Problem 1

CE (a) Is the number of molecules in one mole of $N_{2}$ greater than, less than, or equal to the number of molecules in one mole of $\mathrm{O}_{2} ?$ (b) Is the mass of one mole of $\mathrm{N}_{2}$ greater than, less than, or equal to the mass of one mole of $\mathrm{O}_{2} ?$

Hugh R.

Numerade Educator

Problem 2

CE Predict/Explain If you put a helium-filled balloon in the

refrigerator, (a) will its volume increase, decrease, or stay the

same? (b) Choose the best explanation from among the following:

\begin{equation}

\begin{array}{l}{\text { I. Lowering the temperature of an ideal gas at constant pressure }} \\ {\text { results in a reduced volume. }} \\ {\text { II. The same amount of gas is in the balloon; therefore, its vol- }} \\ {\text { ume remains the same. }} \\ {\text { III. The balloon can expand more in the cool air of the refrigera- }} \\ {\text { tor, giving an increased volume. }}\end{array}

\end{equation}

Hugh R.

Numerade Educator

Problem 3

CE Two containers hold ideal gases at the same temperature. Container $A$ has twice the volume and half the number of molecules as container B. What is the ratio $P_{N} / P_{1}$ , where $P_{A}$ is the pressure in container $\mathrm{A}$ and $P_{\mathrm{B}}$ is the pressure in container $\mathrm{B} ?$

Hugh R.

Numerade Educator

Problem 4

Standard temperature and pressure (STP) is defined as a temperature of $0^{\circ} \mathrm{C}$ and a pressure of 101.3 $\mathrm{kPa}$ . What is the volume occupied by one mole of an ideal gas at STP?

Hugh R.

Numerade Educator

Problem 5

BIO After emptying her lungs, a person inhales 4.3 L of air at $0.0^{\circ} \mathrm{C}$ and holds her breath. How much does the volume of the air increase as it warms to her body temperature of $35^{\circ} \mathrm{C} ?$

Hugh R.

Numerade Educator

Problem 6

An automobile tire has a volume of 0.0185 $\mathrm{m}^{3} .$ At a temperature of 294 $\mathrm{K}$ the absolute pressure in the tire is 212 $\mathrm{kPa}$ . How many moles of air must be pumped into the tire to increase its pressure to 252 $\mathrm{kPa}$ , given that the temperature and volume of the tire remain constant?

Hugh R.

Numerade Educator

Problem 7

Amount of Helium in a Blimp The Goodyear blimp Spirit of Akron is 62.6 $\mathrm{m}$ long and contains 7023 $\mathrm{m}^{3}$ of helium. When the temperature of the helium is $285 \mathrm{K},$ its absolute pressure is 112 $\mathrm{kPa}$ . Find the mass of the helium in the blimp.

Hugh R.

Numerade Educator

Problem 8

A compressed-air tank holds 0.500 $\mathrm{m}^{3}$ of air at a temperature of 295 $\mathrm{K}$ and a pressure of 820 $\mathrm{kPa}$ . What volume would the air occupy if it were released into the atmosphere, where the pressure is 101 $\mathrm{kPa}$ and the temperature is 303 $\mathrm{K} ?$

Hugh R.

Numerade Educator

Problem 9

CE Four ideal gases have the following pressures, $P,$ volumes, $V$ and mole numbers, $n$ gas $A, P=100 \mathrm{kPa}, V=1 \mathrm{m}^{3}, n=10 \mathrm{mol}$ gas $\mathrm{B}, P=200 \mathrm{kPa}, V=2 \mathrm{m}^{3}, n=20 \mathrm{mol} ;$ gas $\mathrm{C}, P=50 \mathrm{kPa}$ $V=1 \mathrm{m}^{3}, n=50 \mathrm{mol} ;$ gas $\mathrm{D}, P=50 \mathrm{kPa}, V=4 \mathrm{m}^{3}, n=5 \mathrm{mol}$ . Rank these gases in order of increasing temperature. Indicate ties where appropriate.

Hugh R.

Numerade Educator

Problem 10

A balloon contains 3.9 liters of nitrogen gas at a temperature

of 84 $\mathrm{K}$ and a pressure of 101 $\mathrm{kPa}$ . If the temperature of the gas is

allowed to increase to $28^{\circ} \mathrm{C}$ and the pressure remains constant,

what volume will the gas occupy?

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Problem 11

Predict/Calculate A balloon is filled with helium at a pressure of $2.4 \times 10^{5} \mathrm{Pa}$ . The balloon is at a temperature of $18^{\circ} \mathrm{C}$ and has a radius of 0.25 $\mathrm{m}$ . (a) How many helium atoms are contained in the balloon? (b) Suppose we double the number of helium atoms in the balloon, keeping the pressure and the temperature fixed. By what factor does the radius of the balloon increase? Explain.

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Problem 12

Predict/Calculate A bicycle tire with a volume of 0.00212 $\mathrm{m}^{3}$ is filled to its recommended absolute pressure of 495 kPa on a cold winter day when the tire's temperature is $-15^{\circ} \mathrm{C}$ . The cyclist then brings his bicycle into a hot laundry room at $32^{\circ} \mathrm{C}$ (a) If the tire warms up while its volume remains constant, will the pressure increase be greater than, less than, or equal to the manufacturer's stated 10$\%$ overpressure limit? (b) Find the absolute pressure in

the tire when it warms to $32^{\circ} \mathrm{C}$ at constant volume.

