University Physics with Modern Physics

Hugh D. Young

Chapter 18

Thermal Properties of Matter - all with Video Answers

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Chapter Questions

Problem 1

A 20.0 -L tank contains $4.86 \times 10^{-4} \mathrm{kg}$ of helium at $18.0^{\circ} \mathrm{C} .$ The molar mass of helium is 4.00 $\mathrm{g} / \mathrm{mol} .$ (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

Shital Rijal

Shital Rijal

Numerade Educator

Problem 2

Helium gas with a volume of $2.60 \mathrm{L},$ under a pressure of 0.180 atm and at a temperature of $41.0^{\circ} \mathrm{C},$ is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is 4.00 $\mathrm{g} / \mathrm{mol} .$

Hend Hamed

Numerade Educator

Problem 3

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 $\mathrm{m}^{3}$ of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 $\mathrm{m}^{3} .$ If the temperature remains constant, what is the final value of the pressure?

Shital Rijal

Numerade Educator

Problem 4

A $3.00-$ L tank contains air at 3.00 atm and $20.0^{\circ} \mathrm{C} .$ The tank is sealed and cooled until the pressure is 1.00 atm. (a) What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. (b) If the temperature is kept at the value found in part (a) and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm?

TP

Tuan Pham

University of Wisconsin - Madison

Problem 5

Planetary Atmospheres. (a) Calculate the density of the atmosphere at the surface of Mars (where the pressure is 650 $\mathrm{Pa}$ and the temperature is typically $253 \mathrm{K},$ with a $\mathrm{CO}_{2}$ atmosphere), Venus (with an average temperature of 730 $\mathrm{K}$ and pressure of 92 atm, with a $\mathrm{CO}_{2}$ atmosphere), and Saturn's moon Titan (where the pressure is 1.5 atm and the temperature is $-178^{\circ} \mathrm{C},$ with a $\mathrm{N}_{2}$
atmosphere). (b) Compare each of these densities with that of the earth's atmosphere, which is 1.20 $\mathrm{kg} / \mathrm{m}^{3} .$ Consult the periodic chart in Appendix D to determine molar masses.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 6

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 L. The pressure of the gas inside the balloon equals air pressure 1.00 atm). (a) If the air inside the balloon is at a constant temperature of $22.0^{\circ} \mathrm{C}$ and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts? (b) Repeat part (a) if the
gas is helium rather than air.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 7

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 $\mathrm{cm}^{3}$ of air at atmospheric pressure $\left(1.01 \times 10^{5} \mathrm{Pa}\right)$ and a temperature of $27.0^{\circ} \mathrm{C}$ At the end of the stroke, the air has been compressed to a volume of 46.2 $\mathrm{cm}^{3}$ and the gauge pressure has increased to $2.72 \times 10^{6}$ Pa. Compute the final temperature.

Shital Rijal

Numerade Educator

Problem 8

A welder using a tank of volume 0.0750 $\mathrm{m}^{3}$ fills it with oxygen (molar mass 32.0 $\mathrm{g} / \mathrm{mol} )$ at a gauge pressure of 3.00 $\mathrm{x}$ $10^{5} \mathrm{Pa}$ and temperature of $37.0^{\circ} \mathrm{C} .$ The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is $22.0^{\circ} \mathrm{C},$ the gauge pressure of the oxygen in the tank is $1.80 \times 10^{5} \mathrm{Pa} .$ Find (a) the initial mass of oxygen and (b) the
mass of oxygen that has leaked out.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 9

A large cylindrical tank contains 0.750 $\mathrm{m}^{3}$ of nitrogen gas at $27^{\circ} \mathrm{C}$ and $7.50 \times 10^{3}$ Pa (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to 0.480 $\mathrm{m}^{3}$ and the temperature is increased to $157^{\circ} \mathrm{C} ?$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 10

An empty cylindrical canister 1.50 $\mathrm{m}$ long and 90.0 $\mathrm{cm}$ in diameter is to be filled with pure oxygen at $22.0^{\circ} \mathrm{C}$ to store in a space station. To hold as much gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 $\mathrm{g} / \mathrm{mol} .$ (a) How many moles of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilograms does this gas increase the mass to be lifted?

Averell Hause

Carnegie Mellon University

Problem 11

The gas inside a balloon will always have a pressure nearly equal to atmospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas) to a volume of 0.600 L at a temperature of $19.0^{\circ} \mathrm{C} .$ What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen $(77.3 \mathrm{K}) ?$

Yaqub Khan

Numerade Educator

Problem 12

Deviations from the Ideal-Gas Equation. For carbon dioxide gas $\left(\mathrm{CO}_{2}\right),$ the constants in the van der Waals equation are $a=0.364 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}$ and $b=4.27 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol} .$ (a) If 1.00
mol of $\mathrm{CO}_{2}$ gas at 350 $\mathrm{K}$ is confined to a volume of $400 \mathrm{cm}^{3},$ find the pressure of the gas using the ideal-gas equation and the van der
Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the van der Waals equation result from the ideal-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 $\mathrm{cm}^{3} .$ Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideal-gas equation if $n / V$ is small.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 13

If a certain amount of ideal gas occupies a volume $V$ at STP on earth, what would be its volume (in terms of $V )$ on Venus, where the temperature is $1003^{\circ} \mathrm{C}$ and the pressure is 92 atm?

Shoukat Ali

Other Schools

Problem 14

A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm $)$ to the surface (where the pressure is 1.00 atm). The temperature at the bottom is $4.0^{\circ} \mathrm{C},$ and the temperature at the surface is $23.0^{\circ} \mathrm{C}$ . (a) What is the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? (b) Would it be safe for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not?

Averell Hause

Carnegie Mellon University

Problem 15

A metal tank with volume 3.10 $\mathrm{L}$ will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at a temperature of $23.0^{\circ} \mathrm{C},$ to what temperature can the gas be warmed before the tank ruptures? You can ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.

