Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 13

Temperature and the Ideal Gas - all with Video Answers

Educators

+ 3 more educators

Chapter Questions

Problem 1

On a warm summer day, the air temperature is $84^{\circ} \mathrm{F}$. Express this temperature in (a) ${ }^{\circ} \mathrm{C}$ and (b) kelvins.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 2

The temperature at which liquid nitrogen boils (at atmospheric pressure) is $77 \mathrm{K}$. Express this temperature in (a) ${ }^{\circ} \mathrm{C}$ and $(\mathrm{b}){ }^{\circ} \mathrm{F}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 3

(a) At what temperature (if any) does the numerical value of the temperature in Celsius degrees equal its numerical value in Fahrenheit degrees? (b) At what temperature (if any) does the numerical value of the temperature in kelvins equal its numerical value in Fahrenheit degrees?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 4

A room air conditioner causes a temperature change of $-6.0^{\circ} \mathrm{C}$ (a) What is the temperature change in kelvins? (b) What is the temperature change in ${ }^{\circ} \mathrm{F} ?$

Shoukat Ali

Other Schools

Problem 5

Aliens from the planet Jeenkah have based their temperature scale on the boiling and freezing temperatures of ethyl alcohol. These temperatures are $78^{\circ} \mathrm{C}$ and $-114^{\circ} \mathrm{C}$, respectively. The people of Jeenkah have six digits on each hand, so they use a base-12 number system and have decided to have $144^{\circ} \mathrm{J}$ between the freezing and boiling temperatures of ethyl alcohol. They set the freezing point to $0^{\circ} \mathrm{J} .$ How would you convert from ${ }^{\circ} \mathrm{J}$ to ${ }^{\circ} \mathrm{C} ?$

Prabhat Tyagi

Numerade Educator

Problem 6

Five slabs with temperature coefficients of expansion $\alpha$ have lengths $L$ at $T_{\mathrm{i}}=20^{\circ} \mathrm{C} .$ Their temperatures then rise to $T_{\mathrm{f}}$. Rank them in order of how much their lengths increase, greatest to smallest.
(a) $L=90 \mathrm{cm}, T_{\mathrm{f}}=40^{\circ} \mathrm{C}, \alpha=8 \times 10^{-6} \mathrm{K}^{-1}$ (granite)
(b) $L=90 \mathrm{cm}, T_{\mathrm{f}}=50^{\circ} \mathrm{C}, \alpha=8 \times 10^{-6} \mathrm{K}^{-1}$ (granite)
(c) $L=60 \mathrm{cm}, T_{\mathrm{f}}=40^{\circ} \mathrm{C}, \alpha=8 \times 10^{-6} \mathrm{K}^{-1}$ (granite)
(d) $L=90 \mathrm{cm}, T_{\mathrm{f}}=40^{\circ} \mathrm{C}, \alpha=12 \times 10^{-6} \mathrm{K}^{-1}$ (concrete)
(e) $L=60 \mathrm{cm}, T_{\mathrm{f}}=50^{\circ} \mathrm{C}, \alpha=12 \times 10^{-6} \mathrm{K}^{-1}$ (concrete)

Shoukat Ali

Other Schools

Problem 7

A $2.4 \mathrm{m}$ length of copper pipe extends directly from a water heater in a basement to a faucet on the first floor of a house. If the faucet isn't fixed in place, how much will it rise when the pipe is heated from $20.0^{\circ} \mathrm{C}$ to $90.0^{\circ} \mathrm{C} ?$ Ignore any increase in the size of the faucet itself or of the water heater.

Narayan Hari

Numerade Educator

Problem 8

Two $35.0 \mathrm{cm}$ metal rods, one made of copper and one made of aluminum, are placed end to end, touching each other. One end is fixed, so that it cannot move. The rods are heated from $0.0^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C}$. How far does the other end of the system of rods move?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 9

Steel railroad tracks of length $18.30 \mathrm{m}$ are laid at $10.0^{\circ} \mathrm{C}$. How much space should be left between the track sections if they are to just touch when the temperature is $50.0^{\circ} \mathrm{C} ?$

Narayan Hari

Numerade Educator

Problem 10

A highway is made of concrete slabs that are 15 m long at $20.0^{\circ} \mathrm{C}$ (a) If the temperature range at the location of the highway is from $-20.0^{\circ} \mathrm{C}$ to $+40.0^{\circ} \mathrm{C},$ what size expansion gap should be left (at $20.0^{\circ} \mathrm{C}$ ) to prevent buckling of the highway? (b) How large are the gaps at $-20.0^{\circ} \mathrm{C} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 11

A lead rod and a common glass rod both have the same length when at $20.0^{\circ} \mathrm{C} .$ The lead rod is heated to $50.0^{\circ} \mathrm{C}$. To what temperature must the glass rod be heated so that they are again at the same length?

Dominador Tan

Numerade Educator

Problem 12

The coefficient of linear expansion of brass is $1.9 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1} .$ At $20.0^{\circ} \mathrm{C},$ a hole in a sheet of brass has an area of $1.00 \mathrm{mm}^{2} .$ How much larger is the area of the hole at $30.0^{\circ} \mathrm{C} ?$

Shoukat Ali

Other Schools

Problem 13

Aluminum rivets used in airplane construction are made slightly too large for the rivet holes to be sure of a tight fit. The rivets are cooled with dry ice $\left(-78.5^{\circ} \mathrm{C}\right)$ before they are driven into the holes. If the holes have a diameter of $0.6350 \mathrm{cm}$ at $20.5^{\circ} \mathrm{C},$ what should be the diameter of the rivets at $20.5^{\circ} \mathrm{C}$ if they are to just fit when cooled to the temperature of dry ice?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 14

The George Washington Bridge crosses the Hudson River between New York and New Jersey. The span of the steel bridge is about $1.6 \mathrm{km}$. If the temperature can vary from a low of $-15^{\circ} \mathrm{F}$ in winter to a high of $105^{\circ} \mathrm{F}$ in summer, by how much might the length of the span change over an entire year?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 15

$950 \mathrm{km} / \mathrm{h} .$ Suppose we wanted to heat a full-size model of the airbus made of aluminum to cause the same increase in circumference without changing the pressure. What would be the increase in temperature needed?