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Problem 13

$\ \mathrm{A} 515-\mathrm{cm}^{3}$ flask contains 0.460 $\mathrm{g}$ of a gas at a pressure of 153 $\mathrm{kPa}$ and a temperature of 322 $\mathrm{K}$ . What is the molecular mass of this gas?

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Problem 14

Predict/Calculate The Atmosphere of Mars On Mars, the average temperature is $-64^{\circ} \mathrm{F}$ and the average atmospheric pressure is 0.92 $\mathrm{kP}$ . (a) What is the number of molecules per volume in the Martian atmosphere? (b) Is the number of molecules per volume on the Earth greater than, less than, or equal to the number per volume on Mars? Explain your reasoning. (c) Estimate the number of molecules per volume in Earth's atmosphere.

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Problem 15

The air inside a hot-air balloon has an average temperature of $79.2^{\circ} \mathrm{C}$ . The outside air has a temperature of $20.3^{\circ} \mathrm{C}$ . What is the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere?

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Problem 16

A cylindrical flask is fitted with an airtight piston that is free to slide up and down, as shown in FIGURE 17$\cdot 33 .$ A mass rests on top of the piston. The initial temperature of the system is 313 $\mathrm{K}$ and the pressure of the gas is held constant at 137 kPa. The temperature is now

increased until the height of the piston rises from 23.4 $\mathrm{cm}$ to 26.0 $\mathrm{cm} .$ What is the final temperature of the gas?

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Problem 17

Consider the system described in the previous problem. Contained within the flask is an ideal gas at a constant temperature of 313 $\mathrm{K}$ . Initially the pressure applied by the piston and the mass is

137 $\mathrm{kPa}$ and the height of the piston above the base of the flask is 23.4 cm. When additional mass is added to the piston, the height of the piston decreases to 20.0 $\mathrm{cm} .$ Find the new pressure applied by the piston.

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Problem 18

Predict/Calculate A gas has a temperature of 310 $\mathrm{K}$ and a pressure of 101 $\mathrm{kPa}$ (a) Find the volume occupied by 1.25 $\mathrm{mol}$ of this gas, assuming it is ideal. (b) Assuming the gas molecules can be approximated as small spheres of diameter $2.5 \times 10^{-10} \mathrm{m},$

determine the fraction of the volume found in part (a) that is occupied by the molecules. (c) In determining the properties of an ideal gas, we assume that molecules are points of zero volume. Discuss the validity of this assumption for the case considered here.

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Problem 19

ce Predict/Explain The air in your room is composed mostly of oxygen $\left(\mathrm{O}_{2}\right)$ and nitrogen $\left(\mathrm{N}_{2}\right)$ molecules. The oxygen molecules are more massive than the nitrogen molecules. (a) Is the speed of the $\mathrm{O}_{2}$ molecules greater than, less than, or equal to the rms speed of the $\mathrm{N}_{2}$ molecules? (b) Choose the best explanation from

among the following:

\begin{equation}

\begin{array}{l}{\text { I. The more massive oxygen molecules have greater momentum }} \\ {\text { and therefore greater speed. }} \\ {\text { II. Equal temperatures for the oxygen and nitrogen molecules }} \\ {\text { imply they have equal rms speeds. }}\\{\text { III. The temperature is the same for both molecules, and hence }} \\ {\text { their average kinetic energies are equal. As a result, the more }} \\ {\text { massive oxygen molecules have lower speeds. }}\end{array}

\end{equation}

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Problem 20

CE If the translational speed of molecules in an ideal gas is doubled, by what factor does the Kelvin temperature change? Explain.

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Problem 21

At what temperature is the rms speed of $\mathrm{H}_{2}$ equal to the rms

speed that $\mathrm{O}_{2}$ has at 303 $\mathrm{K}$ ?

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Problem 22

Suppose a planet has an atmosphere of pure ammonia at $5.5^{\circ} \mathrm{C}$. What is the rms speed of the ammonia molecules? (The molecular weight of ammonia, $\mathrm{NH}_{3},$ is 17.03 $\mathrm{g} / \mathrm{mol.)}$ .

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Problem 23

Predict/Calculate Three moles of oxygen gas (that is, 3.0 mol of $\mathrm{O}_{2} )$ are placed in a portable container with a volume of 0.0035 $\mathrm{m}^{3} .$ If the temperature of the gas is $295^{\circ} \mathrm{C},$ find $(\mathrm{a})$ the pressure of the gas and (b) the average kinetic energy of an oxygen molecule. (c) Suppose the volume of the gas is doubled, while the temperature and number of moles are held constant. By what factor do your answers to parts (a) and (b) change? Explain.

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Problem 24

Predict/Calculate The rms speed of $\mathrm{O}_{2}$ is 1550 $\mathrm{m} / \mathrm{s}$ at a given

temperature. (a) Is the rms speed of $\mathrm{H}_{2} \mathrm{O}$ at this temperature

greater than, less than, or equal to 1550 $\mathrm{m} / \mathrm{s} ?$ Explain. (b) Find the

rms speed of $\mathrm{H}_{2} \mathrm{O}$ at this temperature.

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Problem 25

Predict/Calculate An ideal gas is kept in a container of constant volume. The pressure of the gas is also kept constant. (a) If the number of molecules in the gas is doubled, does the rms speed increase, decrease, or stay the same? Explain. (b) If the initial rms speed is $1300 \mathrm{m} / \mathrm{s},$ what is the final rms speed?