Ajay Singhal

Numerade Educator

Problem 16

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.200 $\mathrm{m} .$ (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is
$20.0^{\circ} \mathrm{C} ?$ (b) What is the force when the temperature of the gas is increased to $100.0^{\circ} \mathrm{C} ?$

Ceren Uzun

Texas Tech University

Problem 17

With the assumptions of Example 18.4 (Section $18.1 ),$ at what altitude above sea level is air pressure 90$\%$ of the pressure at sea level?

Farhanul Hasan

Numerade Educator

Problem 18

Make the same assumptions as in Example 18.4 (Section 18.1). How does the percentage decrease in air pressure in going from sea level to an altitude of 100 $\mathrm{m}$ compare to that when going
from sea level to an altitude of 1000 $\mathrm{m} ?$ If your second answer is not 10 times your first answer, explain why.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 19

(a) Calculate the mass of nitrogen present in a volume of 3000 $\mathrm{cm}^{3}$ if the temperature of the gas is $22.0^{\circ} \mathrm{C}$ and the absolute pressure of $2.00 \times 10^{-13} \mathrm{atm}$ is a partial vacuum easily obtained in laboratories. (b) What is the density (in $\mathrm{kg} / \mathrm{m}^{3} )$ of the $\mathrm{N}_{2}$ ?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 20

With the assumption that the air temperature is a uniform $0.0^{\circ} \mathrm{C}($ as in Example 18.4$),$ what is the density of the air at an altitude of 1.00 $\mathrm{km}$ as a percentage of the density at the surface?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 21

At an altitude of $11,000 \mathrm{m}$ (a typical cruising altitude for a jet airliner), the air temperature is $-56.5^{\circ} \mathrm{C}$ and the air density is 0.364 $\mathrm{kg} / \mathrm{m}^{3} .$ What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example 18.4 in Section 18.1 doesn't apply.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

A large organic molecule has a mass of $1.41 \times 10^{-21} \mathrm{kg}$ What is the molar mass of this compound?

Averell Hause

Carnegie Mellon University

Problem 23

Suppose you inherit 3.00 mol of gold from your uncle (an eccentric chemist) at a time when this metal is selling for $14.75$ per gram. Consult the periodic table in Appendix $D$ and Table $12.1 .$ (a) To the nearest dollar, what is this gold worth? (b) If you have your gold formed into a spherical nugget, what is its diameter?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 24

Modern vacuum pumps make it easy to attain pressures of the order of $10^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of $9.00 \times 10^{-14}$ atm and an ordinary temperature of $300.0 \mathrm{K},$ how many molecules are present in a volume of 1.00 $\mathrm{cm}^{3} ?$ (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 25

The Lagoon Nebula (Fig. E18.25) is a cloud of hydrogen gas located 3900 light-years from the earth. The cloud is about 45 light-years in diameter and glows because of its high temperature of 7500 $\mathrm{K} .$ (The gas is raised to this temperature by the stars that lie within the nebula.) The cloud is also very thin; there are only 80 molecules per cubic centimeter. (a) Find the gas pressure ( in
atmospheres) in the Lagoon Nebula. Compare it to the laboratory pressure referred to in Exercise 18.24 . (b) Science-fiction films sometimes show starships being buffeted by turbulence as they fly
through gas clouds such as the Lagoon Nebula. Does this seem realistic? Why or why not?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 26

In a gas at standard conditions, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about $6 \times 10^{9}$ people)?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 27

How many moles are in a 1.00 -kg bottle of water? How many molecules? The molar mass of water is 18.0 $\mathrm{g} / \mathrm{mol}$ .

Shital Rijal

Numerade Educator

Problem 28

How Close Together Are Gas Molecules? Consider an ideal gas at $27^{\circ} \mathrm{C}$ and 1.00 atm pressure. To get some idea how close these molecules are to each other, on the average, imagine
them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap? (b) How does this distance compare with the diameter of a typical molecule? (c) How does their separation compare with the spacing of atoms in solids, which typically are about 0.3 $\mathrm{nm}$ apart?

NR

Nathaniel Riche

Numerade Educator

Problem 29

Consider 5.00 mol of liquid water. (a) What volume is occupied by this amount of water? The molar mass of water is 18.0 $\mathrm{g} / \mathrm{mol} .$ (b) Imagine the molecules to be, on average, uniformly spaced, with each molecule at the center of a small cube. What is the length of an edge of each small cube if adjacent cubes touch but don't overlap? (c) How does this distance compare with the diameter of a molecule?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 30

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: The periodic table in Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 31

Gaseous Diffusion of Uranium. (a) A process called gaseous diffusion is often used to separate isotopes of uranium that is, atoms of the elements that have different masses, such as 235 $\mathrm{U}$ and 238 $\mathrm{U} .$ The only gaseous compound of uranium at ordinary temperatures is uranium hexafluoride, UF $_{6}$ . Speculate on how 235 $\mathrm{UF}_{6}$ and $^{238} \mathrm{UF}_{6}$ molecules might be separated by diffusion. (b) The molar masses for $^{235} \mathrm{UF}_{6}$ and 238 $\mathrm{UF}_{6}$ molecules are 0.349 $\mathrm{kg} / \mathrm{mol}$ and $0.352 \mathrm{kg} / \mathrm{mol},$ respectively. If uranium hexafluoride acts as an ideal gas, what is the ratio of the root-meansquare speed of $^{235} \mathrm{UF}_{6}$ molecules to that of $^{238} \mathrm{UF}_{6}$ molecules if the temperature is uniform?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 32

The ideas of average and root-mean-square value can be applied to any distribution. A class of 150 students had the following scores on a 100 -point quiz: (a) Find the average score for the class. (b) Find the root-mean-square score for the class.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 33

We have two equal-size boxes, $A$ and $B$ . Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box $A$ is at a temperature of $50^{\circ} \mathrm{C}$ while the gas in box $B$ is at $10^{\circ} \mathrm{C}$ . This is all we know about the gas
in the boxes. Which of the following statements must be true? Which could be true? (a) The pressure in $A$ is higher than in $B$ . (b) There are more molecules in $A$ than in $B$ . (c) $A$ and $B$ do not
contain the same type of gas. (d) The molecules in $A$ have more average kinetic energy per molecule than those in $B$ . (e) The molecules in $A$ are moving faster than those in $B .$ Explain the reasoning behind your answers.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 34

A container with volume 1.48 $\mathrm{L}$ is initially evacuated. Then it is filled with 0.226 $\mathrm{g}$ of $\mathrm{N}_{2} .$ Assume that the pressure of the gas is low enough for the gas to obey the ideal-gas law to high degree of accuracy. If the root-mean-square speed of the gas molecules is $182 \mathrm{m} / \mathrm{s},$ what is the pressure of the gas?