Narayan Hari

Numerade Educator

Problem 16

Suppose you have a filling in one of your teeth, and, while eating some ice cream, you suddenly realize that the filling came out. One of the reasons the filling may have become detached from your tooth is the differential contraction of the filling relative to the rest of the tooth due to the temperature change. (a) Find the change in volume for a metallic dental filling due to the difference between body temperature $\left(37^{\circ} \mathrm{C}\right)$ and the temperature of the ice cream you ate $\left(-5^{\circ} \mathrm{C}\right) .$ The initial volume of the filling is $30 \mathrm{mm}^{3},$ and its expansion coefficient is $\alpha=42 \times 10^{-6} \mathrm{K}^{-1} .$ (b) Find the change in volume of the cavity. The expansion coefficient of the tooth is $\alpha=$ $17 \times 10^{-6} \mathrm{K}^{-1}$.

Shoukat Ali

Other Schools

Problem 17

A cylindrical brass container with a base of $75.0 \mathrm{cm}^{2}$ and height of $20.0 \mathrm{cm}$ is filled to the brim with water when the system is at $25.0^{\circ} \mathrm{C}$. How much water overflows when the temperature of the water and the container is raised to $95.0^{\circ} \mathrm{C}$ ?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 18

An ordinary drinking glass is filled to the brim with water $(268.4 \mathrm{mL})$ at $2.0^{\circ} \mathrm{C}$ and placed on the sunny pool deck for a swimmer to enjoy. If the temperature of the water rises to $32.0^{\circ} \mathrm{C}$ before the swimmer reaches for the glass, how much water will have spilled over the top of the glass? Assume the glass does not expand.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 19

Consider the situation described in Problem 18. (a) Take into account the expansion of the glass and calculate how much water will spill out of the glass. Compare your answer with the case where the expansion of the glass was not considered. (b) By what percentage has the answer changed when the expansion of the glass is considered?

Narayan Hari

Numerade Educator

Problem 20

A steel sphere with radius $1.0010 \mathrm{cm}$ at $22.0^{\circ} \mathrm{C}$ must slip through a brass ring that has an internal radius of $1.0000 \mathrm{cm}$ at the same temperature. To what temperature must the brass ring be heated so that the sphere, still at $22.0^{\circ} \mathrm{C},$ can just slip through?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 21

A square brass plate, $8.00 \mathrm{cm}$ on a side, has a hole cut into its center of area $4.90874 \mathrm{cm}^{2}$ (at $\left.20.0^{\circ} \mathrm{C}\right)$. The hole in the plate is to slide over a cylindrical steel shaft of cross-sectional area $4.91000 \mathrm{cm}^{2}$ (also at $20.0^{\circ} \mathrm{C}$ ). To what temperature must the brass plate be heated so that it can just slide over the steel cylinder (which remains at $\left.20.0^{\circ} \mathrm{C}\right) ?[$ Hint: The steel cylinder is not heated so it does not expand; only the brass plate is heated.]

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 22

A copper washer is to be fit in place over a steel bolt. Both pieces of metal are at $20.0^{\circ} \mathrm{C}$. If the diameter of the bolt is $1.0000 \mathrm{cm}$ and the inner diameter of the washer is $0.9980 \mathrm{cm},$ to what temperature must the washer be raised so it will fit over the bolt? Only the copper washer is heated.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 23

Repeat Problem 22 , but now the copper washer and the steel bolt are both raised to the same temperature. At what temperature will the washer fit on the bolt?

Shoukat Ali

Other Schools

Problem 24

A steel rule is calibrated for measuring lengths at $20.00^{\circ} \mathrm{C} .$ The rule is used to measure the length of a Vycor glass brick; when both are at $20.00^{\circ} \mathrm{C},$ the brick is found to be $25.00 \mathrm{cm}$ long. If the rule and the brick are both at $80.00^{\circ} \mathrm{C},$ what would be the length of the brick as measured by the rule?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 25

A flat square of side $s_{0}$ at temperature $T_{0}$ expands by $\Delta s$ in both length and width when the temperature increases by $\Delta T .$ The original area is $s_{0}^{2}=A_{0}$ and the final area is $\left(s_{0}+\Delta s\right)^{2}=A .$ Show that if $\Delta s \ll s_{0}$
$$
\frac{\Delta A}{A_{0}}=2 \alpha \Delta T
$$
(Although we derive this relation for a square plate, it applies to a flat area of any shape.)

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 26

The volume of a solid cube with side $s_{0}$ at temperature $T_{0}$ is $V_{0}=s_{0}^{3} .$ Show that if $\Delta s \ll s_{0},$ the change in volume $\Delta V$ due to a change in temperature $\Delta T$ is given by
$$\frac{\Delta V}{V_{0}}=3 \alpha \Delta T$$
and therefore that $\beta=3 \alpha$. (Although we derive this relation for a cube, it applies to a solid of any shape.)

Narayan Hari

Numerade Educator

Problem 27

Use the definition that 1 mol of ${ }^{12} \mathrm{C}$ (carbon-12) atoms has a mass of exactly 12 g, along with Avogadro's number, to derive the conversion between atomic mass units and kg.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 28

Find the molar mass of ammonia ( $\mathrm{NH}_{3}$ ).