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Problem 26

What is the temperature of a gas of $\mathrm{CO}_{2}$ molecules whose rms speed is 309 $\mathrm{m} / \mathrm{s}$ ?

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Problem 27

The rms speed of a sample of gas is increased by 1$\%$ . (a) What is the percent change in the temperature of the gas? (b) What is the percent change in the pressure of the gas, assuming its volume is held constant?

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Problem 28

The rms speed of a sample of gas is increased by 1$\%$ . (a) What is the percent change in the temperature of the gas? (b) What is the percent change in the pressure of the gas, assuming its volume is held constant? Enriching Uranium In naturally occurring uranium atoms, 99.3$\%$

are $^{238} \mathrm{U}$ (atomic mass $=238 \mathrm{u},$ where $\mathrm{u}=1.6605 \times 10^{-27} \mathrm{kg} )$ and only 0.7$\%$ are $^{235} \mathrm{U}$ (atomic mass $=235 \mathrm{u} ) .$ Uranium-fueled reactors require an enhanced proportion of $^{235} \mathrm{U}$ . Since both isotopes of uranium have identical chemical properties, they can be separated only by methods that depend on their differing masses. One such method is gaseous diffusion, in which uranium hexafluoride $\left(\mathrm{UF}_{6}\right)$ gas diffuses through a series of porous barriers. The lighter $^{235}\mathrm{UF}_{6}$ molecules have a slightly higher rms speed at a given temperature than the heavier $^{238}\mathrm{UF}_{6}$ molecules, and this allows the two isotopes to be separated. Find the ratio of the rms speeds of the two isotopes at $23.0^{\circ} \mathrm{C}$ .

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Problem 29

A 380 -mL spherical flask contains 0.065 mol of an ideal gas at a temperature of 283 $\mathrm{K}$ . What is the average force exerted on the walls of the flask by a single molecule?

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Problem 30

CE Predict/ Explain A hollow cylindrical rod (rod 1) and a solid cylindrical rod (rod 2$)$ are made of the same material. The two rods have the same length and the same outer radius. If the same

compressional force is applied to each rod, (a) is the change in length of rod 1 greater than, less than, or equal to the change in length of rod 2$?($ b) Choose the best explanation from among the

following:

\begin{equation}

\begin{array}{l}{\text { I. The solid rod has the larger effective cross-sectional area, since }} \\ {\text { the empty part of the hollow rod doesn't resist compression. }} \\ {\text { Therefore, the solid rod has the smaller change in length. }} \\ {\text { II. The rods have the same outer radius and hence the same cross- }} \\ {\text { sectional area. As a result, their change in length is the same. }}\\{\text { III. The walls of the hollow rod are hard and resist compression }} \\ {\text { more than the uniform material in the solid rod. Therefore }} \\ {\text { the hollow rod has the smaller change in length. }}\end{array}

\end{equation}

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Problem 31

A rock climber hangs freely from a nylon rope that is 16 $\mathrm{m}$ long and has a diameter of 8.1 $\mathrm{mm}$ . If the rope stretches $4.2 \mathrm{cm},$ what is the mass of the climber?

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Problem 32

BIO To stretch a relaxed biceps muscle 2.5 $\mathrm{cm}$ requires a force of 25 N. Find the Young's modulus for the muscle tissue, assuming it to be a uniform cylinder of length 0.24 mand cross-sectional area 47 $\mathrm{cm}^{2}$ .

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Problem 33

A 22 -kg chimpanzee hangs from the end of a horizontal, broken branch 1.1 $\mathrm{m}$ long, as shown in Fi6uRE $17-34 .$ The branch is a uniform cylinder 4.6 $\mathrm{cm}$ in diameter, and the end of the branch supporting the chimp sags downward through a vertical distance of 13 $\mathrm{cm} .$ What is the shear modulus for this branch?

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Problem 34

The Marianas Trench The deepest place in all the oceans is the Marianas Trench, where the depth is 10.9 $\mathrm{km}$ and the pressure is $1.10 \times 10^{8}$ Pa. If a copper ball 15.0 $\mathrm{cm}$ in diameter is taken to the bottom of the trench, by how much does its volume decrease?

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Problem 35

CE Four cylindrical rods with various cross-sectional areas and initial lengths are stretched by an applied force, as in Figure $17-11$ . The resulting change in length of each rod is given in the following table. Rank these rods in order of increasing Young's modulus. Indicate ties where appropriate.

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Problem 36

Predict/Calculate A steel wire 4.1 $\mathrm{m}$ long stretches 0.13 $\mathrm{cm}$ when it is given a tension of 380 $\mathrm{N}$ . (a) What is the diameter of the wire? (b) If it is desired that the stretch be less than 0.13 $\mathrm{cm},$ should its diameter be increased or decreased? Explain.

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Problem 37

BIO Spiderweb An orb weaver spider with a mass of 0.26 g hangs vertically by one of its threads. The thread has a Young's modulus of $4.7 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$ and a radius of $9.8 \times 10^{-6} \mathrm{m}$ . (a) What is the fractional increase in the thread's length caused by the spider?

(b) Suppose a 76 -kg person hangs vertically from a nylon rope. What radius must the rope have if its fractional increase in length is to be the same as that of the spider's thread?

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Problem 38

Predict/Calculate Two rods of equal length $(0.55 \mathrm{m})$ and diameter $(1.7 \mathrm{cm})$ are placed end to end. One rod is aluminum, the other is brass. If a compressive force of 8400 $\mathrm{N}$ is applied to the rods, (a) how much does their combined length decrease? (b) Which of the rods changes its length by the greatest amount? Explain.

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Problem 39

A piano wire 0.82 m long and 0.93 $\mathrm{mm}$ in diameter is fixed on one end. The other end is wrapped around a tuning peg 3.5 $\mathrm{mm}$ in diameter. Initially the wire, whose Young's modulus

is $2.4 \times 10^{10} \mathrm{N} / \mathrm{m}^{2},$ has a tension of 14 $\mathrm{N}$ . Find the tension in the wire after the tuning peg has been turned through one complete revolution.

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Problem 40

CE The formation of ice from water is accompanied by which of the following: (a) an absorption of heat by the water; (b) an increase in temperature; (c) a decrease in volume; (d) a removal of heat from the water; (e) a decrease in temperature?

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Problem 41

Vapor Pressure for Water FIGURE $17-35$ shows a portion of the vapor-pressure curve for water. Referring to the figure, estimate the pressure that would be required for water to boil at $30^{\circ} \mathrm{C}$ .

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Problem 42

Using the vapor-pressure curve given in Figure $17-35,$ find the temperature at which water boils when the pressure is 1.5 kPa.

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Problem 43

Dew-Point Temperature One measure of the amount of water in the atmosphere is the dew point, the temperature at which the atmosphere's water content would be in equilibrium with liquid

water and droplets of dew would begin to form. (a) On a warm summer day when the dew-point temperature is $68^{\circ} \mathrm{F}$ , what is the pressure of water in the atmosphere? Refer to Figure $17-35$ and note that the pressure of the water vapor (called the partial pressure

of water) is equal to the vapor pressure at the dew-point temperature. (b) If the weather changes and the partial pressure of water is reduced to 1.7 $\mathrm{kPa}$ , what is the new dew-point temperature?

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Problem 44

Relative Humidity One measure of the amount of water in the

atmosphere is the relative humidity, the ratio of the actual partial

pressure of water vapor to the maximum possible partial pressure

of water. If the partial pressure of water vapor exceeds the vapor

pressure (Figure $17-35$ ) at that temperature, the vapor condenses

to liquid and either fog or dew begins to form. In that case the

relative humidity would be 100$\%$ . Referring to Figure $17-35$ , if the

partial pressure of water in the atmosphere is 1.7 kPa when the air

temperature is $25^{\circ} \mathrm{C},$ what is the relative humidity?

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Problem 45

Predict/Calculate The Vapor Pressure of $C O_{2}$ A portion of the

vapor-pressure curve for carbon dioxide is given in FlGURE $17-36$

(a) Estimate the pressure at which $\mathrm{CO}_{2}$ boils at $0^{\circ} \mathrm{C}$ . (b) If the temperature is increased, does the boiling pressure increase, decrease,

or stay the same? Explain.

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Problem 46

Phase Diagram for Water The phase diagram for water is shown in FIGURE $17-37$ . (a) What is the temperature $T_{1}$ on the phase diagram? (b) What is the temperature $T_{2}$ on the phase diagram? (c) What happens to the melting/freezing temperature of water if atmospheric pressure is decreased? Justify your answer by referring to the phase diagram. (d) What happens to the boiling/condensation temperature of water if atmospheric pressure is increased? Justify your answer by referring to the phase diagram.

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Problem 47

Phase Diagram for $\mathrm{CO}_{2}$ The phase diagram for $\mathrm{CO}_{2}$ is shown in FlGURE $17-38 .$ (a) What is the phase of $\mathrm{CO}_{2}$ at $T=20^{\circ} \mathrm{C}$ and $P=500 \mathrm{kPa}$ ? (b) What is the phase of $\mathrm{CO}_{2}$ at $T=-80^{\circ} \mathrm{C}$ and

$P=120 \mathrm{kPa} ?$ (c) For reasons of economy and convenience, bulk

$\mathrm{CO}_{2}$ is often transported in liquid form in pressurized tanks. Using

the phase diagram, determine the minimum pressure required

keep $\mathrm{CO}_{2}$ in the liquid phase at $20^{\circ} \mathrm{C} .$

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Problem 48

A sample of liquid water at atmospheric pressure has a temperature just above the freezing point. Refer to Figure $17-37$ to answer the following questions. (a) What phase changes occur if the temperature of the system is increased while the pressure is held constant? (b) Suppose, instead, that the temperature of the system is held constant just above the freezing point while the pressure is decreased. What phase changes occur now?

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Problem 49

How much heat must be removed from 1.96 $\mathrm{kg}$ of water at $0^{\circ} \mathrm{C}$ to

make ice cubes at $0^{\circ} \mathrm{C} ?$

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Problem 50

A heat transfer of $9.5 \times 10^{5} \mathrm{J}$ is required to convert a block of ice

at $-15^{\circ} \mathrm{C}$ to water at $15^{\circ} \mathrm{C}$ . What was the mass of the block of ice?

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Problem 51

How much heat must be added to 2.55 $\mathrm{kg}$ of copper to change it from a solid at 1358 $\mathrm{K}$ to a liquid at 1358 $\mathrm{K}$ ?

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Problem 52

An ammonia refrigeration cycle involves the conversion of 0.85 $\mathrm{kg}$ of liquid ammonia into vapor every minute at the boiling-point temperature. At what rate does the ammonia absorb energy?

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Problem 53

CE Four liquids are at their freezing temperature. Heat is now removed from each of the liquids until it becomes completely solidified. The amount of heat that must be removed, Q, and the mass, $m,$ of each of the liquids are as follows: liquid $\mathrm{A}, Q=33,500 \mathrm{J}, m=0.100 \mathrm{kg} ;$ liquid $\mathrm{B}, \quad Q=166,000 \mathrm{J}$

$m=0.500 \mathrm{kg} ;$ liquid $\mathrm{C}, Q=31,500 \mathrm{J}, m=0.250 \mathrm{kg} ;$ liquid $\mathrm{D},$

$Q=5400 \mathrm{J}, m=0.0500 \mathrm{kg} .$ Rank these liquids in order of increasing latent heat of fusion. Indicate ties where appropriate.

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Problem 54

Predict/Calculate A 1.1$\cdot \mathrm{kg}$ block of ice is initially at a temperature of $-5.0^{\circ} \mathrm{C}$ (a) If $5.2 \times 10^{5} \mathrm{J}$ of heat are added to the ice, what is the final temperature of the system? Find the amount of ice, if any, that remains. (b) Suppose the amount of heat added to the ice block is doubled. By what factor must the mass of the ice be increased if the system is to have the same final temperature? Explain.

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Problem 55

Predict/Calculate Referring to the previous problem, suppose the amount of heat added to the block of ice is reduced by a factor of 2 to $2.6 \times 10^{5}$ J. Note that this amount of heat is still sufficient

to melt at least some of the ice. (a) Do you expect the temperature increase in this case to be one-half that found in the previous problem? Explain. (b) What is the final temperature of the system in this case? Find the amount of ice, if any, that remains.

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Problem 56

Figure $17-30$ shows a temperature-versus-heat plot for 1.000 $\mathrm{kg}$ of water. (a) Calculate the heat corresponding to the points A, B, C, and D. (b) Calculate the slope of the line from point $B$ to point C. Show that this slope is equal to $1 / c,$ where $c$ is the specific heat of liquid water.

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Problem 57

Predict/Calculate Suppose the 1.000 $\mathrm{kg}$ of water in Figure $17-30$ starts at point $\mathrm{A}$ at time zero. Heat is added to this system at the rate of $12,250 \mathrm{J} / \mathrm{s}$ . How much time does it take for the system to reach (a) point $\mathrm{B},$ (b) point $\mathrm{C},$ and $(\mathrm{c})$ point $\mathrm{D} ?$ (d) Describe the physical state of the system at time $t=63.00 \mathrm{s}$ .

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Problem 58

BIO In Conceptual Example $17-15$ we pointed out that steam can cause more serious burns than water at the same temperature. Here we examine this effect quantitatively, noting that flesh

becomes badly damaged when its temperature reaches $50.0^{\circ} \mathrm{C}$ .

(a) Calculate the heat released as 12.5 $\mathrm{g}$ of liquid water at $100^{\circ} \mathrm{C}$

is cooled to $50.0^{\circ} \mathrm{C} .$ (b) Calculate the heat released when 12.5 $\mathrm{g}$ of

stearn at $100^{\circ} \mathrm{C}$ is condensed and cooled to $50.0^{\circ} \mathrm{C} .(\mathrm{c})$ Find the mass of flesh that can be heated from $37.0^{\circ} \mathrm{C}$ (normal body temperature) to $50.0^{\circ} \mathrm{C}$ for the cases considered in parts (a) and (b).

(The average specific heat of flesh is 3500 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ .)

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Problem 59

When you go out to your car one cold winter morning you dis- cover a 0.58 -cm-thick layer of ice on the windshield, which has an area of 1.6 $\mathrm{m}^{2} .$ If the temperature of the ice is $-2.0^{\circ} \mathrm{C},$ and its density is $917 \mathrm{kg} / \mathrm{m}^{3},$ find the heat required to melt all the ice.

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Problem 60

A large punch bowl holds 3.99 kg of lemonade (which is essentially water) at $20.5^{\circ} \mathrm{C}$ A 0.0550 -kge at $-10.2^{\circ} \mathrm{C}$ is placed in the lemonade. What are the final temperature of the system, and the amount of ice (if any) remaining? Ignore any heat exchange with the bowl or the surroundings.

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Problem 61

A 155 -g aluminum cylinder is removed from a liquid nitrogen bath, where it has been cooled to $-196^{\circ} \mathrm{C}$ . The cylinder is immediately placed in an insulated cup containing 80.0 $\mathrm{g}$ of water at $15.0^{\circ} \mathrm{C} .$ What is the equilibrium temperature of this system? If your answer is $0^{\circ} \mathrm{C}$ , determine the amount of water that has frozen.The average specific heat of aluminum over this temperature range is 653 $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$

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Problem 62

An 825 -g iron block is heated to $352^{\circ} \mathrm{C}$ and placed in an insulated container (of negligible heat capacity) containing 40.0 $\mathrm{g}$ of water at $20.0^{\circ} \mathrm{C}$ . What is the equilibrium temperature of this system? If your answer is $100^{\circ} \mathrm{C}$ , determine the amount of water that has vaporized. The average specific heat of iron over this temperature range is 560 $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ .

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Problem 63

Party Planning You are expecting to serve 32 cups of soft drinks to your guests tonight. Each cup will hold 285 gof a soft drink that has a specific heat of 4186 $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ and an initial temperature of $24^{\circ} \mathrm{C} .$ If each guest would like to enjoy the drink at $3.0^{\circ} \mathrm{C},$ how much ice (in kg) should you buy? Assume the initial temperature of the ice is $0^{\circ} \mathrm{C},$ and ignore the heat exchange with the plastic cups and the surroundings.

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Problem 64

Predict/Calculate $A 35-$ g ice cube at $0.0^{\circ} \mathrm{C}$ is added to 110 $\mathrm{g}$ of water in a 62 -g aluminum cup. The cup and the water have an initial temperature of $23^{\circ} \mathrm{C}$ (a) Find the equilibrium temperature of the cup and its contents. (b) Suppose the aluminum cup is

replaced with one of equal mass made from silver. Is the equilibrium temperature with the silver cup greater than, less than, or the same as with the aluminum cup? Explain.

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Problem 65

$\ \mathrm{A} \ 48$ -g block of copper at $-12^{\circ} \mathrm{C}$ is added to 110 $\mathrm{g}$ of water in a 75 -g aluminum cup. The cup and the water have an initial temperature of $4.1^{\circ} \mathrm{C}$ (a) Find the equilibrium temperature of the cup and its contents. (b) What mass of ice, if any, is present when the system reaches equilibrium?

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Problem 66

A 0.075 -kg ice cube at $0.0^{\circ} \mathrm{C}$ is dropped into a Styrofoam cup holding 0.33 $\mathrm{kg}$ of water at $14^{\circ} \mathrm{C}$ (a) Find the final temperature of the system and the amount of ice (if any) remaining. Assume the cup and the surroundings can be ignored. (b) Find the initial temperature of the water that would be enough to just barely melt all of the ice.

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Problem 67

To help keep her barn warm on cold days, a farmer stores 865 $\mathrm{kg}$ of warm water in the barn. How many hours would a 2.00 -kilowatt electric heater have to operate to provide the same amount of heat as is given off by the water as it cools from $20.0^{\circ} \mathrm{C}$ to $0^{\circ} \mathrm{C}$ and then freezes at $0^{\circ} \mathrm{C} ?$

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Problem 68

CE As you go up in altitude, do you expect the ratio of oxygen to nitrogen in the atmosphere to increase, decrease; or stay the same? Explain.

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Problem 69

CE Predict/Explain Suppose the Celsius temperature of an ideal gas is doubled from $100^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$ (a) Does the average kinetic energy of the molecules in this gas increase by a factor that is greater than, less than, or equal to 2? (b) Choose the best explanation from among the following:

\begin{equation}

\begin{array}{l}{\text { I. Changing the temperature from } 100^{\circ} \mathrm{C} \text { to } 200^{\circ} \mathrm{C} \text { goes beyond }} \\ {\text { the boiling point, which will increase the kinetic energy by }} \\ {\text { more than a factor of } 2 .} \\ {\text { The average kinetic energy is directly proportional to the tem- }} \\ {\text { perature, so doubling the temperature doubles the kinetic }} \\ {\text { energy. }}\\{\text { III. Doubling the Celsius temperature from } 100^{\circ} \mathrm{C} \text { to } 200^{\circ} \mathrm{C}} \\ {\text { changes the Kelvin temperature from } 373.15 \mathrm{K} \text { to } 473.15 \mathrm{K} \text { , }} \\ {\text { which is an increase of less than a factor of } 2 .}\end{array}

\end{equation}

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Problem 70

CE Predict/ Explain Suppose the absolute temperature of an ideal gas is doubled from 100 $\mathrm{K}$ to 200 $\mathrm{R}$ (a) Does the average speed of the molecules in this gas increase by a factor that is greater than, less than, or equal to 2$?(\mathrm{b})$ Choose the best explanation from among the following:

\begin{equation}

\begin{array}{l}{\text { I. Doubling the Kelvin temperature doubles the average kinetic }} \\ {\text { energy, but this implies an increase in the average speed by a }} \\ {\text { factor of } \sqrt{2}=1.414 \ldots, \text { which is less than } 2 \text { . }} \\ {\text { II. The Kelvin temperature is the one we use in the ideal-gas law, }} \\ {\text { and therefore doubling it also doubles the average speed of }} \\ {\text { the molecules. }}\\{\text { III. The change in average speed depends on the mass of the mol- }} \\ {\text { ecules in the gas, and hence doubling the Kelvin temperature }} \\ {\text { generally results in an increase in speed that is greater than a }} \\ {\text { factor of } 2 \text { . }}\end{array}

\end{equation}

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Problem 71

Largest Raindrops Atmospheric scientists studying clouds in the Marshall Islands have observed what they believe to be the world's largest raindrops, with a radius of 0.52 $\mathrm{cm} .$ How many molecules are in these monster drops?

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Problem 72

Cooling Computers Researchers are developing "heat exchangers" for laptop computers that take heat from the laptop-to keep it from being damaged by overheating-and use it to vaporize methanol. Given that 5100 $\mathrm{J}$ of heat is removed from the laptop when 4.6 $\mathrm{g}$ of methanol is vaporized, what is the latent heat of vaporization for methanol?

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Problem 73

A 1.10 -kg block of ice is initially at a temperature of $-5.0^{\circ} \mathrm{C}$ . (a) If $3.1 \times 10^{5} \mathrm{J}$ of thermal energy are added to the ice, what is the amount of ice that remains? (b) How much additional thermal energy must be added to this system to convert it to 1.10 $\mathrm{kg}$ of water at $5.0^{\circ} \mathrm{C}$ ?

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Problem 74

Scuba Tanks In scuba diving circles, "an 80" refers to a scuba tank that holds 80 cubic feet of air, a standard amount for recreational diving. Given that a scuba tank is a cylinder 2 feet long and half a

foot in diameter, determine (a) the volume of a tank and (b) the pressure in a tank when 80 cubic feet of air is compressed into its relatively small volume. (c) What is the mass of air in a tank that holds 80 cubic feet of air? Assume the temperature is $21^{\circ} \mathrm{C}$ and that the walls of the tank are of negligible thickness.

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Problem 75

Evaporating Atmosphere Hydrogen gas evaporates into space even though its rms speed is less than one-fifth of the gravitational escape speed. This is because the distribution of molecular speeds

at equilibrium (see Figure $17-27$ shows that some of the molecules do have speeds that exceed the escape speed. To what temperature must the atmosphere be heated for the air around us to have an

rms speed that is one-fifth of the gravitational escape speed? Treat the air as having a molecular mass of 0.029 $\mathrm{kg} / \mathrm{mol} .$

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Problem 76

To make steam, you add $5.60 \times 10^{5} \mathrm{J}$ of thermal energy to 0.220 $\mathrm{kg}$ of water at an initial temperature of $50.0^{\circ} \mathrm{C} .$ Find the final temperature of the steam.

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Problem 77

A Boiling Geyser (a) The column of water that forms as a geyser erupts is 2100 $\mathrm{m}$ tall. What is the pressure at the bottom of the column of water? (b) The water vapor-pressure curve (Figure

$17-18 )$ near the bottom of the column can be approximated by $P=\left(1.86 \times 10^{5}\right) T-4.80 \times 10^{7},$ where $P$ is the pressure in Pa and $T$ is the Celsius temperature. What is the boiling-point temperature for water at the bottom of the column?

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Problem 78

A Melting Glacier (a) A glacier is made of ice of density 850 $\mathrm{kg} /$

$\mathrm{m}^{3}$ and is 92 $\mathrm{m}$ thick. Treating the glacial ice as if it were a liquid, what is the pressure at the bottom of the ice? (b) The water solid-liquid curve (Figure $17-21$ (b)) near the bottom of the glacier can be approximated by $P=\left(-1.31 \times 10^{7}\right) T+1.01 \times 10^{5},$ where $P$ is the pressure in Pa and $T$ is the Celsius temperature. What is the melting-point temperature for water at the bottom of the glacier?

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Problem 79

Peter catches a 4.2 -kg striped bass on a fishing line 0.55 $\mathrm{mm}$ in diameter and begins to reel it in. He fishes from a pier well above the water, and his fish hangs vertically from the line out of the

water. The fishing line has a Young's modulus of 5.1 $\times 10^{9} \mathrm{N} / \mathrm{m}^{2}$ .

(a) What is the fractional increase in length of the fishing line if the fish is at rest? (b) What is the fractional increase in the fishing line's length when the fish is pulled upward with a constant speed

of 2.5 $\mathrm{m} / \mathrm{s} ?$ (c) What is the fractional increase in the fishing line's length when the fish is pulled upward with a constant acceleration of 2.5 $\mathrm{m} / \mathrm{s}^{2}$ ?

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Problem 80

A steel ball (density $=7860 \mathrm{kg} / \mathrm{m}^{3}$ ) with a diameter of 6.4 $\mathrm{cm}$ is tied to an aluminum wire 82 $\mathrm{cm}$ long and 2.5 $\mathrm{mm}$ in diameter.

The ball is whirled about in a vertical circle with a tangential speed of 7.8 $\mathrm{m} / \mathrm{s}$ at the top of the circle and 9.3 $\mathrm{m} / \mathrm{s}$ at the bottom of the circle. Find the amount of stretch in the wire (a) at the top and (b) at the bottom of the circle.

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Problem 81

A lead brick with the dimensions shown in FIGURE $17-39$ rests on a rough solid surface. A force of 2400 $\mathrm{N}$ is applied as indicated. Find (a) the change in height of the brick and (b) the amount of shear deformation.

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Problem 82

(a) Find the amount of heat that must be extracted from 1.3 $\mathrm{kg}$ of

steam at $120^{\circ} \mathrm{C}$ to convert it to ice at $0.0^{\circ} \mathrm{C}$ (b) What speed would

this 1.3 -kg block of ice have if its translational kinetic energy were

equal to the thermal energy calculated in part (a)?

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Problem 83

Mighty lce Lift A tremendous force is generated when water freezes into ice and expands in volume by 9.0$\%$ . Suppose a $1.000-$ $\mathrm{m}^{3}$ cube of liquid water freezes into ice that is 1.000 $\mathrm{m}$ on a side by 1.090 $\mathrm{m}$ tall. How many 68 -kg students would the ice be able to lift? Determine this by calculating the amount of force on the top 1.000 -m' face that would be required to squeeze 1.090 $\mathrm{m}^{3}$ of ice back to 1.000 $\mathrm{m}^{3}$ , assuming all of the volume change occurs along the vertical direction.

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Problem 84

Orthopedic Implants Metals such as titanium and stainless steel are frequently used for orthopedic implants such as artificial hip and knee joints. As with most metals, their elastic properties are significantly different from those of bone. Recently, metal "foams" made from aluminum and steel have been shown to have promising properties for use as implants. (a) $\mathrm{A} 0.5$ -m-long

piece of bone with a certain cross-sectional area shortens by 0.10 $\mathrm{mm}$ under a given compressive force. By how much does a piece of steel with the same length and cross-sectional area shorten if the same force is applied? (b) In order to determine whether a material such as the aluminum-steel foam behaves similarly to bone, plots of measured stress (force per area) versus strain $\left(\Delta L / L_{0}\right)$ such as the ones shown in FIGURE $17-40$ may be used. Which one of these plots corresponds to a material with elastic properties equal to those of bone in compression? (c) What is the value of Young's modulus for the material represented by curve D?

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Problem 85

Students on a spring break picnic bring a cooler that contains

5.1 kg of ice at $0.0^{\circ} \mathrm{C}$ . The cooler has walls that are 3.8 $\mathrm{cm}$ thick and are made of Styrofoam, which has a thermal conductivity of 0.030 $\mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right) .$ The surface area of the cooler is $1.5 \mathrm{m}^{2},$ and it

rests in the shade where the air temperature is $21^{\circ} \mathrm{C}$ (a) Find the

rate at which heat flows into the cooler. (b) How much time does

it take for the ice in the cooler to melt?

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Problem 86

A $5.9-$ -kg block of ice at $-1.5^{\circ} \mathrm{C}$ slides on a horizontal surface with a coefficient of kinetic friction equal to $0.069 .$ The initial speed of the block is 7.1 $\mathrm{m} / \mathrm{s}$ and its final speed is 5.3 $\mathrm{m} / \mathrm{s}$ . Assuming that all the energy dissipated by kinetic friction goes into melting a small mass $m$ of the ice, and that the rest of the ice block remains at $-1.5^{\circ} \mathrm{C}$ , determine the value of $m .$

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Problem 87

A cylindrical copper rod 37 $\mathrm{cm}$ long and 7.5 $\mathrm{cm}$ in diameter is placed upright on a hot plate held at a constant temperature of $120^{\circ} \mathrm{C},$ as indicated in FlGuRE $17-41 .$ A small depression on top of the rod holds a $25-\mathrm{g}$ ice cube at an initial temperature of $0.0^{\circ} \mathrm{C}$ . How much time does it take for the ice cube to melt? Assume there is no heat loss through the vertical surface of the rod, and that the thermal conductivity of copper is 390 $\mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right) .$

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Problem 88

What pressure did the bathysphere experience at its record depth?

\begin{equation}\begin{array}{ll}{\text { A. } 9.37 \text { atm }} & {\text { B. } 89.6 \text { atm }} \\ {\text { C. } 91.9 \text { atm }} & {\text { D. } 92.9 \text { atm }}\end{array}

\end{equation}

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Problem 89

How many moles of air did the bathysphere contain when it was sealed at the surface, assuming a temperature of 297 $\mathrm{K}$ and ignoring the thickness of the metal shell? (Note: A resting person

breathes roughly 0.5 mol of air per minute.)

\begin{equation}

\begin{array}{ll}{\text { A. } 65.2 \mathrm{mol}} & {\text { B. } 270 \mathrm{mol}} \\ {\text { C. } 392 \mathrm{mol}} & {\text { D. } 523 \mathrm{mol}}\end{array}

\end{equation}

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Problem 90

How much did the volume of the bathysphere decrease as it was lowered to its record depth? (For simplicity, treat the bathysphere as a solid metal sphere.)

\begin{equation}

\begin{array}{ll}{\text { A. } 9.0 \times 10^{-5} \mathrm{m}^{3}} & {\text { B. } 9.2 \times 10^{-5} \mathrm{m}^{3}} \\ {\text { C. } 1.1 \times 10^{-4} \mathrm{m}^{3}} & {\text { D. } 3.8 \times 10^{-4} \mathrm{m}^{3}}\end{array}

\end{equation}

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Problem 91

Suppose the bathysphere and its occupants had a combined mass of $12,700$ kg. How much did the cable stretch when the bathysphere was at a depth of 923 $\mathrm{m} ?$ (Neglect the weight of the

cable itself, but include the effects of the bathysphere's buoyancy.)

\begin{equation}

\begin{array}{ll}{\text { A. } 47 \mathrm{cm}} & {\text { B. } 48 \mathrm{cm}} \\ {\text { C. } 52 \mathrm{cm}} & {\text { D. } 53 \mathrm{cm}}\end{array}

\end{equation}

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Problem 92

REFERRING TO EXAMPLE $17-17$ (a) Find the final temperature of the system if $t w o 0.0450$ -kg ice cubes are added to the warm lemonade. The temperature of the ice is $0^{\circ} \mathrm{C} ;$ the temperature and mass of the warm lemonade are $20.0^{\circ} \mathrm{C}$ and 3.95 $\mathrm{kg}$ , respectively. (b) How many 0.0450 -kg ice cubes at $0^{\circ} \mathrm{C}$ must be added to the original

warm lemonade if the final temperature of the system is to be at least as cold as $15.0^{\circ} \mathrm{C} ?$

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Problem 93

REFERRING TO EXAMPLE $17-17$ (a) Find the final temperature of the system if a single $0.045-\mathrm{kg}$ ice cube at $0^{\circ} \mathrm{C}$ is added to 2.00 $\mathrm{kg}$ of lemonade at $1.00^{\circ} \mathrm{C}$ (b) What initial temperature of the lemonade will be just high enough to melt all of the ice in a single ice cube and result in an equilibrium temperature of $0^{\circ} \mathrm{C} ?$ The mass of the lemonade is 2.00 $\mathrm{kg}$ and the temperature of the ice cube is $0^{\circ} \mathrm{C}$ .

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