Averell Hause

Carnegie Mellon University

Problem 35

(a) A deuteron, $_{1}^{2} \mathrm{H},$ is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300
million $\mathrm{K} .$ What is the rms speed of the deuterons? Is this a significant fraction of the speed of light $\left(c=3.0 \times 10^{8} \mathrm{m} / \mathrm{s}\right) ?$ (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10$c ?$

Ceren Uzun

Texas Tech University

Problem 36

Martian Climate. The atmosphere of Mars is mostly $\mathrm{CO}_{2}$ (molar mass 44.0 $\mathrm{g} / \mathrm{mol}$ ) under a pressure of 650 $\mathrm{Pa}$ , which we shall assume remains constant. In many places the temperature varies from $0.0^{\circ} \mathrm{C}$ in summer to $-100^{\circ} \mathrm{C}$ in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the CO $_{2}$ molecules speed equal to 0.10$c ?$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 37

(a) Oxygen (O) has a molar mass of 32.0 $\mathrm{g} / \mathrm{mol} .$ What is the average translational kinetic energy of an oxygen molecule at a temperature of 300 $\mathrm{K}$ ? (b) What is the average value of the square of its speed? (c) What is the root-mean-square speed? (d)
What is the momentum of an oxygen molecule traveling at this speed? (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel
0.10 $\mathrm{m}$ on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.) (f) What is the average force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 atm? (h) Compute the number of oxygen molecules that are actually contained in a vessel of this size at 300 $\mathrm{K}$ and atmospheric pressure. (i) Your answer for part (h) should be three times as large as the answer for part (g). Where does this discrepancy arise?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 38

Calculate the mean free path of air molecules at a pressure of $3.50 \times 10^{-13}$ atm and a temperature of 300 $\mathrm{K}$ . (This pressure is readily attainable in the laboratory; see Exercise $18.24 .$ ) As in Example $18.8,$ model the air molecules as spheres of radius
$2.0 \times 10^{-10} \mathrm{m} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at $20.0^{\circ} \mathrm{C} ?$ (Hint: The periodic table in Appendix D shows the molar mass (in $\mathrm{g} / \mathrm{mol}$ ) of each element under the chemical symbol for that element. The molar mass of $\mathrm{H}_{2}$ is twice the molar mass of hydrogen atoms, and similarly for $\mathrm{N}_{2} .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

Smoke particles in the air typically have masses of the order of $10^{-16} \mathrm{kg} .$ The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the root-mean-square speed of Brownian motion for a particle with a mass of $3.00 \times 10^{-16} \mathrm{kg}$ in air at 300 $\mathrm{K}$ . (b) Would the root-mean-square speed be different if the particle were in hydrogen gas at the same temperature? Explain.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 41

(a) How much heat does it take to increase the temperature of 2.50 mol of a diatomic ideal gas by 50.0 $\mathrm{K}$ near room temperature if the gas is held at constant volume? (b) What is the answer to the question in part (a) if the gas is monatomic rather than diatomic?

Keshav Singh

Numerade Educator

Problem 42

Perfectly rigid containers each hold $n$ moles of ideal gas, one being hydrogen $\left(\mathrm{H}_{2}\right)$ and other being neon (Ne). If it takes 300 $\mathrm{J}$ of heat to increase the temperature of the hydrogen by $\mathrm{by}$ $2.50^{\circ} \mathrm{C},$ by how many degrees will the same amount of heat raise the temperature of the neon?

Averell Hause

Carnegie Mellon University

Problem 43

(a) Compute the specific heat at constant volume of nitrogen $\left(\mathrm{N}_{2}\right)$ gas, and compare it with the specific heat of liquid water. The molar mass of $\mathrm{N}_{2}$ is 28.0 $\mathrm{g} / \mathrm{mol}$ . (b) You warm 1.00 $\mathrm{kg}$ of water at a constant volume of 1.00 $\mathrm{L}$ from $20.0^{\circ} \mathrm{C}$ to $30.0^{\circ} \mathrm{C}$ in a kettle. For the same amount of heat, how many kilograms of $20.0^{\circ} \mathrm{C}$ air would you be able to warm to $30.0^{\circ} \mathrm{C} ?$ What volume (in liters) would this air occupy at $20.0^{\circ} \mathrm{C}$ and a pressure of 1.00 atm? Make the simplifying assumption that air is 100$\% \mathrm{N}_{2}$ .

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 44

(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 $\mathrm{g} / \mathrm{mol} .$ (b) The actual specific heat of water vapor at low pressures is about 2000 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . Compare this with your calculation and comment on the actual role of vibrational motion.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 45

(a) Use Eq. 18.28 to calculate the specific heat at constant volume of aluminum in units of $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . Consult the periodic table in Appendix D. (b) Compare the answer in part (a) with the value given in Table $17.3 .$ Try to explain any disagreement between these two values.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 46

For a gas of nitrogen molecules $\left(\mathrm{N}_{2}\right),$ what must the temperature be if 94.7$\%$ of all the molecules have speeds less than (a) $1500 \mathrm{m} / \mathrm{s} ;$ (b) $1000 \mathrm{m} / \mathrm{s} ;$ (c) 500 $\mathrm{m} / \mathrm{s} ?$ Use Table $18.2 .$ The molar mass of $\mathrm{N}_{2}$ is 28.0 $\mathrm{g} / \mathrm{mol} .$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 47

For diatomic carbon dioxide gas $\left(\mathrm{CO}_{2},$ molar mass \right. 44.0 $\mathrm{g} / \mathrm{mol} )$ at $T=300 \mathrm{K}$ , calculate (a) the most probable speed
$v_{\mathrm{mp}} ;$ (b) the average speed $v_{\mathrm{av}} ;(\mathrm{c})$ the root-mean-square speed $v_{\mathrm{rms}}$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 48