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 29

Find the mass (in $\mathrm{kg}$ ) of one molecule of $\mathrm{CO}_{2}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 30

The mass of 1 mol of ${ }^{13} \mathrm{C}$ (carbon-13) is $13.003 \mathrm{g}$.
(a) What is the mass in $u$ of one ${ }^{13} \mathrm{C}$ atom?
(b) What is the mass in kilograms of one ${ }^{13} \mathrm{C}$ atom?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 31

Estimate the number of $\mathrm{H}_{2} \mathrm{O}$ molecules in a human body of mass $80.2 \mathrm{kg}$. Assume that, on average, water makes up about $62 \%$ of the mass of a human body.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 32

The mass density of diamond (a crystalline form of carbon) is $3500 \mathrm{kg} / \mathrm{m}^{3} .$ How many carbon atoms per cubic centimeter are there?

Narayan Hari

Numerade Educator

Problem 33

How many hydrogen atoms are present in 684.6 g of sucrose $\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right) ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 34

How many moles of He are in $13 \mathrm{g}$ of He?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 35

The principal component of natural gas is methane $\left(\mathrm{CH}_{4}\right) .$ How many moles of $\mathrm{CH}_{4}$ are present in $144.36 \mathrm{g}$ of methane?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 36

What is the mass of one gold atom in kilograms?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 37

Air at room temperature and atmospheric pressure has a mass density of $1.2 \mathrm{kg} / \mathrm{m}^{3} .$ The average molecular mass of air is 29.0 u. How many molecules are in $1.0 \mathrm{cm}^{3}$ of air?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 38

At $0.0^{\circ} \mathrm{C}$ and 1.00 atm, $1.00 \mathrm{mol}$ of a gas occupies a volume of $0.0224 \mathrm{m}^{3} .$ (a) What is the number density? (b) Estimate the average distance between the molecules. (c) If the gas is nitrogen $\left(\mathrm{N}_{2}\right)$, the principal component of air, what is the total mass and mass density?

Rashmi Sinha

Numerade Educator

Problem 39

Sand is composed of $\mathrm{SiO}_{2} .$ Find the order of magnitude of the number of silicon (Si) atoms in a grain of sand. Approximate the sand grain as a sphere of diameter $0.5 \mathrm{mm}$ and an $\mathrm{SiO}_{2}$ molecule as a sphere of diameter $0.5 \mathrm{nm}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 40

A flight attendant wants to change the temperature of the air in the cabin from $18.0^{\circ} \mathrm{C}$ to $21.0^{\circ} \mathrm{C}$ without changing the pressure. What fractional change in the number of moles of air in the cabin would be required?

Shoukat Ali

Other Schools

Problem 41

A cylinder in a car engine takes $V_{\mathrm{i}}=4.50 \times 10^{-2} \mathrm{m}^{3}$ of air into the chamber at $30^{\circ} \mathrm{C}$ and at atmospheric pressure. The piston then compresses the air to one-ninth of the original volume $\left(0.111 \mathrm{V}_{\mathrm{i}}\right)$ and to $20.0 \mathrm{times}$ the original pressure $\left(20.0 P_{i}\right) .$ What is the new temperature of the air?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 42

A tire with an inner volume of $0.0250 \mathrm{m}^{3}$ is filled with air at a gauge pressure of $36.0 \mathrm{lb} / \mathrm{in}^{2} .$ If the tire valve is opened to the atmosphere, what volume outside of the tire does the escaping air occupy? Some air remains within the tire occupying the original volume, but now that remaining air is at atmospheric pressure. Assume the temperature of the air does not change.

Narayan Hari

Numerade Educator

Problem 43

Six cylinders contain ideal gases (not necessarily the same gas) with the properties given $(P=$ pressure, $V=$ volume, $N=$ number of molecules). Rank them in order of temperature, highest to lowest.
(a) $P=100 \mathrm{kPa}, V=4 \mathrm{L}, N=6 \times 10^{23}$
(b) $P=200 \mathrm{kPa}, V=4 \mathrm{L}, N=6 \times 10^{23}$
(c) $P=50 \mathrm{kPa}, V=8 \mathrm{L}, N=6 \times 10^{23}$
(d) $P=100 \mathrm{kPa}, V=4 \mathrm{L}, N=3 \times 10^{23}$
(e) $P=100 \mathrm{kPa}, V=2 \mathrm{L}, N=3 \times 10^{23}$
(f) $P=50 \mathrm{kPa}, V=4 \mathrm{L}, N=3 \times 10^{23}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 44

What fraction of the air molecules in a house must be pushed outside while the furnace raises the inside temperature from $16.0^{\circ} \mathrm{C}$ to $20.0^{\circ} \mathrm{C} ?$ The pressure does not change since the house is not airtight.

Narayan Hari

Numerade Educator

Problem 45

A patient with emphysema is breathing pure $\mathrm{O}_{2}$ through a face mask. The cylinder of $\mathrm{O}_{2}$ contains $0.0170 \mathrm{m}^{3}$ of $\mathrm{O}_{2}$ gas at a pressure of $15.2 \mathrm{MPa}$. (a) What volume would the oxygen occupy at atmospheric pressure (and the same temperature)? (b) If the patient takes in $8.0 \mathrm{L} / \mathrm{min}$ of $\mathrm{O}_{2}$ at atmospheric pressure, how long will the cylinder last?

Narayan Hari

Numerade Educator

Problem 46

Incandescent lightbulbs are filled with an inert gas to lengthen the filament life. With the current off (at $\left.T=20.0^{\circ} \mathrm{C}\right),$ the gas inside a lightbulb has a pressure of $115 \mathrm{kPa} .$ When the bulb is burning, the temperature rises to $70.0^{\circ} \mathrm{C} .$ What is the pressure at the higher temperature?

