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Chapter 14

HEAT

Educators

AH

Problem 1

(I) To what temperature will 8200 J of heat raise 3.0 kg of water that is initially at 10.0$^\circ$C?

Zachary W.
Numerade Educator

Problem 2

(I) How much heat (in joules) is required to raise the temperature of 34.0 kg of water from 15$^\circ$C to 95$^\circ$C?

AH
Averell H.
Carnegie Mellon University

Problem 3

(II) When a diver jumps into the ocean, water leaks into the gap region between the diver's skin and her wetsuit, forming a water layer about 0.5 mm thick. Assuming the total surface area of the wetsuit covering the diver is about 1.0 m$^2$, and that ocean water enters the suit at 10$^\circ$C and is
warmed by the diver to skin temperature of 35$^\circ$C, estimate how much energy (in units of candy bars $=$ 300 kcal) is required by this heating process.

Zachary W.
Numerade Educator

Problem 4

(II) An average active person consumes about 2500 Cal a day. ($a$) What is this in joules? ($b$) What is this in kilowatthours? ($c$) If your power company charges about 10 cents per kilowatt-hour, how much would your energy cost per day if you bought it from the power company? Could you feed
yourself on this much money per day?

AH
Averell H.
Carnegie Mellon University

Problem 5

(II) A British thermal unit (Btu) is a unit of heat in the British system of units. One Btu is defined as the heat needed to raise 1 lb of water by 1 F$^\circ$. Show that

$$1 \, \mathrm{Btu} = 0.252 \, \mathrm{kcal} = 1056 \, \mathrm{J}.$$

Zachary W.
Numerade Educator

Problem 6

(II) How many joules and kilocalories are generated when the brakes are used to bring a 1300-kg car to rest from a speed of 95 km/h?

AH
Averell H.
Carnegie Mellon University

Problem 7

(II) A water heater can generate 32,000 kJ/h. How much water can it heat from 12$^\circ$C to 42$^\circ$C per hour?

Zachary W.
Numerade Educator

Problem 8

(II) A small immersion heater is rated at 375W. Estimate how long it will take to heat a cup of soup (assume this is 250 mL of water) from 15$^\circ$C to 75$^\circ$C.

AH
Averell H.
Carnegie Mellon University

Problem 9

(I) An automobile cooling system holds 18 L of water. How much heat does it absorb if its temperature rises from 15$^\circ$C to 95$^\circ$C?

Zachary W.
Numerade Educator

Problem 10

(I) What is the specific heat of a metal substance if 135 kJ of heat is needed to raise 4.1 kg of the metal from 18.0$^\circ$C to 37.2$^\circ$C?

AH
Averell H.
Carnegie Mellon University

Problem 11

(II) ($a$) How much energy is required to bring a 1.0-L pot of water at 20$^\circ$C to 100$^\circ$C? ($b$) For how long could this amount of energy run a 60-W lightbulb?

Zachary W.
Numerade Educator

Problem 12

(II) Samples of copper, aluminum, and water experience the same temperature rise when they absorb the same amount of heat.What is the ratio of their masses? [$Hint$: See Table 14$-$1.]

AH
Averell H.
Carnegie Mellon University

Problem 13

(II) How long does it take a 750-W coffeepot to bring to a boil 0.75 L of water initially at 11$^\circ$C? Assume that the part of the pot which is heated with the water is made of 280 g of aluminum, and that no water boils away.

Zachary W.
Numerade Educator

Problem 14

(II) What will be the equilibrium temperature when a 265-g block of copper at 245$^\circ$C is placed in a 145-g aluminum calorimeter cup containing 825 g of water at 12.0$^\circ$C?

AH
Averell H.
Carnegie Mellon University

Problem 15

(II) A 31.5-g glass thermometer reads 23.6$^\circ$C before it is placed in 135 mL of water. When the water and thermometer come to equilibrium, the thermometer reads 41.8$^\circ$C. What was the original temperature of the water? Ignore the mass of fluid inside the glass thermometer.

Zachary W.
Numerade Educator

Problem 16

(II) A 0.40-kg iron horseshoe, just forged and very hot (Fig. 14$-$16), is dropped into 1.25 L of water in a 0.30-kg iron pot initially at 20.0$^\circ$C. If the final equilibrium temperature is 25.0$^\circ$C, estimate the initial temperature of the hot horseshoe.

AH
Averell H.
Carnegie Mellon University

Problem 17

(II) When a 290-g piece of iron at 180$^\circ$C is placed in a 95-g aluminum calorimeter cup containing 250 g of glycerin at 10$^\circ$C, the final temperature is observed to be 38$^\circ$C. Estimate the specific heat of glycerin.

Zachary W.
Numerade Educator

Problem 18

(II) The $heat \, capacity$, $C$, of an object is defined as the amount of heat needed to raise its temperature by 1 C$^\circ$. Thus, to raise the temperature by requires heat $Q$ given by $\Delta T$

$$Q = C \, \Delta \, T$$

($a$) Write the heat capacity C in terms of the specific heat, $c$, of the material. ($b$) What is the heat capacity of 1.0 kg of water? ($c$) Of 45 kg of water?

AH
Averell H.
Carnegie Mellon University

Problem 19

(II) The 1.20-kg head of a hammer has a speed of 7.5 m/s just before it strikes a nail (Fig. 14$-$17) and is brought to rest. Estimate the temperature rise of a 14-g iron nail generated by eight such hammer blows done in quick succession. Assume the nail absorbs all the energy.

Zachary W.
Numerade Educator

Problem 20

(II) A 215-g sample of a substance is heated to 330$^\circ$C and then plunged into a 105-g aluminum calorimeter cup containing 185 g of water and a 17-g glass thermometer at 10.5$^\circ$C. The final temperature is 35.0$^\circ$C. What is the specific heat of the substance? (Assume no water boils away.)

AH
Averell H.
Carnegie Mellon University

Problem 21

(II) A 0.095-kg aluminium sphere is dropped from the roof of a 55-m-high building. If 65$\%$ of the thermal energy produced when it hits the ground is absorbed by the sphere, what is its temperature increase?

Zachary W.
Numerade Educator

Problem 22

(II) Estimate the Calorie content of 65 g of candy from the following measurements. A 15-g sample of the candy is placed in a small aluminum container of mass 0.325 kg filled with oxygen. This container is placed in 1.75 kg of water in an aluminum calorimeter cup of mass 0.624 kg at an initial temperature of 15.0$^\circ$C. The oxygen$-$candy mixture in the small container (a "bomb calorimeter") is ignited,
and the final temperature of the whole system is 53.5$^\circ$C.

AH
Averell H.
Carnegie Mellon University

Problem 23

(II) Determine the energy content of 100 g of Karen's fudge cookies from the following measurements. A 10-g sample of a cookie is allowed to dry before putting it in a $bomb \, calorimeter$ (page 396). The aluminum bomb has a mass of 0.615 kg and is placed in 2.00 kg of water contained in an
aluminum calorimeter cup of mass 0.524 kg. The initial temperature of the system is 15.0$^\circ$C, and its temperature after ignition is 36.0$^\circ$C.

Zachary W.
Numerade Educator

Problem 24

(I) If 3.40 $\times$ 105 J of energy is supplied to a container of liquid oxygen at $-$183$^\circ$C, how much oxygen can evaporate?

AH
Averell H.
Carnegie Mellon University

Problem 25

(II) How much heat is needed to melt 23.50 kg of silver that is initially at 25$^\circ$C?

Zachary W.
Numerade Educator

Problem 26

(II) During exercise, a person may give off 185 kcal of heat in 25 min by evaporation of water (at 20$^\circ$C) from the skin. How much water has been lost? [$Hint$: See page 399.]

AH
Averell H.
Carnegie Mellon University

Problem 27

(II) What mass of steam at 100$^\circ$C must be added to 1.00 kg of ice at 0$^\circ$C to yield liquid water at 30$^\circ$C?

Zachary W.
Numerade Educator

Problem 28

(II) A 28-g ice cube at its melting point is dropped into an insulated container of liquid nitrogen. How much nitrogen evaporates if it is at its boiling point of 77 K and has a latent heat of vaporization of 200 kJ/kg? Assume for simplicity that the specific heat of ice is a constant and is equal to its value near its melting point.

AH
Averell H.
Carnegie Mellon University

Problem 29

(II) High-altitude mountain climbers do not eat snow, but always melt it first with a stove. To see why, calculate the energy absorbed from your body if you: ($a$) eat 1.0 kg of $-15^\circ$C snow which your body warms to body temperature of 37$^\circ$C; ($b$) melt 1.0 kg of $-15^\circ$C snow using a stove and drink the resulting 1.0 kg of water at 2$^\circ$C, which your body has to warm to 37$^\circ$C.

Zachary W.
Numerade Educator

Problem 30

(II) An iron boiler of mass 180 kg contains 730 kg of water at 18$^\circ$C. A heater supplies energy at the rate of 58,000 kJ/h. How long does it take for the water ($a$) to reach the boiling point, and ($b$) to all have changed to steam?

AH
Averell H.
Carnegie Mellon University

Problem 31

(II) Determine the latent heat of fusion of mercury using the following calorimeter data: 1.00 kg of solid Hg at its melting point of $-39.0^\circ$C is placed in a 0.620-kg aluminum calorimeter with 0.400 kg of water at 12.80$^\circ$C; the resulting equilibrium temperature is 5.06$^\circ$C.

Zachary W.
Numerade Educator

Problem 32

(II) At a crime scene, the forensic investigator notes that the 6.2-g lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was shot at room temperature (20$^\circ$C), what does the investigator calculate as the minimum muzzle velocity of the gun?

AH
Averell H.
Carnegie Mellon University

Problem 33

(II) A 64-kg ice-skater moving at 7.5 m/s glides to a stop. Assuming the ice is at 0$^\circ$C and that 50$\%$ of the heat generated by friction is absorbed by the ice, how much ice melts?

Zachary W.
Numerade Educator

Problem 34

(II) A cube of ice is taken from the freezer at $-$8.5$^\circ$C and placed in an 85-g aluminum calorimeter filled with 310 g of water at room temperature of 20.0$^\circ$C. The final situation is
all water at 17.0$^\circ$C. What was the mass of the ice cube?

AH
Averell H.
Carnegie Mellon University

Problem 35

(II) A 55-g bullet traveling at 250 m/s penetrates a block of ice at 0$^\circ$C and comes to rest within the ice. Assuming that the temperature of the bullet doesn't change appreciably, how much ice is melted as a result of the collision?

Zachary W.
Numerade Educator

Problem 36

(I) Calculate the rate of heat flow by conduction through the windows of Example 14$-$8, assuming that there are strong gusty winds and the external temperature is $-5^\circ$C.

AH
Averell H.
Carnegie Mellon University

Problem 37

(I) One end of a 56-cm-long copper rod with a diameter of 2.0 cm is kept at 460$^\circ$C, and the other is immersed in water at 22$^\circ$C. Calculate the heat conduction rate along the rod.

Zachary W.
Numerade Educator

Problem 38

(II) (a) How much power is radiated by a tungsten sphere (emissivity $\epsilon =$ 0.35) of radius 19 cm at a temperature of 25$^\circ$C? (b) If the sphere is enclosed in a room whose walls are kept at $-5^\circ$C, what is the $net$ flow rate of energy out of the sphere?

AH
Averell H.
Carnegie Mellon University

Problem 39

(II) How long does it take the Sun to melt a block of ice at 0$^\circ$C with a flat horizontal area 1.0 m$^2$ and thickness 1.0 cm? Assume that the Sun's rays make an angle of 35$^\circ$ with the
vertical and that the emissivity of ice is 0.050.

Zachary W.
Numerade Educator

Problem 40

(II) $Heat \, conduction\, to \, skin$. Suppose 150 W of heat flows by conduction from the blood capillaries beneath the skin to the body's surface area of 1.5 m$^2$. If the temperature difference is 0.50 C$^\circ$, estimate the average distance of capillaries below the skin surface.

AH
Averell H.
Carnegie Mellon University

Problem 41

(II) Two rooms, each a cube 4.0 m per side, share a 14-cm thick brick wall. Because of a number of 100-W lightbulbs in one room, the air is at 30$^\circ$C, while in the other room it is at 10$^\circ$C. How many of the 100-W bulbs are needed to maintain the temperature difference across the wall?

Zachary W.
Numerade Educator

Problem 42

(II) A 100-W lightbulb generates 95 W of heat, which is dissipated through a glass bulb that has a radius of 3.0 cm and is 0.50 mm thick. What is the difference in temperature between the inner and outer surfaces of the glass?

AH
Averell H.
Carnegie Mellon University

Problem 43

(III) Approximately how long should it take 8.2 kg of ice at 0$^\circ$C to melt when it is placed in a carefully sealed Styrofoam ice chest of dimensions 25 cm $\times$ 35 cm $\times$ 55 cm whose walls are 1.5 cm thick? Assume that the conductivity of Styrofoam is double that of air and that the outside temperature is 34$^\circ$C.

Zachary W.
Numerade Educator

Problem 44

(III) A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end (Fig. 14$-$18). The copper end is placed in a furnace maintained at a constant temperature of 205$^\circ$C. The aluminum end is placed in an ice bath held at a constant temperature of 0.0$^\circ$C. Calculate the temperature at the point where the two rods are joined.

AH
Averell H.
Carnegie Mellon University

Problem 45

(III) Suppose the insulating qualities of the wall of a house come mainly from a 4.0-in. layer of brick and an $R$-19 layer of insulation, as shown in Fig. 14$-$19.What is the total rate of heat loss through
such a wall, if its total area is 195 ft$^2$ and the temperature difference across it is 35 F$^\circ$?

Zachary W.
Numerade Educator

Problem 46

A soft-drink can contains about 0.35 kg of liquid at 5$^\circ$C. Drinking this liquid can actually consume some of the fat in the body, since energy is needed to warm the liquid to body temperature (37$^\circ$C). How many food Calories should the drink have so that it is in perfect balance with the heat needed to warm the liquid (essentially water)?

AH
Averell H.
Carnegie Mellon University

Problem 47

($a$) Estimate the total power radiated into space by the Sun, assuming it to be a perfect emitter at $T =$ 5500 K. The Sun's radius is $7.0 \times 10^8$ m. ($b$) From this, determine the power per unit area arriving at the Earth, $1.5 \times 10^{11}$ m away (Fig. 14$-$20).

Zachary W.
Numerade Educator

Problem 48

To get an idea of how much thermal energy is contained in the world's oceans, estimate the heat liberated when a cube of ocean water, 1 km on each side, is cooled by 1K. (Approximate the ocean water as pure water for this estimate.)

AH
Averell H.
Carnegie Mellon University

Problem 49

What will be the final result when equal masses of ice at 0$^\circ$C and steam at 100$^\circ$C are mixed together?

Zachary W.
Numerade Educator

Problem 50

A mountain climber wears a goose-down jacket 3.5 cm thick with total surface area 0.95 m$^2$. The temperature at the surface of the clothing is $-18^\circ$C and at the skin is 34$^\circ$C. Determine the rate of heat flow by conduction through the jacket assuming ($a$) it is dry and the thermal conductivity $k$ is that of goose down, and ($b$) the jacket is wet, so $k$ is that of water and the jacket has matted to 0.50 cm thickness.

AH
Averell H.
Carnegie Mellon University

Problem 51

During light activity, a 70-kg person may generate 200 kcal/h. Assuming that 20$\%$ of this goes into useful work and the other 80$\%$ is converted to heat, estimate the temperature rise of the body after 45 min if none of this heat is transferred to the environment.

Zachary W.
Numerade Educator

Problem 52

Estimate the rate at which heat can be conducted from the interior of the body to the surface. As a model, assume that the thickness of tissue is 4.0 cm, that the skin is at 34$^\circ$C and the interior at 37$^\circ$C, and that the surface area is 1.5 m$^2$. Compare this to the measured value of about 230 W that must be dissipated by a person working lightly. This clearly shows the necessity of convective cooling by the blood.

AH
Averell H.
Carnegie Mellon University

Problem 53

A bicyclist consumes 9.0 L of water over the span of 3.5 hours during a race. Making the approximation that 80$\%$ of the cyclist's energy goes into evaporating this water (at 20$^\circ$C)
as sweat, how much energy in kcal did the rider use during the ride? [$Hint$: See page 399.]

Zachary W.
Numerade Educator

Problem 54

If coal gives off 30 MJ/kg when burned, how much coal is needed to heat a house requiring 2.0 $\times$ 10$^5$ MJ for the whole winter? Assume that 30$\%$ of the heat is lost up the chimney.

AH
Averell H.
Carnegie Mellon University

Problem 55

A 15-g lead bullet is tested by firing it into a fixed block of wood with a mass of 35 kg. The block and imbedded bullet together absorb all the heat generated. After thermal equilibrium has been reached, the system has a temperature rise measured as 0.020 C$^\circ$. Estimate the bullet's entering speed.

Zachary W.
Numerade Educator

Problem 56

A 310-kg marble boulder rolls off the top of a cliff and falls a vertical height of 120 m before striking the ground. Estimate the temperature rise of the rock if 50$\%$ of the heat generated remains in the rock.

AH
Averell H.
Carnegie Mellon University

Problem 57

A 2.3-kg lead ball is placed in a 2.5-L insulated pail of water initially at 20.0$^\circ$C. If the final temperature of the water$-$lead combination is 32.0$^\circ$C, what was the initial temperature of the lead ball?

Zachary W.
Numerade Educator

Problem 58

A microwave oven is used to heat 250 g of water. On its maximum setting, the oven can raise the temperature of the liquid water from 20$^\circ$C to 100$^\circ$C in 1 min 45 s ($=$ 105 s). ($a$) At
what rate does the oven put energy into the liquid water? ($b$) If the power input from the oven to the water remains constant, determine how many grams of water will boil away if the oven is operated for 2min (rather than just 1 min 45 s).

AH
Averell H.
Carnegie Mellon University

Problem 59

In a typical squash game (Fig. 14$-$21), two people hit a soft rubber ball at a wall. Assume that the ball hits the wall at a velocity of 22 m/s and bounces back at a velocity of 12 m/s, and that the kinetic energy lost in the process heats the ball. What will be the temperature increase of the ball after one
bounce? (The specific heat of rubber is about 1200 J/kg$\cdot$C$^\circ$.)

Zachary W.
Numerade Educator

Problem 60

The temperature within the Earth's crust increases about 1.0 C$\circ$ for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s $\cdot$C$^\circ$m. ($a$) Determine the heat transferred
from the interior to the surface for the entire Earth in 1.0 h. ($b$) Compare this heat to the 1000W/m$^2$ that reaches the Earth's surface in 1.0 h from the Sun.

AH
Averell H.
Carnegie Mellon University

Problem 61

An iron meteorite melts when it enters the Earth's atmosphere. If its initial temperature was $-105^\circ$C outside of Earth's atmosphere, calculate the minimum velocity the meteorite must have had before it entered Earth's atmosphere.

Zachary W.
Numerade Educator

Problem 62

The temperature of the glass surface of a 75-W lightbulb is 75$^\circ$C when the room temperature is 18$^\circ$C. Estimate the temperature of a 150-W lightbulb with a glass bulb the same size. Consider only radiation, and assume that 90$\%$ of the energy is emitted as heat.

AH
Averell H.
Carnegie Mellon University

Problem 63

In a cold environment, a person can lose heat by conduction and radiation at a rate of about 200W. Estimate how long it would take for the body temperature to drop from 36.6$^\circ$C to 35.6$^\circ$C if metabolism were nearly to stop. Assume a mass of 65 kg. (See Table 14$-$1.)

Zachary W.
Numerade Educator

Problem 64

A 12-g lead bullet traveling at 220 m/s passes through a thin wall and emerges at a speed of 160 m/s. If the bullet absorbs 50$\%$ of the heat generated, ($a$) what will be the temperature rise of the bullet? ($b$) If the bullet's initial temperature was 20$^\circ$C, will any of the bullet melt, and if so, how much?

AH
Averell H.
Carnegie Mellon University

Problem 65

A leaf of area 40 cm$^2$ and mass $4.5 \times 10^{-4}$ kg directly faces the Sun on a clear day. The leaf has an emissivity of 0.85 and a specific heat of 0.80 kcal/kg$\cdot$K.($a$) Estimate the energy
absorbed per second by the leaf from the Sun, and then ($b$) estimate the rate of rise of the leaf's temperature. ($c$) Will the temperature rise continue for hours? Why or why not? ($d$) Calculate the temperature the leaf would reach if it lost all its heat by radiation to the surroundings at 24$^\circ$C. ($e$) In what other ways can the heat be dissipated by the leaf?

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Problem 66

Using the result of part ($a$) in Problem 65, take into account radiation from the leaf to calculate how much water must be transpired (evaporated) by the leaf per hour to maintain a temperature of 35$^\circ$C.

AH
Averell H.
Carnegie Mellon University

Problem 67

After a hot shower and dishwashing, there seems to be no hot water left in the 65-gal (245-L) water heater. This suggests that the tank has emptied and refilled with water at roughly 10$^\circ$C. (a) How much energy does it take to reheat the water to 45$^\circ$C? (b) How long would it take if the heater output is 9500 W?

Zachary W.
Numerade Educator

Problem 68

A house thermostat is normally set to 22$^\circ$C, but at night it is turned down to 16$^\circ$C for 9.0 h. Estimate how much more heat would be needed (state as a percentage of daily usage) if the thermostat were not turned down at night. Assume that the outside temperature averages 0$^\circ$C for the 9.0 h at night and 8$^\circ$C for the remainder of the day, and that the heat loss from the house is proportional to the temperature difference inside and out. To obtain an estimate from the data, you must make other simplifying assumptions; state what these are.

AH
Averell H.
Carnegie Mellon University