CALC Prove that $f(v)$ as given by Eq. $(18.33)$ is maximum for $\epsilon=k T .$ Use this result to obtain Eq. $(18.34)$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 49

Solid water (ice) is slowly warmed from a very low temperature. (a) What minimum external pressure $p_{1}$ must be applied to
the solid if a melting phase transition is to be observed? Describe
the sequence of phase transitions that occur if the applied pressure $p$ is such that $p<p_{1}$ . (b) Above a certain maximum pressure $p_{2},$ no
boiling transition is observed. What is this pressure? Describe the
sequence of phase transitions that occur if $p_{1}<p<p_{2}$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 50

Puffy cumulus clouds, which are made of water droplets, occur at lower altitudes in the atmosphere. Wispy cirrus clouds, which are made of ice crystals, occur only at higher altitudes. Find the altitude $y$ (measured from sea level) above which only cirrus
clouds can occur. On a typical day and at altitudes less than 11 $\mathrm{km}$ , the temperature at an altitude $y$ is given by $T=T_{0}-\alpha y,$ where
$T_{0}=15.0^{\circ} \mathrm{C}$ and $\alpha=6.0 \mathrm{C}^{\circ} / 1000 \mathrm{m} .$

Bryce Samwel

Numerade Educator

Problem 51

The atmosphere of the planet Mars is 95.3$\%$ carbon dioxide $\left(\mathrm{CO}_{2}\right)$ and about 0.03$\%$ water vapor. The atmospheric pressure is only about 600 $\mathrm{Pa}$ , and the surface temperature varies from $-30^{\circ} \mathrm{C}$ to $-100^{\circ} \mathrm{C}$ . The polar ice caps contain both $\mathrm{CO}_{2}$ ice and water ice. Could there be liquid $\mathrm{CO}_{2}$ on the surface of Mars? Could there be liquid water? Why or why not?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 52

A physics lecture room has a volume of 216 $\mathrm{m}^{3} .$ (a) For a pressure of 1.00 atm and a temperature of $27.0^{\circ} \mathrm{C},$ use the ideal-gas law to estimate the number of air molecules in the room. Assume all the air is $\mathrm{N}_{2}$ . (b) Calculate the particle density $-$ that is, the number of $\mathrm{N}_{2}$ molecules per cubic centimeter. (c) Calculate the mass of the air in the room.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 53

CP Blo The Effect of Altitude on the Lungs. (a) Calculate the change in air pressure you will experience if you climb a 1000 -m mountain, assuming that the temperature and air density do not change over this distance and that they were $22^{\circ} \mathrm{C}$ and $1.2 \mathrm{kg} / \mathrm{m}^{3},$ respectively, at the bottom of the mountain. (Note that the result of Example 18.4 doesn't apply, since the expression derived in that example accounts for the variation of air density with altitude and we are told to ignore that in this problem.) If you took a $0.50-$ breath at the foot of the mountain and managed to hold it until you reached the top, what would be the volume of this breath when you exhaled it there?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 54

CP BIO The Bends. If deep-sea divers rise to the surface too quickly, nitrogen bubbles in their blood can expand and prove fatal. This phenomenon is known as the bends. If a scuba diver rises quickly from a depth of 25 $\mathrm{m}$ in Lake Michigan (which is fresh water), what will be the volume at the surface of an $\mathrm{N}_{2}$ bubble that occupied 1.0 $\mathrm{mm}^{3}$ in his blood at the lower depth? Does it seem that this difference is large enough to be a problem? (Assume that the pressure difference is due only to the changing water pressure, not to any temperature difference, an assumption
that is reasonable, since we are warm-blooded creatures.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 55

CP A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500.0 $\mathrm{m}^{3}$ and the surrounding air is at $15.0^{\circ} \mathrm{C},$ what must the temperature of the air in the balloon be for it to lift a total load of 290 $\mathrm{kg}$ (in addition to the mass of the hot air)? The density of air at $15.0^{\circ} \mathrm{C}$ and atmospheric pressure is1.23 $\mathrm{kg} / \mathrm{m}^{3} .$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 56

(a) Use Eq. $(18.1)$ to estimate the change in the volume of a solid steel sphere of volume 11 $\mathrm{L}$ when the temperature and pressure increase from $21^{\circ} \mathrm{C}$ and $1.013 \times 10^{5}$ Pa to $42^{\circ} \mathrm{C}$ and $2.10 \times 10^{7}$ Pa. (Hint: Consult Chapters 11 and 17 to determine the values of $\beta$ and $k .$ (b) In Example 18.3 the change in volume
of an $11-$ L steel scuba tank was ignored. Was this a good approximation? Explain.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 57

A cylinder 1.00 m tall with inside diameter 0.120 $\mathrm{m}$ is used to hold propane gas (molar mass 44.1 $\mathrm{g} / \mathrm{mol}$ ) for use in a barbecue. It is initially filled with gas until the gauge pressure is $1.30 \times 10^{6} \mathrm{Pa}$ and the temperature is $22.0^{\circ} \mathrm{C} .$ The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is $2.50 \times 10^{5}$ Pa. Calculate the mass of propane that has been used.

Supratim Pal

Numerade Educator

Problem 58

During a test dive in $1939,$ prior to being accepted by the U.S. Navy, the submarine Squalus sank at a point where the depth of water was 73.0 m. The temperature at the surface was $27.0^{\circ} \mathrm{C},$ and at the bottom it was $7.0^{\circ} \mathrm{C}$ . The density of seawater is
1030 $\mathrm{kg} / \mathrm{m}^{3} .$ (a) A diving bell was used to rescue 33 trapped crewmen from the Squalus. The diving bell was in the form of a circular cylinder 2.30 $\mathrm{m}$ high, open at the bottom and closed at the top. When the diving bell was lowered to the bottom of the sea, to what height did water rise within the diving bell? (Hint: You may ignore the relatively small variation in water pressure between the bottom of the bell and the surface of the water within the bell.) (b) At what gauge pressure must compressed air have been supplied to the bell while on the bottom to expel all the water from it?

Averell Hause

Carnegie Mellon University

Problem 59

Atmosphere of Titan. Titan, the largest satellite of Saturn, has a thick nitrogen atmosphere. At its surface, the pressure is 1.5 earth-atmospheres and the temperature is 94 $\mathrm{K}$ (a) What is the
surface temperature in $^{\circ} \mathrm{C} ?$ (b) Calculate the surface density in Titan's atmosphere in molecules per cubic meter. (c) Compare the density of Titan's surface atmosphere to the density of earth's atmosphere at $22^{\circ} \mathrm{C}$ . Which body has denser atmosphere?

Shital Rijal

Numerade Educator

Problem 60

Pressure on Venus. At the surface of Venus the average temperature is a balmy $460^{\circ} \mathrm{C}$ due to the greenhouse effect (global warming! , the pressure is 92 earth-atmospheres, and the acceleration due to gravity is 0.894$g_{\text { earth }} .$ The atmosphere is nearly all $\mathrm{CO}_{2}$ (molar mass 44.0 $\mathrm{g} / \mathrm{mol}$ ) and the temperature remains remarkably constant. We shall assume that the temperature does not change at all with altitude. (a) What is the atmospheric pressure 1.00 $\mathrm{km}$ above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres. (b) What is the root-mean-square speed of
the $\mathrm{CO}_{2}$ molecules at the surface of Venus and at an altitude of 1.00 $\mathrm{km} ?$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 61

An automobile tire has a volume of 0.0150 $\mathrm{m}^{3}$ on a cold day when the temperature of the air in the tire is $5.0^{\circ} \mathrm{C}$ and atmospheric pressure is 1.02 atm. Under these conditions the gauge pressure is measured to be 1.70 atm (about 25 $\mathrm{lb} / \mathrm{in.} .$ ). After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to $45.0^{\circ} \mathrm{C}$ and the volume has risen to 0.0159 $\mathrm{m}^{3}$ . What then is the gauge pressure?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 62

A flask with a volume of $1.50 \mathrm{L},$ provided with a stop cock, contains ethane gas $\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)$ at 300 $\mathrm{K}$ and atmospheric pressure $\left(1.013 \times 10^{5} \mathrm{Pa}\right) .$ The molar mass of ethane is 30.1 $\mathrm{g} / \mathrm{mol}$ . The system is warmed to a temperature of $490 \mathrm{K},$ with the stop cock open to the atmosphere. The stopcock is then closed, and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 63

A balloon whose volume is 750 $\mathrm{m}^{3}$ is to be filled with hydrogen at atmospheric pressure $\left(1.01 \times 10^{5} \mathrm{Pa}\right) .$ (a) If the hydrogen is stored in cylinders with volumes of 1.90 $\mathrm{m}^{3}$ at a gauge pressure of $1.20 \times 10^{6} \mathrm{Pa},$ how many cylinders are required? Assume that the temperature of the hydrogen remains constant.
(b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if the gas in the balloon and the surrounding air are both at $15.0^{\circ} \mathrm{C}$ ? The molar mass of hydrogen $\left(\mathrm{H}_{2}\right)$ is 2.02 $\mathrm{g} / \mathrm{mol}$ . The density of air at $15.0^{\circ} \mathrm{C}$ and atmospheric pressure is 1.23 $\mathrm{kg} / \mathrm{m}^{3} .$ See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 $\mathrm{g} / \mathrm{mol}$ ) instead of hydrogen, again
at $15.0^{\circ} \mathrm{C}$ ?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 64

A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.500 atm at $20.0^{\circ} \mathrm{C}$ . The round part of the tank has a radius of $10.0 \mathrm{cm},$ and the gas is supporting a piston that can move up and down in the cylinder without friction. There is a vacuum above the piston. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 65

A large tank of water has a hose connected to it, as shown in Fig. P18.65. The tank is sealed at the top and has compressed air between the water surface and the top. When the water height $h$ has the value $3.50 \mathrm{m},$ the absolute pressure $p$ of the compressed air is $4.20 \times$
$10^{5}$ Pa. Assume that the air above the water expands at constant temperature, and take the
atmospheric pressure to be $1.00 \times 10^{5}$ Pa. (a) What is the speed with which water flows out of the hose when $h=3.50 \mathrm{m} ?$ (b) As water flows out of the tank, $h$ decreases. Calculate the speed of flow for $h=3.00 \mathrm{m}$ and for $h=2.00 \mathrm{m} .$ (c) At what value of $h$ does the flow stop?

Shital Rijal

Numerade Educator

Problem 66

A person at rest inhales 0.50 $\mathrm{L}$ of air with each breath at a pressure of 1.00 atm and a temperature of $20.0^{\circ} \mathrm{C}$ . The inhaled air is 21.0$\%$ oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of 2000 $\mathrm{m}$ but the temperature is still $20.0^{\circ} \mathrm{C}$ . Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some
people report "shortness of breath" at high elevations.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 67

How Many Atoms Are You? Estimate the number of atoms in the body of a $50-\mathrm{kg}$ physics student. Note that the human body is mostly water, which has molar mass 18.0 $\mathrm{g} / \mathrm{mol}$ and that each water molecule contains three atoms.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 68

The size of an oxygen molecule is about 2.0 $\times 10^{-10} \mathrm{m}$ Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from ideal-gas behavior at ordinary temperatures $(T=300 \mathrm{K})$ .