Narayan Hari

Numerade Educator

Problem 47

What is the mass density of air at $P=1.0$ atm and $T=(a)-10^{\circ} \mathrm{C}$ and (b) $30^{\circ} \mathrm{C}$ ? The average molecular mass of air is approximately 29 u.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 48

A constant volume gas thermometer containing helium is immersed in boiling ammonia $\left(-33^{\circ} \mathrm{C}\right)$, and the pressure is read once equilibrium is reached. The thermometer is then moved to a bath of boiling water $\left(100.0^{\circ} \mathrm{C}\right) .$ After the manometer was adjusted to keep the volume of helium constant, by what factor was the pressure multiplied?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 49

A hydrogen balloon at Earth's surface has a volume of $5.0 \mathrm{m}^{3}$ on a day when the temperature is $27^{\circ} \mathrm{C}$ and the pressure is $1.00 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} .$ The balloon rises and expands as the pressure drops. What would the volume of the same number of moles of hydrogen be at an altitude of $40 \mathrm{km}$ where the pressure is $0.33 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}$ and the temperature is $-13^{\circ} \mathrm{C} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 50

An ideal gas that occupies $1.2 \mathrm{m}^{3}$ at a pressure of $1.0 \times 10^{5} \mathrm{Pa}$ and a temperature of $27^{\circ} \mathrm{C}$ is compressed to a volume of $0.60 \mathrm{m}^{3}$ and heated to a temperature of $227^{\circ} \mathrm{C} .$ What is the new pressure?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 51

In intergalactic space, there is an average of about one hydrogen atom per cubic centimeter and the temperature is $3 \mathrm{K}$. What is the absolute pressure?

Narayan Hari

Numerade Educator

Problem 52

A tank of compressed air of volume $1.0 \mathrm{m}^{3}$ is pressurized to $20.0 \mathrm{atm}$ at $T=273 \mathrm{K}$. A valve is opened, and air is released until the pressure in the tank is 15.0 atm. How many molecules were released?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 53

A mass of $0.532 \mathrm{kg}$ of molecular oxygen is contained in a cylinder at a pressure of $1.0 \times 10^{5} \mathrm{Pa}$ and a temperature of $0.0^{\circ} \mathrm{C} .$ What volume does the gas occupy?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 54

Verify, using the ideal gas law, the assertion in Problem 38 that $1.00 \mathrm{mol}$ of a gas at $0.0^{\circ} \mathrm{C}$ and 1.00 atm occupies a volume of $0.0224 \mathrm{m}^{3}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 55

A bubble rises from the bottom of a lake of depth $80.0 \mathrm{m},$ where the temperature is $4^{\circ} \mathrm{C} .$ The water temperature at the surface is $18^{\circ} \mathrm{C} .$ If the bubble's initial diameter is $1.00 \mathrm{mm},$ what is its diameter when it reaches the surface? (Ignore the surface tension of water. Assume the bubble warms as it rises to the same temperature as the water and retains a spherical shape. Assume $P_{\mathrm{atm}}=1.0 \mathrm{atm}.$)

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 56

Consider the expansion of an ideal gas at constant pressure. The initial temperature is $T_{0}$ and the initial volume is $V_{0}$. (a) Show that $\Delta V / V_{0}=\beta \Delta T,$ where $\beta=$ $1 / T_{0} \cdot$ (b) Compare the coefficient of volume expansion $\beta$ for an ideal gas at $20^{\circ} \mathrm{C}$ to the values for liquids and gases listed in Table $13.3 .$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 57

What is the temperature of an ideal gas whose molecules have an average translational kinetic energy of $3.20 \times 10^{-20} \mathrm{J} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 58

What is the total translational kinetic energy of the gas molecules of air at atmospheric pressure that occupies a volume of $1.00 \mathrm{L} ?$

Narayan Hari

Numerade Educator

Problem 59

What is the kinetic energy per unit volume in an ideal gas at (a) $P=1.00$ atm and (b) $P=300.0$ atm?

Vipender Yadav

Numerade Educator

Problem 60

Show that, for an ideal gas,
$$P=\frac{1}{3} \rho v_{\mathrm{rms}}^{2}$$
where $P$ is the pressure, $\rho$ is the mass density, and $v_{\mathrm{rms}}$ is the rms speed of the gas molecules.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 61

Rank the six gases of Problem 43 in order of the total translational kinetic energy, greatest to least.

Shoukat Ali

Other Schools

Problem 62

What is the total internal kinetic energy of $1.0 \mathrm{mol}$ of an ideal gas at $0.0^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 63

If $2.0 \mathrm{mol}$ of nitrogen gas $\left(\mathrm{N}_{2}\right)$ are placed in a cubic box, $25 \mathrm{cm}$ on each side, at $1.6 \mathrm{atm}$ of pressure, what is the rms speed of the nitrogen molecules?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 64

There are two identical containers of gas at the same temperature and pressure, one containing argon and the other neon. What is the ratio of the rms speed of the argon atoms to that of the neon atoms? The atomic mass of argon is twice that of neon.

Narayan Hari

Numerade Educator

Problem 65

A smoke particle has a mass of $1.38 \times 10^{-17} \mathrm{kg}$, and it is randomly moving about in thermal equilibrium with room temperature air at $27^{\circ} \mathrm{C} .$ What is the rms speed of the particle?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 66

Find the rms speed in air at $0.0^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm}$ of $(\mathrm{a})$ the $\mathrm{N}_{2}$ molecules, (b) the $\mathrm{O}_{2}$ molecules, and (c) the $\mathrm{CO}_{2}$ molecules.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 67

What are the rms speeds of helium atoms, and nitrogen, hydrogen, and oxygen molecules at $25^{\circ} \mathrm{C} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 68

If the upper atmosphere of Jupiter has a temperature of $160 \mathrm{K}$ and the escape speed is $60 \mathrm{km} / \mathrm{s},$ would an astronaut expect to find much hydrogen there?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 69

What is the temperature of an ideal gas whose molecules in random motion have an average translational kinetic energy of $4.60 \times 10^{-20} \mathrm{J} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 70

On a cold day, you take a breath, inhaling 0.50 L of air whose initial temperature is $-10^{\circ} \mathrm{C}$. In your lungs, its temperature is raised to $37^{\circ} \mathrm{C}$. Assume that the pressure is $101 \mathrm{kPa}$ and that the air may be treated as an ideal gas. What is the total change in translational kinetic energy of the air you inhaled?

Shoukat Ali

Other Schools

Problem 71

Show that the rms speed of a molecule in an ideal gas at absolute temperature $T$ is given by
$$v_{\mathrm{rms}}=\sqrt{\frac{3 k_{\mathrm{B}} T}{m}}$$
where $m$ is the mass of a molecule.

Narayan Hari

Numerade Educator

Problem 72

Show that the rms speed of a molecule in an ideal gas at absolute temperature $T$ is given by
$$v_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}$$
where $M$ is the molar mass $-$ the mass of the gas per mole.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 73

The reaction rate for the prepupal development of male Drosophila is temperature-dependent. Assuming that the reaction rate is exponential as in Eq. $(13-41)$, the activation energy for this development is $2.81 \times 10^{-19}$ J. A Drosophila is originally at $10.00^{\circ} \mathrm{C}$, and its temperature is increasing. If the rate of development has increased $3.50 \%$, how much has its temperature increased?

Shoukat Ali

Other Schools

Problem 74

The reaction rate for the hydrolysis of benzoyl-Larginine amide by trypsin at $10.0^{\circ} \mathrm{C}$ is 1.878 times faster than that at $5.0^{\circ} \mathrm{C}$. Assuming that the reaction rate is exponential as in Eq. $(13-41),$ what is the activation energy?

Shoukat Ali

Other Schools

Problem 75

At high altitudes, water boils at a temperature lower than $100.0^{\circ} \mathrm{C}$ due to the lower air pressure. A rule of thumb states that the time to hard-boil an egg doubles for every $10.0^{\circ} \mathrm{C}$ drop in temperature. What activation energy does this rule imply for the chemical reactions that occur when the egg is cooked?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 76

Estimate the mean free path of a $\mathrm{N}_{2}$ molecule in air at (a) sea level $(P \approx 100 \mathrm{kPa}$ and $T \approx 290 \mathrm{K}),$ (b) the top of Mt. Everest (altitude $=8.8 \mathrm{km}, P \approx 50 \mathrm{kPa},$ and $T \approx 230 \mathrm{K}),$ and (c) an altitude of $30 \mathrm{km}(P \approx 1 \mathrm{kPa}$ and $T \approx 230 \mathrm{K})$. For simplicity, assume that air is pure nitrogen gas. The diameter of a $\mathrm{N}_{2}$ molecule is approximately $0.3 \mathrm{nm}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 77

About how long will it take a perfume molecule to diffuse a distance of $5.00 \mathrm{m}$ in one direction in a room if the diffusion constant is $1.00 \times 10^{-5} \mathrm{m}^{2} / \mathrm{s} ?$ Assume that the air is perfectly still-there are no air currents.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 78

Estimate the time it takes a sucrose molecule to move $5.00 \mathrm{mm}$ in one direction by diffusion in water. Assume there are no currents in the water.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 79

Your friend is $3.0 \mathrm{m}$ away from you in a room. There are no significant air currents. She opens a bottle of perfume, and you first smell it 20 s later. How long would it have taken for you to smell it if she had been $6.0 \mathrm{m}$ away instead?

Narayan Hari

Numerade Educator

Problem 80

Platelet cells in blood play an essential role in the formation of clots and exist in normal human blood at the level of about 200000 per cubic millimeter. In order to illustrate that diffusion alone is not responsible for transporting platelets, consider the following situation. The diffusion constant for platelets in blood is approximately $5 \times 10^{-10} \mathrm{m}^{2} / \mathrm{s}$. About how long would it take a platelet to diffuse from the center of an artery (diameter $8.0 \mathrm{mm}$ ) to a clot forming on one wall of the artery?

Shoukat Ali

Other Schools

Problem 81

In plants, water diffuses out through small openings known as stomatal pores. If $D=2.4 \times 10^{-5} \mathrm{m}^{2} / \mathrm{s}$ for water vapor in air, and the length of the pores is $2.5 \times$ $10^{-5} \mathrm{m},$ how long does it take for a water molecule to diffuse out through the pore?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 82

Agnes Pockels $(1862-1935)$ was able to determine Avogadro's number using only a few household chemicals, in particular oleic acid, whose formula is $\mathrm{C}_{18} \mathrm{H}_{34} \mathrm{O}_{2}$. (a) What is the molar mass of this acid? (b) The mass of one drop of oleic acid is $2.3 \times 10^{-5} \mathrm{g}$ and the volume is $2.6 \times 10^{-5} \mathrm{cm}^{3}$ How many moles of oleic acid are there in one drop? (c) When oleic acid is spread out on water, it lines up in a layer one molecule thick. If the base of the molecule of oleic acid is a square of side $d$, the height of the molecule is known to be $7 d$. Pockels spread out one drop of oleic acid on some water, and measured the area to be $70.0 \mathrm{cm}^{2}$. Using the volume and the area of oleic acid, what is $d ?$ (d) If we assume that this film is one molecule thick, how many molecules of oleic acid are there in the drop? (e) What value does this give you for Avogadro's number?

Shoukat Ali

Other Schools

Problem 83

As a Boeing 747 gains altitude, the passenger cabin is pressurized. However, the cabin is not pressurized fully to atmospheric $\left(1.01 \times 10^{5} \mathrm{Pa}\right),$ as it would be at sea level, but rather pressurized to $7.62 \times 10^{4} \mathrm{Pa}$. Suppose a 747 takes off from sea level when the temperature in the airplane is $25.0^{\circ} \mathrm{C}$ and the pressure is $1.01 \times 10^{5} \mathrm{Pa}$. (a) If the cabin temperature remains at $25.0^{\circ} \mathrm{C}$, what is the percentage change in the number of moles of air in the cabin? (b) If instead, the number of moles of air in the cabin does not change, what would the temperature be?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 84

For divers going to great depths, the composition of the air in the tank must be modified. The ideal composition is to have approximately the same number of $\mathrm{O}_{2}$ molecules per unit volume as in surface air (to avoid oxygen poisoning), and to use helium instead of nitrogen for the remainder of the gas (to avoid nitrogen narcosis, which results from nitrogen dissolving in the bloodstream). Of the molecules in dry surface air, $78 \%$ are $\mathrm{N}_{2}, 21 \%$ are $\mathrm{O}_{2},$ and $1 \%$ are $\mathrm{Ar}$. (a) How many $\mathrm{O}_{2}$ molecules per cubic meter are there in surface air at $20.0^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm} ?$ (b) For a diver going to a depth of $100.0 \mathrm{m},$ what percentage of the gas molecules in the tank should be $\mathrm{O}_{2} ?$ (Assume that the density of seawater is $1025 \mathrm{kg} / \mathrm{m}^{3}$ and the temperature is $20.0^{\circ} \mathrm{C} .$ )

Shoukat Ali

Other Schools

Problem 85

If you wanted to make a scale model of air al $0.0^{\circ} \mathrm{C}$ and 1.00 atm, using Ping-Pong balls (diameter, $3.75 \mathrm{cm}$ ) to represent the $\mathrm{N}_{2}$ molecules (diameter, $0.30 \mathrm{nm}$ ), (a) how far apart on average should the Ping-Pong balls be at any instant? (b) How far would a Ping-Pong ball travel on average before colliding with another?

Shoukat Ali

Other Schools

Problem 86

A Pyrex container is filled to the very top with $4.00 \mathrm{L}$ of water. Both the container and the water are at a temperature of $90.0^{\circ} \mathrm{C} .$ When the temperature has cooled to $20.0^{\circ} \mathrm{C},$ how much additional water can be added to the container?

Shoukat Ali

Other Schools

Problem 87

A hot air balloon with a volume of $12.0 \mathrm{m}^{3}$ is initially filled with air at a pressure of 1.00 atm and a temperature of 19.0 $^{\circ} \mathrm{C} .$ When the balloon air is heated, the volume and the pressure of the balloon remain constant because the balloon is open to the atmosphere at the bottom. How many moles are in the balloon when the air is heated to $40.0^{\circ} \mathrm{C} ?$

Shoukat Ali

Other Schools

Problem 88

In a certain bimetallic strip (see Fig. 13.7) the brass strip is $0.100 \%$ longer than the steel strip at a temperature of $275^{\circ} \mathrm{C} .$ At what temperature do the two strips have the same length?

Shoukat Ali

Other Schools

Problem 89

The driver from Practice Problem 13.3 fills his $18.9 \mathrm{L}$ steel gasoline can in the morning when the temperature of the can and the gasoline is $15.0^{\circ} \mathrm{C}$ and the pressure is $1.0 \mathrm{atm},$ but this time he remembers to replace the tightly fitting cap after filling the can. Assume that the can is completely full of gasoline (no air space) and that the cap does not leak. The temperature climbs to $30.0^{\circ} \mathrm{C}$. Ignoring the expansion of the steel can, what would be the pressure of the gasoline? The bulk modulus for gasline is $1.00 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$

Shoukat Ali

Other Schools

Problem 90

An iron bridge girder $\left(Y=2.0 \times 10^{11} \mathrm{N} / \mathrm{m}^{2}\right)$ is constrained between two rock faces whose spacing doesn't change. At $20.0^{\circ} \mathrm{C}$ the girder is relaxed. How large a stress develops in the iron if the sun heats the girder to $40.0^{\circ} \mathrm{C} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 91

Consider the sphere and ring of Problem 20. What must the final temperature be if both the ring and the sphere are heated to the same final temperature?

Narayan Hari

Numerade Educator

Problem 92

Suppose due to a bad break of your femur, you require the insertion of a titanium rod to help the fracture heal. The coefficient of linear expansion for titanium is $\alpha=8.6 \times 10^{-6} \mathrm{K}^{-1},$ and the length of the rod when it is in equilibrium with the leg bone and muscle at $37^{\circ} \mathrm{C}$ is $5.00 \mathrm{cm} .$ How much shorter was the rod at room temperature $\left(20^{\circ} \mathrm{C}\right) ?$

Narayan Hari

Numerade Educator

Problem 93

A certain acid has a molecular mass of 63 u. By mass, it consists of $1.6 \%$ hydrogen, $22.2 \%$ nitrogen, and $76.2 \%$ oxygen. What is the chemical formula for this acid?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 94

The data in the following table are from a constantvolume gas thermometer experiment. The volume of the gas was kept constant, while the temperature was changed. The resulting pressure was measured. Plot the data on a pressure versus temperature diagram. Based on these data, estimate the value of absolute zero in Celsius.
$$\begin{array}{rl}\hline T\left({ }^{\circ} \mathrm{C}\right) & P(\mathrm{atm}) \\0 & 1.00 \\20 & 1.07 \\100 & 1.37 \\-33 & 0.88 \\-196 & 0.28 \\\hline\end{array}$$

Shoukat Ali

Other Schools

Problem 95

At a normal body temperature of $37.0^{\circ} \mathrm{C},$ (a) what is the average kinetic energy of the gas molecules in the lungs? (b) If a fever increases the temperature to $37.8^{\circ} \mathrm{C},$ by what percentage does the average kinetic energy of the molecules increase?

Narayan Hari

Numerade Educator

Problem 96

The volume of air taken in by a warm-blooded vertebrate in the Andes Mountains is 210 L/day at standard temperature and pressure (i.e., $0^{\circ} \mathrm{C}$ and 1 atm). If the air in the lungs is at $39^{\circ} \mathrm{C}$, under a pressure of $450 \mathrm{mm} \mathrm{Hg},$ and we assume that the vertebrate takes in an average volume of $100 \mathrm{cm}^{3}$ per breath at the temperature and pressure of its lungs, how many breaths does this vertebrate take per day?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 97

An iron cannonball of radius $0.08 \mathrm{m}$ has a cavity of radius $0.05 \mathrm{m}$ that is to be filled with gunpowder. If the measurements were made at a temperature of $22^{\circ} \mathrm{C}$, how much extra volume of gunpowder, if any, will be required to fill 500 cannonballs when the temperature is $30^{\circ} \mathrm{C} ?$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 98

Ten students take a test and get the following scores: $83,62,81,77,68,92,88,83,72,$ and $75 .$ What are the average value, the rms value, and the most probable value, respectively, of these test scores?

Narayan Hari

Numerade Educator

Problem 99

A hand pump is being used to inflate a bicycle tire that has a gauge pressure of $40.0 \mathrm{lb} / \mathrm{in}^{2} .$ If the pump is a cylinder of length 18.0 in. with a cross-sectional area of 3.00 in. $^{2},$ how far down must the piston be pushed before air will flow into the tire? Assume the air re- mains at constant temperature.

Narayan Hari

Numerade Educator

Problem 100

An ideal gas in a constant-volume gas thermometer (Fig. 13.11) is held at a volume of 0.500 L. As the temperature of the gas is increased by $20.0^{\circ} \mathrm{C}$, the mercury level on the right side of the manometer must rise by $8.00 \mathrm{mm}$ in order to keep the gas volume constant. (a) What is the slope of a graph of $P$ versus $T$ for this gas (in $\left.\mathrm{mmHg} /{ }^{\circ} \mathrm{C}\right) ?$ (b) How many moles of gas are present?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 101

A cylinder with an interior cross-sectional area of $70.0 \mathrm{cm}^{2}$ has a moveable piston of mass $5.40 \mathrm{kg}$ at the top that can move up and down without friction. The cylinder contains $2.25 \times 10^{-3} \mathrm{mol}$ of an ideal gas at 23.0 $^{\circ} \mathrm{C}$. (a) What is the volume of the gas when the piston is in equilibrium? Assume the air pressure outside the cylinder is 1.00 atm. (b) By what factor does the volume change if the gas temperature is raised to $223.0^{\circ} \mathrm{C}$ and the piston moves until it is again in equilibrium?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 102

Estimate the average distance between molecules in air at $0.0^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm}$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 103

Show that, in two gases at the same temperature, the rms speeds are inversely proportional to the square root of the molecular masses:
$$
\frac{\left(v_{\mathrm{rms}}\right)_{1}}{\left(v_{\mathrm{rms}}\right)_{2}}=\sqrt{\frac{m_{2}}{m_{1}}}
$$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 104

Ge alveoli (see Section 13.8 ) have an average radius of $0.125 \mathrm{mm}$ and are approximately spherical. If the pressure in the sacs is $1.00 \times 10^{5} \mathrm{Pa}$, and the temperature is $310 \mathrm{K}$ (average body temperature), how many air molecules are in an alveolus?

Narayan Hari

Numerade Educator

Problem 105

A $10.0 \mathrm{L}$ vessel contains $12 \mathrm{g}$ of $\mathrm{N}_{2}$ gas at $20^{\circ} \mathrm{C}$. (a) Estimate the nearest-neighbor distance. (b) Can the gas be considered to be dilute? [Hint: Compare the nearest-neighbor distance to the diameter of an $\mathrm{N}_{2}$ molecule, about $0.3 \mathrm{nm} .$]

Narayan Hari

Numerade Educator

Problem 106

During hibernation, an animal's metabolism slows down, and its body temperature lowers. For example, a California ground squirrel's body temperature lowers from $40.0^{\circ} \mathrm{C}$ to $10.0^{\circ} \mathrm{C}$ during hibernation. If we assume that the air in the squirrel's lungs is $75.0 \% \mathrm{N}_{2}$ and $25.0 \%$ $\mathrm{O}_{2},$ by how much will the rms speed of the air molecules in the lungs have decreased during hibernation?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 107

A steel ring of inner diameter $7.00000 \mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ is to be heated and placed over a brass shaft of outer diameter $7.00200 \mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$. (a) To what temperature must the ring be heated to fit over the shaft? The shaft remains at $20.0^{\circ} \mathrm{C}$. (b) Once the ring is on the shaft and has cooled to $20.0^{\circ} \mathrm{C},$ to what temperature must the ring plus shaft combination be cooled to allow the ring to slide off the shaft again?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 108

The inner tube of a Pyrex glass mercury thermometer has a diameter of $0.120 \mathrm{mm} .$ The bulb at the bottom of the thermometer contains $0.200 \mathrm{cm}^{3}$ of mercury. How far will the thread of mercury move for a change of $1.00^{\circ} \mathrm{C} ?$ Remember to take into account the expansion of the glass.

Narayan Hari

Numerade Educator

Problem 109

A wine barrel has a diameter at its widest point of $134.460 \mathrm{cm}$ at a temperature of $20.0^{\circ} \mathrm{C} .$ A circular iron band, of diameter $134.448 \mathrm{cm},$ is to be placed around the barrel at the widest spot. The iron band is $5.00 \mathrm{cm}$ wide and $0.500 \mathrm{cm}$ thick. (a) To what temperature must the band be heated to be able to fit it over the barrel? (b) Once the band is in place and cools to $20.0^{\circ} \mathrm{C}$, what will be the tension in the band?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 110

A bimetallic strip is made from metals with expansion coefficients $\alpha_{1}$ and $\alpha_{2}\left(\right.$ with $\left.\alpha_{2}>\alpha_{1}\right) .$ The thickness of each layer is $s .$ At some temperature $T_{0}$, the bimetallic strip is relaxed and straight. (a) Show that, at temperature $T_{0}+\Delta T,$ the radius of curvature of the strip is
$$R \approx \frac{s}{\left(\alpha_{2}-\alpha_{1}\right) \Delta T}$$
[Hint: At $T_{0},$ the lengths of the two layers are the same. At temperature $T_{0}+\Delta T,$ the layers form circular arcs of radii $R$ and $R+s,$ which subtend the same angle $\theta .$ Assume a small $\Delta T$ so that $\alpha \Delta T \ll 1$ (for either value of $\alpha$ ). $]$ (b) If the layers are made of iron and brass, with $s=$ $0.1 \mathrm{mm},$ what is $R$ for $\Delta T=20.0^{\circ} \mathrm{C} ?$

Shoukat Ali

Other Schools

Problem 111

Michael has set the gauge pressure of the tires on his car to $36.0 \mathrm{lb} / \mathrm{in}^{2} .$ He draws chalk lines around the edges of the tires where they touch the driveway surface to measure the area of contact between the tires and the ground. Each front tire has a contact area of 24.0 in. $^{2}$ and each rear tire has a contact area of 20.0 in. $^{2}$ (a) What is the weight (in lb) of the car? (b) The centerto-center distance between front and rear tires is $7.00 \mathrm{ft}$. Taking the straight line between the centers of the tires on the left side (driver's side) to be the $y$ -axis with the origin at the center of the front left tire (positive direction pointing forward), what is the $y$ -coordinate of the car's CM?

Manish Jain

Numerade Educator

Problem 112

(a) Calculate Earth's escape speed- -the minimum speed needed for an object near the surface to escape Earth's gravitational pull. [Hint: Use conservation of energy and ignore air resistance.] (b) Calculate the average speed of a hydrogen molecule $\left(\mathrm{H}_{2}\right)$ at $0^{\circ} \mathrm{C}$. (c) Calculate the average speed of an oxygen molecule $\left(\mathrm{O}_{2}\right)$ at $0^{\circ} \mathrm{C}$ (d) Use your answers from parts (a) through (c) along with what you know about the distribution of molecular speeds to explain why Earth's atmosphere contains plenty of oxygen but almost no hydrogen.

Shoukat Ali

Other Schools

Problem 113

A long, narrow steel rod of length $2.5000 \mathrm{m}$ at $25^{\circ} \mathrm{C}$ is oscillating as a pendulum about a horizontal axis through one end. If the temperature changes to $0^{\circ} \mathrm{C},$ what will be the fractional change in its period?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 114

A temperature change $\Delta T$ causes a volume change $\Delta V$ but has no effect on the mass of an object. (a) Show that the change in density $\Delta \rho$ is given by $\Delta \rho=-\beta \rho \Delta T$ (b) Find the fractional change in density $(\Delta \rho / \rho)$ of a brass sphere when the temperature changes from $32^{\circ} \mathrm{C}$ to $-10.0^{\circ} \mathrm{C}$.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 115

A diver rises quickly to the surface from a $5.0 \mathrm{m}$ depth. If she did not exhale the gas from her lungs before rising, by what factor would her lungs expand? Assume the temperature to be constant and the pressure in the lungs to match the pressure outside the diver's body. The density of seawater is $1.03 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}.$

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 116

A scuba diver has an air tank with a volume of $0.010 \mathrm{m}^{3} .$ The air in the tank is initially at a pressure of $1.0 \times 10^{7}$ Pa. Assuming that the diver breathes $0.500 \mathrm{L} / \mathrm{s}$ of air, find how long the lank will last at depths of (a) $2.0 \mathrm{m}$ and $(\mathrm{b}) 20.0 \mathrm{m} .$ (Make the same assumptions as in Example $13.6 .$

Narayan Hari

Numerade Educator

Problem 117

A sealed cylinder contains a sample of ideal gas at a pressure of 2.0 atm. The rms speed of the molecules is $v_{0} \cdot$ (a) If the rms speed is then reduced to $0.90 v_{0},$ what is the pressure of the gas? (b) By what percentage does the speed of sound in the gas change?

Manish Jain

Numerade Educator

Problem 118

Estimate the percentage of the $\mathrm{O}_{2}$ molecules in air at $30^{\circ} \mathrm{C}$ that are moving faster than the speed of sound in air at that temperature (see Fig. 13.13 ).

Manish Jain

Numerade Educator

Problem 119

The diameter of an oxygen $\left(\mathrm{O}_{2}\right)$ molecule is approximately $0.3 \mathrm{nm} .$ For an oxygen molecule in air at atmospheric pressure and $20^{\circ} \mathrm{C},$ estimate the average magnitudes of these quantities during a $1.0 \mathrm{s}$ time interval: (a) the distance traveled between collisions with other molecules; (b) the number of collisions; (c) the total distance traveled; (d) the displacement.

Manish Jain

Numerade Educator

Problem 120

A $12.0 \mathrm{cm}$ cylindrical chamber has an $8.00 \mathrm{cm}$ diameter piston attached to one end. The piston is connected to an ideal spring as shown. Initially, the gas inside the chamber is at atmospheric pressure and $20.0^{\circ} \mathrm{C}$ and the spring is not compressed. When a total of $6.50 \times 10^{-2} \mathrm{mol}$ of gas is added to the chamber at $20.0^{\circ} \mathrm{C},$ the spring compresses a distance of $\Delta x=5.40 \mathrm{cm} .$ What is the spring constant of the spring?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator