text{°}\text{C}$. What is the rate of collisions (collisions/s) on one wall of the box?

|| FIGURE P20.57 shows the thermal energy of 0.14 mol of gas as a function of temperature. What is $C_{\text{V}}$ for this gas?

FIGURE P20.57

|| A $100{\text{cm}}^3$ box contains helium at a pressure of 2.0 atm and a temperature of $100\text{°}\text{C}$. It is placed in thermal contact with a $200{\text{cm}}^3$ box containing argon at a pressure of 4.0 atm and a temperature of $400\text{°}\text{C}$.

1. What is the initial thermal energy of each gas?

2. What is the final thermal energy of each gas?

3. How much heat energy is transferred, and in which direction?

4. What is the final temperature?

5. What is the final pressure in each box?

|| A gas of $1.0\times {10}^{20}$ atoms or molecules has $1.0\text{J}$ of thermal energy. Its molar specific heat at constant pressure is $20.8\text{J}\text{/}\text{mol}\text{K}$. What is the temperature of the gas?

|| 2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K.

1. What is the initial thermal energy of each gas?

2. What is the final thermal energy of each gas?

3. How much heat energy is transferred, and in which direction?

4. What is the final temperature?

||

1. INT What is the total rotational kinetic energy of 1.0 mol of nitrogen gas at 300 K?

2. A nitrogen molecule consists of two nitrogen atoms separated by 0.11 nm, the bond length. Treat the molecule as a rotating dumbbell and find the rms angular velocity at this temperature of a nitrogen molecule around the z-axis, as shown in Figure 20.10.

| A monatomic gas and a diatomic gas have equal numbers of moles and equal temperatures. Both are heated at constant pressure until their volume doubles. What is the ratio $Q_{\text{diatomic}}\text{/}{Q}_{\text{monatomic}}?$

|| A water molecule has its three atoms arranged in a “V” shape, so it has rotational kinetic energy around any of three mutually perpendicular axes. However, like diatomic molecules, its vibrational modes are not active at temperatures below 1000 K. What is the thermal energy of 2.0 mol of steam at a temperature of 160°C?

|| In the discussion following Equation 20.43 it was said that $Q_1=-Q_2$. Prove that this is so.

|| INT A monatomic gas is adiabatically compressed to $\frac{1}{8}$ of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not?

1. The rms speed.

2. The mean free path.

3. The thermal energy of the gas.

4. The molar specific heat at constant volume.

|| n moles of a diatomic gas with $C_{\text{V}}=\frac{5}{2}R$ has initial pressure $p_{\text{i}}$ and volume $V_{\text{i}}$. The gas undergoes a process in which the pressure is directly proportional to the volume until the rms speed of the molecules has doubled.

1. Show this process on a pV diagram.

2. How much heat does this process require? Give your answer in terms of $n\text{,}{p}_{\text{i}}$, and $V_{\text{i}}$.

|| 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.

|| INT The rms speed of the molecules in 1.0 g of hydrogen gas is $1800\text{m}\text{/}\text{s}$.

1. What is the total translational kinetic energy of the gas molecules?

2. What is the thermal energy of the gas?

3. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

|| $n_1$ moles of a monatomic gas and $n_2$ moles of a diatomic gas are mixed together in a container.

1. Derive an expression for the molar specific heat at constant volume of the mixture.

2. Show that your expression has the expected behavior if $n_1\to 0$ or $n_2\to 0$.

|| CALC The molar heat capacities of gases are not perfectly constant, as we’ve assumed, but increase slowly with temperature. An empirical formula for the molar heat capacity at constant volume of nitrogen is $C_{\text{V}}=(20.6-1.6\times {10}^{-3}T+8.0\times {10}^{-6}{T}^2)\text{J}\text{/}\text{mol}\text{K}$, where T is in K. What is the entropy increase of the gas if 5.0 g of nitrogen in a rigid container are slowly heated from $300°\text{C}$ to $500°\text{C}$?

|| A thin partition divides a container of volume V into two parts. One side contains $n_{\text{A}}$ moles of gas A in a fraction $f_{\text{A}}$ of the container; that is, $V_{\text{A}}=f_{\text{A}}V$. The other side contains $n_{\text{B}}$ moles of a different gas B at the same temperature in a fraction $f_{\text{B}}$ of the container. The partition is removed, allowing the gases to mix. Find an expression for the change of entropy. This is called the entropy of mixing.

||| 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?

||| 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 $T_{1\text{i}}$. The diatomic gas has initial temperature $T_{2\text{i}}$.

1. Show that the equilibrium thermal energies are

   $$\begin{array}{ccc}{E}_{1\text{f}}& =& {\displaystyle \frac{3n_1}{3n_1+5n_2}}(E_{1\text{i}}+E_{2\text{i}})\\ E_{2\text{f}}& =& {\displaystyle \frac{5n_2}{3n_1+5n_2}}(E_{1\text{i}}+E_{2\text{i}})\end{array}$$

2. Show that the equilibrium temperature is

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

3. 2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K. What is the final temperature? How much heat energy is transferred, and in which direction?

||| Equation 20.57 is the entropy change of an ideal gas in terms of temperature and volume.

1. Find an expression for the entropy change of an ideal gas in terms of pressure and volume. Both $C_{\text{V}}$ and $C_{\text{P}}$ should appear in your answer.

2. A $250{\text{cm}}^3$ container filled with 1.0 g of helium at 1.0 atm slowly expands to $500{\text{cm}}^3$ as the pressure increases to 3.0 atm. What is the change of entropy of the helium?

||| A 2.0 mol sample of oxygen gas in a rigid, 15 L container is slowly cooled from $250\text{°}\text{C}$ to $50\text{°}\text{C}$ by being in thermal contact with a large bath of $50\text{°}\text{C}$ water. What is the entropy change of (a) the gas and (b) the universe?