Averell Hause

Carnegie Mellon University

Problem 69

You have two identical containers, one containing gas $A$ and the other gas $B .$ The masses of these molecules are $m_{A}=3.34 \times 10^{-27} \mathrm{kg}$ and $m_{B}=5.34 \times 10^{-26} \mathrm{kg} .$ Both gases are under the same pressure and are at $10.0^{\circ} \mathrm{C}$ . (a) Which molecules $(A$ or $B)$ have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once
you have accomplished your goal, which molecules $(A$ or $B)$ now have greater average translational kinetic energy per molecule?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 70

Insect Collisions. A cubical cage 1.25 $\mathrm{m}$ on each side contains 2500 angry bees, each flying randomly at 1.10 $\mathrm{m} / \mathrm{s} .$ We can model these insects as spheres 1.50 $\mathrm{cm}$ in diameter. On the average, (a) how far does a typical bee travel between collisions,
(b) what is the average time between collisions, and (c) how many collisions per second does a bee make?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 71

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is $22.0^{\circ} \mathrm{C}$ . Assume that all the gas is $\mathrm{N}_{2},$ of molar mass 28.0 $\mathrm{g} / \mathrm{mol}$ , (a) Find the mass of a single $\mathrm{N}_{2}$ molecule. (b) How much translational kinetic energy does an average $\mathrm{N}_{2}$ molecule have? (c) How many $\mathrm{N}_{2}$ molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?

Shital Rijal

Numerade Educator

Problem 72

(a) Compute the increase in gravitational potential energy for a nitrogen molecule (molar mass 28.0 $\mathrm{g} / \mathrm{mol} )$ for an increase in elevation of 400 $\mathrm{m}$ near the earth's surface. (b) At what temperature is this equal to the average kinetic energy of a nitrogen molecule? (c) Is it possible that a nitrogen molecule near sea level where $T=15.0^{\circ} \mathrm{C}$ could rise to an altitude of 400 $\mathrm{m} ?$ Is it likely that it could do so without hitting any other molecules along the way? Explain.

Harrison Stuckey

Harrison Stuckey

Numerade Educator

Problem 73

The Lennard-Jones Potential. A commonly used potential-energy function for the interaction of two molecules (see Fig. 18.8$)$ is the Lennard-Jones $6-12$ potential: $$U(r)=U_{0}\left[\left(\frac{R_{0}}{r}\right)^{12}-2\left(\frac{R_{0}}{r}\right)^{6}\right]$$
where $r$ is the distance between the centers of the molecules and
$U_{0}$ and $R_{0}$ are positive constants. The corresponding force $F(r)$ is given in Eq. $(14.26) .$ (a) Graph $U(r)$ and $F(r)$ versus $r .$ (b) Let $r_{1}$ be the value of $r$ at which $U(r)=0,$ and let $r_{2}$ be the value of $r$ at which $F(r)=0 .$ Show the locations of $r_{1}$ and $r_{2}$ on your graphs of $U(r)$ and $F(r) .$ Which of these values represents the equilibrium separation between the molecules? (c) Find the values of $r_{1}$ and $r_{2}$ in terms of $R_{0},$ and find the ratio $r_{1} / r_{2}$ . (d) If the molecules are located a distance $r_{2}$ apart [as calculated in part (c)], how much
work must be done to pull them apart so that $r \rightarrow \infty ?$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 74

(a) What is the total random translational kinetic energy of 5.00 L of hydrogen gas (molar mass 2.016 $\mathrm{g} / \mathrm{mol} )$ with pressure $1.01 \times 10^{5} \mathrm{Pa}$ and temperature 300 $\mathrm{K} ?$ (Hint: Use the procedure of Problem 18.71 as a guide.) (b) If the tank containing the gas is placed on a swift jet moving at $300.0 \mathrm{m} / \mathrm{s},$ by what percentage is the total kinetic energy of the gas increased? (c) since the kinetic energy of the gas molecules is greater when it is on the jet, does this mean that its temperature has gone up? Explain.

Averell Hause

Carnegie Mellon University

Problem 75

The speed of propagation of a sound wave in air at $27^{\circ} \mathrm{C}$ is about 350 $\mathrm{m} / \mathrm{s} .$ Calculate, for comparison, (a) $v_{\mathrm{rms}}$ for nitrogen molecules and (b) the rms value of $v_{x}$ at this temperature. The molar mass of nitrogen $\left(\mathrm{N}_{2}\right)$ is 28.0 $\mathrm{g} / \mathrm{mol} .$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 76

Hydrogen on the Sun. The surface of the sun has a temperature of about 5800 $\mathrm{K}$ and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is $1.67 \times 10^{-27}$ kg. (b) The escape speed for a particle to leave the gravitational influence of the sun is given by $(2 G M / R)^{1 / 2},$ where $M$ is the sun's mass, $R$ its radius, and $G$ the gravitational constant $($ see Example 13.5 of Section 13.3$) .$ Use the data in Appendix $F$ to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 77

(a) Show that a projectile with mass $m$ can "escape" from the surface of a planet if it is launched vertically upward with a kinetic energy greater than $m g R_{p},$ where $g$ is the acceleration due
to gravity at the planet's surface and $R_{p}$ is the planet's radius. Ignore air resistance. (See Problem $18.76 . )$ (b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 $\mathrm{g} / \mathrm{mol} )$ equal that required to escape? What about a hydrogen molecule (molar mass 2.02 $\mathrm{g} / \mathrm{mol} ) ?(\mathrm{c})$ (c) Repeat part (b) for the moon, for which $g=$ 1.63 $\mathrm{m} / \mathrm{s}^{2}$ and $R_{\mathrm{p}}=1740 \mathrm{km} .$ (d) While the earth and the moon have similar average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 78

Planetary Atmospheres. (a) The temperature near the top of Jupiter's multicolored cloud layer is about 140 $\mathrm{K}$ . The temperature at the top of the earth's troposphere, at an altitude of about $20 \mathrm{km},$ is about 220 $\mathrm{K}$ . Calculate the rms speed of hydrogen molecules in both these environments. Give your answers in $\mathrm{m} / \mathrm{s}$ and as a fraction of the escape speed from the respective planet (see Problem 18.76 ). (b) Hydrogen gas $\left(\mathrm{H}_{2}\right)$ is a rare element in the earth's atmosphere. In the atmosphere of Jupiter, by contrast, 89$\%$ of all molecules are $\mathrm{H}_{2} .$ Explain why, using your results from part (a). (c) Suppose an astronomer claims to have discovered an oxygen
$\left(\mathrm{O}_{2}\right)$ atmosphere on the asteroid Ceres. How likely is this? Ceres
has a mass equal to 0.014 times the mass of the moon, a density of $2400 \mathrm{kg} / \mathrm{m}^{3},$ and a surface temperature of about 200 $\mathrm{K}$ .

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 79

(a) For what mass of molecule or particle is $v_{\mathrm{rms}}$ equal to 1.00 $\mathrm{mm} / \mathrm{s}$ at 300 $\mathrm{K} ?$ (b) If the particle is an ice crystal, how many molecules does it contain? The molar mass of water is 18.0 $\mathrm{g} / \mathrm{mol}$ . (c) Calculate the diameter of the particle if it is a spherical piece of ice. Would it be visible to the naked eye?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 80

In describing the heat capacities of solids in Section 18.4 we stated that the potential energy $U=\frac{1}{2} k x^{2}$ of a harmonic oscillator averaged over one period of the motion is equal to the kinetic
energy $K=\frac{1}{2} m v^{2}$ averaged over one period. Prove this result using Eqs. $(14.13)$ and $(14.15)$ for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position and velocity make the phase angle $\phi$ equal to zero. (Hint: Use the trigonometric identities $\cos ^{2}(\theta)=[1+\cos (2 \theta)] / 2$ and $\sin ^{2}(\theta)=[1-\cos (2 \theta)] / 2 .$ What is the average value of $\cos (2 \omega t)$ over one period?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 81

It is possible to make crystalline solids that are only one layer of atoms thick. Such "two-dimensional" crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature? Give your answer as a multiple of $R$ and in $\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}$ ) At very low temperatures, will the molar heat capacity of a two-dimensional crystal be greater than, less than, or equal to the result you found in part (a)? Explain why.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 82

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 $\mathrm{K}$ (b) Calculate the moment of inertia of an oxygen molecule $\left(\mathrm{O}_{2}\right)$ for rotation about either the $y$ - or $z$ -axis shown in Fig. 18.18 $\mathrm{b}$ . Treat the molecule as
two massive points (representing the oxygen atoms) separated by a distance of $1.21 \times 10^{-10} \mathrm{m}$ . The molar mass of oxygen atoms is 16.0 $\mathrm{g} / \mathrm{mol} .$ (c) Find the rms angular velocity of rotation of an oxygen molecule about either the $y$ - or $z$ -axis shown in Fig. 18.18 $\mathrm{b}$ . How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery $(10,000$ rev/min $)$ ?

Dading Chen

Numerade Educator

Problem 83

For each polyatomic gas in Table $18.1,$ compute the value of the molar heat capacity at constant volume, $C_{V},$ on the assumption that there is no vibrational energy. Compare with the measured values in the table, and compute the fraction of the total heat capacity that is due to vibration for each of the three gases. (Note: $\mathrm{CO}_{2}$ is linear; $\mathrm{SO}_{2}$ and $\mathrm{H}_{2} \mathrm{S}$ are not. Recall that a linear polyatomic molecule has two rotational degrees of freedom, and a nonlinear molecule has three.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 84

(a) Show that $\int_{0}^{\infty} f(v) d v=1,$ where $f(v)$ is the Maxwell-Boltzmann distribution of Eq. $(18.32) .$ (b) In terms of the physical definition of $f(v),$ explain why the integral in part (a) must have this value.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 85

Calculate the integral in Eq. $(18.31), \int_{0}^{\infty} v^{2} f(v) d v$ and compare this result to $\left(v^{2}\right)\text {av as given by Eq. }(18.16) .$ (Hint: You may use the tabulated integral
$$\int_{0}^{\infty} x^{2 n} e^{-\alpha x^{2}} d x=\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot(2 n-1)}{2^{n+1} \alpha^{n}} \sqrt{\frac{\pi}{\alpha}}$$ where $n$ is a positive integer and $\alpha$ is a positive constant.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 86

Calculate the integral in Eq. $(18.30), \int_{0}^{\infty} v f(v) d v$ and compare this result to $v_{\mathrm{av}}$ as given by Eq. $(18.35) .$ (Hint: Make the change of variable $v^{2}=x$ and use the tabulated integral $$\int_{0}^{\infty} x^{n} e^{-\alpha x} d x=\frac{n !}{\alpha^{n+1}}$$ where $n$ is a positive integer and $\alpha$ is a positive constant.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 87

CALC (a) Explain why in a gas of $N$ molecules, the number of molecules having speeds in the finite interval $v$ to $v+\Delta v$ is $\Delta N=N \int_{v}^{v+\Delta v} f(v) d v .($ b) If $\Delta v$ is small, then $f(v)$ is approximately constant over the interval and $\Delta N \approx N f(v) \Delta v .$ For oxygen gas $\left(\mathrm{O}_{2},$ molar mass 32.0 $\mathrm{g} / \mathrm{mol}\right)$ at $T=300 \mathrm{K},$ use this approximation to calculate the number of molecules with speeds
within $\Delta v=20 \mathrm{m} / \mathrm{s}$ of $v_{\mathrm{mp}}$ . Express your answer as a multiple of N. (c) Repeat part (b) for speeds within $\Delta v=20 \mathrm{m} / \mathrm{s}$ of 7$v_{\mathrm{mp}}$ . (d) Repeat parts (b) and (c) for a temperature of 600 $\mathrm{K}$ . (e) Repeat
parts (b) and (c) for a temperature of 150 $\mathrm{K}$ . (f) What do your results tell you about the shape of the distribution as a function of temperature? Do your conclusions agree with what is shown in Fig. 18.23$?$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 88

Meteorology. The vapor pressure is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The relative humidity is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same temperature, expressed as a percentage. The air is saturated when the humidity is 100$\%$ (a) The vapor pressure of water at $20.0^{\circ} \mathrm{C}$ is $2.34 \times 10^{3}$ Pa. If the air temperature is $20.0^{\circ} \mathrm{C}$ and the relative humidity is $60 \%,$ what is the partial pressure of water vapor in the atmosphere (that is, the pressure due to water vapor alone)? (b) Under the conditions of part (a), what is the mass of water in 1.00 $\mathrm{m}^{3}$ of air? (The molar mass of water is 18.0 $\mathrm{g} / \mathrm{mol}$ . Assume that water vapor can be treated as an ideal gas.)

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 89

The Dew Point. The vapor pressure of water (see Problem 18.88 ) decreases as the temperature decreases. If the amount of water vapor in the air is kept constant as the air is cooled, a temperature is reached, called the dew point, at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature. The temperature in a room is $30.0^{\circ} \mathrm{C}$ . A meteorologist cools a metal can by gradually adding cold water. When the can temperature reaches
$16.0^{\circ} \mathrm{C},$ water droplets form on its outside surface. What is the relative humidity of the $30.0^{\circ} \mathrm{C}$ air in the room? The table lists the vapor pressure of water at various temperatures:

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 90

Altitude at Which Clouds Form. On a spring day in the midwestern United States, the air temperature at the surface is $28.0^{\circ} \mathrm{C} .$ Puffy cumulus clouds form at an altitude where the air temperature equals the dew point (see Problem 18.89$) .$ If the air temperature decreases with altitude at a rate of $0.6 \mathrm{C}^{\circ} / 100 \mathrm{m},$ at approximately what height above the ground will clouds form if the relative humidity at the surface is 35$\%$ and 80$\% ?$ (Hint: Use the table in Problem $18.89.)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 91

Dark Nebulae and the Interstellar Medium. The dark area in Fig. $\mathrm{Pl} 8.91$ that appears devoid of stars is a dark nebula, a cold gas cloud in interstellar space that contains enough
material to block out light from the stars behind it. A typical dark nebula is about 20 light-years in diameter and contains about 50 hydrogen atoms per cubic centimeter (monatomic hydrogen, not $\mathrm{H}_{2}$ ) at a temperature of about 20 $\mathrm{K}$ . (A light-year is the distance light
travels in vacuum in one year and is equal to $9.46 \times 10^{15} \mathrm{m.}$ ) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The radius of a hydrogen atom is $5.0 \times 10^{-11} \mathrm{m}$ . (b) Estimate the rms speed of a hydrogen atom and the mean free time (the average time between collisions for a given atom). Based on this result, do you think that atomic collisions, such as those leading to $\mathrm{H}_{2}$ molecule formation, are very important in determining the composition of the nebula? (c) Estimate the pressure inside a dark nebula. (d)
Compare the rms speed of a hydrogen atom to the escape speed at the surface of the nebula (assumed spherical). If the space around the nebula were a vacuum, would such a cloud be stable or would it tend to evaporate? (e) The stability of dark nebulae is explained by the presence of the interstellar medium (ISM), an even thinner gas that permeates space and in which the dark nebulae are embedded. Show that for dark nebulae to be in equilibrium with the ISM, the numbers of atoms per volume $(N / V)$ and the temperatures $(T)$ of dark nebulae and the ISM must be related by $$
\frac{(N / V)_{\text { nebula }}}{(N / V)_{\mathrm{ISM}}}=\frac{T_{\mathrm{ISM}}}{T_{\text { ncbula }}}$$ (f) In the vicinity of the sun, the ISM contains about 1 hydrogen atom per 200 $\mathrm{cm}^{3} .$ Estimate the temperature of the ISM in the vicinity of the sun. Compare to the temperature of the sun's surface, about 5800 $\mathrm{K}$ . Would a spacecraft coasting through interstellar space burn up? Why or why not?

Lainey Roebuck

Numerade Educator

Problem 92

Earth's Atmosphere. In the troposphere, the part of the atmosphere that extends from earth's surface to an altitude of about $11 \mathrm{km},$ the temperature is not uniform but decreases with
increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship $$T=T_{0}-\alpha y$$ where $T_{0}$ is the temperature at the earth's surface and $T$ is the temperature at height $y$ , the pressure $p$ at height $y$ is given by $$\ln \left(\frac{p}{p_{0}}\right)=\frac{M g}{R \alpha} \ln \left(\frac{T_{0}-\alpha y}{T_{0}}\right) $$ where $p_{0}$ is the pressure at the earth's surface and $M$ is the molar mass for air. The coefficient $\alpha$ is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 $\mathrm{C}^{\circ} / 100 \mathrm{m}$ . (b) Show that the above result reduces to the result of Example 18.4$($ Section 18.1$)$ in the limit that $\alpha \rightarrow 0$ . (c) With $\alpha=0.6 \mathrm{C}^{\circ} / 100 \mathrm{m},$ calculate $p$ for $y=8863 \mathrm{m}$ and compare your answer to the result of Example $18.4 .$ Take $T_{0}=288 \mathrm{K}$ and $p_{0}=1.00$ atm.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 93

In Example 18.7$($ Section 18.3$)$ we saw that $v_{\text { ms }}>v_{\text { av. }}$ It is not difficult to show that this is always the case. (The only exception is when the particles have the same speed, in which case $v_{\text { rms }}=v_{\text { av. }} )$ (a) For two particles with speeds $v_{1}$ and $v_{2},$ show that $v_{\mathrm{rms}} \geq v_{\mathrm{av}}$ regardless of the numerical values of $v_{1}$ and $v_{2} .$ Then show that $v_{\mathrm{ms}}>v_{\mathrm{av}}$ if $v_{1} \neq v_{2}$ . (b) Suppose that for a collection of $N$ particles you know that $v_{\mathrm{rms}}>v_{\mathrm{av}}$ . Another particle, with speed $u,$ is added to the collection of particles. If the new rms and average speeds are denoted as $v_{\mathrm{ms}}^{\prime}$ and $v_{\mathrm{av}}^{\prime}$ , show that
$$v_{\mathrm{rms}}^{\prime}=\sqrt{\frac{N v_{\mathrm{rms}}^{2}+u^{2}}{N+1}} \quad \text { and } \quad v_{\mathrm{av}}^{\prime}=\frac{N v_{\mathrm{av}}+u}{N+1}$$ (c) Use the expressions in part (b) to show that $v^{\prime}$ rms $>v^{\prime}$ av regardless of the numerical value of $u .$ (d) Explain why your results for (a) and (c) together show that $v_{\text { rms }}>v_{\text { av }}$ for any collection of particles if the particles do not all have the speed.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator