Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 18

Temperature, Heat, and the First Law of Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

An $8.00 \mathrm{~g}$ ice cube at its melting point is added to $130 \mathrm{~cm}^{3}$ of tea (water) initially at $90.0^{\circ} \mathrm{C}$ in a well-insulated container. When equilibrium is reached, what has been the decrease in the tea's temperature?

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Problem 2

A certain substance has a mass per mole of $50.0 \mathrm{~g} / \mathrm{mol}$. When $325 \mathrm{~J}$ is added as heat to $a$ ?? $0 \mathrm{~g}$ sample the sample's temperature rises from $25.0^{\circ} \mathrm{C}$ to $45.0^{\circ} \mathrm{C}$. What are the (a) specific heat and (b) molar specific heat of this substance? (c) How many moles are in the sample?

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Problem 3

A common window is to be replaced with a better insulated one. The initial window has glass with thickness $4.0 \mathrm{~mm}$. The new window has two such glass layers, separated by $1.2 \mathrm{~cm}$, with the intermediate space containing air. For (a) the common window and (b) the replacement, find the energy loss in watts per square meter when the outside temperature is $-20^{\circ} \mathrm{F}$ and the inside temperature is $+72^{\circ} \mathrm{F}$.

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Problem 4

One way to keep the contents of a garage from becoming too cold on a night when a severe subfreezing temperature is forecast is to put a tub of water in the garage. If the mass of the water is $135 \mathrm{~kg}$ and its initial temperature is $20^{\circ} \mathrm{C}$, (a) how much energy must the water transfer to its surroundings in order to freeze completely and (b) what is the lowest possible temperature of the water and its surroundings until that happens?

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Problem 5

Nonmetric version: (a) How long does a $2.0 \times 10^{5} \mathrm{Btw} / \mathrm{h}$ water heater take to raise the temperature of 65 gal of water from $70^{\circ} \mathrm{F}$ to $100^{\circ} \mathrm{F}$ ? Metric version: (b) How long does a $59 \mathrm{~kW}$ water heater take to raise the temperature of $246 \mathrm{~L}$ of water from $21^{\circ} \mathrm{C}$ to $38^{\circ} \mathrm{C}$ ?

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Problem 6

A lab sample of gas is taken through cycle abca shown in the $p-V$ diagram of Fig. 18-24. The net work done is +1.5J. Along path $a b$, the change in the internal energy is $+3.0 \mathrm{~J}$ and the magnitude of the work done is $5.0 \mathrm{~J}$. Along path ca, the energy trans$5.0 \mathrm{~J}$. Along path $c a$, the cnergy trans- ferred to the gas as heat is $+2.5$ J. How much energy is transferred as heat
Figure 18-24 Problem 6. along (a) path $a b$ and (b) path $b c$ ? 7 When a system is taken from state $i$ to

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Problem 7

When a system is taken from state $i$ to state $f$ along path iaf in Fig. 18-25, $Q$ $=100$ cal and $W=40$ cal. Along path ibf, $Q=72$ cal. (a) What is $W$ along path $i b f ?$ (b) If $W=-26 \mathrm{cal}$ for the return path $f i$, what is $Q$ for this path? $i b$ and (c) path bf?

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Problem 8

Samples $A$ and $B$ are at different initial temperatures when they are placed in a thermally insulated container and allowed to
(a)
(b)
Figure 18-26 Problem 8 .
come to thermal equilibrium. Figure $18-26 a$ gives their temperatures $T$ versus time $t$. Sample $A$ has a mass of $4.0 \mathrm{~kg} ;$ sample $B$ has a mass of $2.0 \mathrm{~kg}$. Figure $18-26 b$ is a general plot for the material of sample $B$. It shows the temperature change $\Delta T$ that the material undergoes when energy is transferred to it as heat $Q$. The change $\Delta T$ is plotted versus the energy $Q$ per unit mass of the material, and the scale of the vertical axis is set by $\Delta T_{x}=4.0(\mathrm{~V}$. What is the specific heat of sample $A$ ?

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Problem 9

A small electric immersion heater is used to heat $170 \mathrm{~g}$ of water for a cup of instant coffee. The heater is labeled "180 watts" (it converts electrical energy to thermal energy at this rate). Calculate the time required to bring all this water from $23.0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$, ignoring any heat losses.

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Problem 10

A thermodynamic system is taken from state $A$ to state $B$ to state $C$, and then back to $A$, as shown in the $p$ - $V$ diagram of Fig. 18-27a. The vertical scale is set by $p_{s}=20 \mathrm{~Pa}$, and the horizontal scale is set by $V_{s}=2.0 \mathrm{~m}^{3}$. (a) $-(\mathrm{g})$ Complete the table in Fig. 18-27b by inserting a plus sign, a minus sign, or a zero in cach indicated cell. (h) What is the net work done by the system as it moves once through the cycle $A B C A$ ?
Figure 18-27 Problem $10 .$

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Problem 11

A $3.0 \mathrm{~cm}$ slab has formed on an outdoor tank of water (Fig. 18-28). The air is at $-14^{\circ} \mathrm{C}$. Find the rate of ice formation (centimeters per hour). The ice has thermal conductivity $0.0040 \mathrm{cal} / \mathrm{s} \cdot \mathrm{cm} \cdot \mathrm{C}^{*}$ and density $0.92 \mathrm{~g} / \mathrm{cm}^{3}$. Assume there is no energy transfer through the walls or bottom.

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Problem 12

The specific heat of a substance varies with temperature according to the function $c=0.20+0.14 T+$ $0.023 T^{2}$, with $T$ in ${ }^{\circ} \mathrm{C}$ and $c$ in $\mathrm{cal} / \mathrm{g} \cdot \mathrm{K}$. Find the energy required to raise the temperature of $1.0 \mathrm{~g}$ of this substance from $5.0^{\circ} \mathrm{C}$ to $15^{\circ} \mathrm{C}$.

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Problem 13

As a result of a temperature rise of $64^{\circ}$, a bar with a crack at its center buckles upward (Fig. 18-29). The fixed distance $L_{0}$ is $3.77 \mathrm{~m}$ and the coefficient of linear expansion of the bar is $25 \times 10^{-6 / C}$. Find the rise $x$ of the center.

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Problem 14

Leidenfrost effect. A water drop will last about 1 son a hot skillet with $\quad$ Water drop will last about 1 s on a hot skillet with a temperature between $100^{\circ} \mathrm{C}$ and about $200^{\circ} \mathrm{C}$ However, if the skillet is much hotter, the drop can last sevFigure 18-30 Problem $14 .$ eral minutes, an effect named after an early investigator. The longer lifetime is due to the support of a thin layer of air and water vapor that separates the drop from the metal (by distance $L$ in Fig. 18-30). Let $L=0.0800 \mathrm{~mm}$, and assume that the drop is flat with height $h=1.50 \mathrm{~mm}$ and bottom face area $A=5.00 \times 10^{-6} \mathrm{~m}^{2}$. Also assume that the skillet has a constant temperature $T_{x}=300^{\circ} \mathrm{C}$ and the drop has a temperature of $100^{\circ} \mathrm{C}$. Water has density $\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}$, and the supporting layer has thermal conductivity $k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. (a) At what rate is energy conducted from the skillet to the drop through the drop's bottom surface?
(b) If conduction is the primary way energy moves from the skillet to the drop, how long will the drop last?

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Problem 15

Ethyl alcohol has a boiling point of $78.0^{\circ} \mathrm{C}$, a freezing point of $-114^{\circ} \mathrm{C}$, a heat of vaporization of $879 \mathrm{~kJ} / \mathrm{kg}$, a heat of fusion of $109 \mathrm{~kJ} / \mathrm{kg}_{n}$ and a specific heat of $2.43 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$. How much energy must be removed from $0.720 \mathrm{~kg}$ of ethyl alcohol that is initially a gas at $78.0^{\circ} \mathrm{C}$ so that it becomes a solid at $-114^{\circ} \mathrm{C}$ ?

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Problem 16

Suppose $200 \mathrm{~J}$ of work is done on a system and $80.0 \mathrm{cal}$ is extracted from the system as heat. In the sense of the first law of thermodynamics, what are the values (including algebraic signs) of (a) $W,(\mathrm{~b}) Q$, and $(\mathrm{c}) \Delta E_{\mathrm{in}} ?$

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Problem 17

In Fig. 18-31, a gas sample expands from $V_{0}$ to $4.0 V_{0}$ while its pressure decreases from $p_{0}$ to $p_{0} / 4.0$. If $V_{0}=1.0 \mathrm{~m}^{3}$ and $p_{0}=60 \mathrm{~Pa}$, how much work is done by the gas if its pressure changes with volume via (a) path $A,(b)$ path $B$, and $(c)$ path $C$ ? 18 Figure $18-32$ shows the cross section of a wall made of three lay- ers. The layer thicknesses are $I$ ers. The layer thicknesses are $L_{1}$, $\quad$ Volume $\left(\mathrm{m}^{3}\right)$ $L_{2}=0.750 L_{1}$, and $L_{3}=0.350 L_{1-} \quad$ Figure 18-31 Problem 17. The thermal conductivities are $k_{1}, k_{2}=0.900 k_{1}$, and $k_{3}=0.800 k_{1}$ The temperatures at the left side and right side of the wall are $T_{H}=30.0^{\circ} \mathrm{C}$ and $T_{C}=-15.0^{\circ} \mathrm{C}$, respectively. Thermal conduction is steady- (a) What is the tem- $T_{H}$ perature difference $\Delta T_{2}$ across layer 2 (between the left and right sides of the (between the left and right sides of the layer)? If $k_{2}$ were, instead, equal to $1.1 k_{1}$.

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Problem 17

In Fig. 18-31, a gas sample expands from $V_{0}$ to $4.0 V_{0}$ while its pressure decreases from $p_{0}$ to $p_{0} / 4.0$. If $V_{0}=1.0 \mathrm{~m}^{3}$ and $p_{0}=60 \mathrm{~Pa}$, how much work is done by the gas if its pressure changes with volume via (a) path of $A$, (b) path $B$, and (c) path $C$ ?

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Problem 18

Figure 18-32 shows the cross section of a wall made of three lay- $\quad 0 \quad V_{0} \quad 4.0 V_{0}$ ers. The layer thicknesses are $L_{1}$, $L_{2}=0.750 L_{1}$, and $L_{3}=0.350 L_{1-} \quad$ Figure 18-31 Problem 17. The thermal conductivities are
$k_{1}, k_{2}=0.900 k_{1}$, and $k_{3}=0.800 k_{1}$ -
The temperatures at the left side and right side of the wall are $T_{H}=30.0^{\circ} \mathrm{C}$ and $T_{C}=-15.0^{\circ} \mathrm{C}$, respectively. Thermal conduction is steady. (a) What is the tem- $T_{H}$ perature difference $\Delta T_{2}$ across layer 2 (between the left and right sides of the layer)? If $k_{2}$ were, instead, equal to $1.1 k_{1}$. Figure 18-32 Problem 18 . (b) would the rate at which energy is conducted through the wall be greater than, less than, or the same as previously, and (c) what would be the value of $\Delta T_{2}$ ?

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Problem 19

Consider the slab shown in Fig. 18-18. Suppose that $L=15.0 \mathrm{~cm}, A=105 \mathrm{~cm}^{2}$, and the material is copper. If $T_{H}=125^{\circ} \mathrm{C}$, $T_{C}=10.0^{\circ} \mathrm{C}$, and a steady state is reached, find the conduction rate through the slab.

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Problem 20

The ceiling of a single-family dwelling in a cold climate is to have an $R$-value of 35 . To give such insulation, how thick would a layer of (a) polyurethane foam and (b) silver have to be?

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Problem 21

What mass of steam at $100^{\circ} \mathrm{C}$ must be mixed with $250 \mathrm{~g}$ of ice at its melting point, in a thermally insulated container, to produce liquid water at $50^{\circ} \mathrm{C} ?$

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Problem 22

Evaporative cooling of beverages A cold beverage can be kept cold even on a warm day if it is slipped into a porous ceramic container that has been soaked in water. Assume that energy lost to evaporation matches the net energy gained via the radiation exchange through the top and side surfaces The container and beverage have temperature $T=10^{\circ} \mathrm{C}$, the environment has temperature $T_{\mathrm{cuv}}=32^{\circ} \mathrm{C}$, and the container is a cylinder with radius $r=2.2 \mathrm{~cm}$ and height $10 \mathrm{~cm}$. Approximate the emissivity as $\varepsilon=1$, and neglect other energy exchanges. At what rate $d m / d t$ is the container losing water mass?

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Problem 23

Calculate the minimum amount of energy, in joules, required to completely melt $85.0 \mathrm{~g}$ of silver initially at $15.0^{\circ} \mathrm{C}$.

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Problem 24

A solid cylinder of radius $r_{1}=2.5 \mathrm{~cm}$, length $h_{1}=5.5 \mathrm{~cm}$, emissivity $0.85$, and temperature $30^{\circ} \mathrm{C}$ is suspended in an environment of temperature $50^{\circ} \mathrm{C}$. (a) What is the cylinder's net thermal radiation transfer rate $P_{1}$ ? (b) If the cylinder is stretched until its radius is $r_{2}=0.50 \mathrm{~cm}$, its net thermal radiation transfer rate becomes $P_{2}$. What is the ratio $P_{2} / P_{1}$ ?

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Problem 25

A gas within a closed chamber undergoes the cycle shown in the $p-V$ diagram of Fig. 18-33. The horizontal scale is set by $V_{x}=8.0 \mathrm{~m}^{3}$. Calculate the net energy added to the system as heat during one complete cycle.

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Problem 26

Two constant-volume gas thermometers are assembled, one with nitrogen and the other with hydrogen. Both contain
enough gas so that $p_{3}=78 \mathrm{kPa}$.
Figure 18-33 Problem $25 .$ between the pressures in the two thermometers if both bulbs are in boiling water pressure?

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Problem 27

Figure 18-34 shows (in cross section) a wall consisting of four layers, with thermal conductivities $k_{1}=0.060 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, k_{3}=0.040$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, and $k_{4}=0.12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ ( $k_{2}$ is not known). The layer thicknesses are $L_{1}=1.2 \mathrm{~cm}, L_{3}=5.6 \mathrm{~cm}$, and $L_{4}=4.0 \mathrm{~cm}\left(L_{2}\right.$ is not known). The known temperatures are $T_{1}=30^{\circ} \mathrm{C}, T_{12}=25^{\circ} \mathrm{C}$, and $T_{4}=-10^{\circ} \mathrm{C}$ Energy transfer through the wall is steady. What is interface temperature $T_{34}$ ?
Figure 18-34 Problem $27 .$

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Problem 28

On a linear $X$ temperature scale, water freezes at $-125.0^{\circ} X$ and boils at $360.0^{\circ} \mathrm{X}$. On a linear $\mathrm{Y}$ temperature scale, water freezes at $-70.00^{\circ} \mathrm{Y}$ and boils at $-30.00^{\circ} \mathrm{Y}$. A temperature of $50.00^{\circ} \mathrm{Y}$ corresponds to what temperature on the X scale?

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Problem 29

A sphere of radius $0.350 \mathrm{~m}$, temperature $27.0^{\circ} \mathrm{C}$, and emissivity $0.850$ is located in an environment of temperature $77.0^{\circ} \mathrm{C}$. At what rate does the sphere (a) emit and (b) absorb thermal radiation? (c) What is the sphere's net change in energy in $3.50 \mathrm{~min}$ ?

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Problem 30

If you were to walk briefly in space without a spacesuit while far from the Sun (as an astronaut does in the movie 2001, A Space Odys3ey), yuu wusld foel the cold of space-while you tadiated energy, you would absorb almost none from your environment. (a) At what rate would you lose energy? (b) How much energy would you lose in 44 s? Assume that your emissivity is $0.90$, and estimate other data needed in the calculations.

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Problem 30

If you were to walk briefly in space without a spacesuit while far from the Sun (as an astronaut does in the movie 2001, A Space Odys3ey), yuu wusld foel the cold of space-while you tadiated energy, you would absorb almost none from your environment. (a) At what rate would you lose energy? (b) How much energy would you lose in 44 s? Assume that your emissivity is $0.90$, and estimate other data needed in the calculations.

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Problem 31

Ice has formed on a shallow pond, and a steady state has been reached, with the air above the ice at $-5.0^{\circ} \mathrm{C}$ and the bottom of the pond at $4.0^{\circ} \mathrm{C}$. If the total depth of ice $+$ water is $2.0 \mathrm{~m}$, how thick is the ice? (Assume that the thermal conductivities of ice and water are $0.40$ and $0.12 \mathrm{cal} / \mathrm{m} \cdot \mathrm{C}^{\circ} \cdot \mathrm{s}$, respectively. $)$

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Problem 32

Penguin huddling. To withstand the harsh weather of the Antarctic, emperor penguins huddle in groups (Fig. 18-35). Assume that a penguin is a circular cylinder with a top surface area $a=0.26 \mathrm{~m}^{2}$ and height $h=90 \mathrm{~cm}$. Let $P_{s}$ be the rate at which an individual penguin radiates energy to the environment (through the top and the sides); thus $N P$, is the rate at which $N$ identical, wellseparated penguins radiate. If the penguins huddle closely to form a huddled cylinder with top surface area Na and height $h$, the cylinder radiates at the rate $P_{h}$ - If $N=1000$, (a) what is the value of the fraction $P_{k} / N P_{r}$ and (b) by what percentage does huddling reduce the total radiation loss?
Alain Torterotot/Peter Amold/Photolibrary
Figure 18-35 Problem 32.

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Problem 33

The volume of a solid aluminum ball with initial radius $20 \mathrm{~cm}$ increases by $347 \mathrm{~cm}^{3}$ when the ball is heated. What is the temperature change?

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Problem 34

The giant hornet Vespa mandarinia japonica preys on Japanese bees However, if one of the hornets attempts to invade a beehive, several hundred of the bees quickly form a compact ball around the hornet to stop it. They don't sting, bite, crush, or suffocate it. Rather they overheat it by quickly raising their body temperatures from the normal $35^{\circ} \mathrm{C}$ to $47^{\circ} \mathrm{C}$ or $48^{\circ} \mathrm{C}$, which is lethal to the hornet but not to the bees (Fig. 18-36). Assume the following: 500 bees form a ball of radius $R=1.8 \mathrm{~cm}$ for a time $t=$ 18 min, the primary loss of energy by the ball is by thermal radiation, the ball's surface has emissivity $\varepsilon=0.80$, and the ball has a uniform temperature. On average, how much additional energy must each bee produce during the 18 min to maintain $47^{\circ} \mathrm{C}$ ?
Figure 18-36
Problem $34 .$

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Problem 35

A certain diet doctor encourages people to diet by drinking ice water. His theory is that the body must burn off enough fat to raise the temperature of the water from $0.00^{\circ} \mathrm{C}$ to the body temperature of $37.0^{\circ} \mathrm{C}$ How many liters of ice water would have to be consumed to burn off $454 \mathrm{~g}$ (about 1 lb) of fat, assuming that burning this much fat requires 3900 Cal be transferred to the ice water? Why is it not advisable to follow this diet? (One liter $=10^{3} \mathrm{~cm}^{3}$. The density of water is $1.00 \mathrm{~g} / \mathrm{cm}^{3}$.)

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Problem 36

What mass of butter, which has a usable energy content of $6.0 \mathrm{Cal} / \mathrm{g}(=6000 \mathrm{cal} / \mathrm{g})$, would be equivalent to the change in gravitational potential energy of a $78.0 \mathrm{~kg}$ man who ascends from sea level to the top of Mt. Everest, at elevation $8.84 \mathrm{~km}$ ? Assume that the average $g$ for the ascent is $9.80 \mathrm{~m} / \mathrm{s}^{2}$.

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Problem 37

(a) Two $40 \mathrm{~g}$ ice cubes are dropped into $200 \mathrm{~g}$ of water in a thermally insulated container. If the water is initially at $25^{\circ} \mathrm{C}$, and the ice comes directly from a freezer at $-15^{\circ} \mathrm{C}$, what is the final temperature at thermal equilibrium? (b) What is the final temperature if only one ice cube is used?

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Problem 38

(a) In 1983 , the temperature at the Soviet Vostok Station in Antarctica reached a record low of $-89.2^{\circ} \mathrm{C}$. What temperature is this on the Fahrenheit scale? (b) The highest officially recorded temperature in the continental United States was $134^{\circ} \mathrm{F}$ in Death Valley, California. What is this temperature on the Celsius scale?

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Problem 39

A vertical glass tube of length $L=1.280000 \mathrm{~m}$ is half filled with a liquid at $20.000000^{\circ} \mathrm{C}$. How much will the height of the liquid column change when the tube and liquid are heated to $30.000000^{\circ} \mathrm{C}$ ? Use coefficients $\alpha_{\text {glas }}=2.000000 \times 10^{-5} / \mathrm{K}$ and $\beta_{\text {liquia }}=4.000000 \times 10^{-5} / \mathrm{K}$.

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Problem 40

As a gas is held within a closed chamber, it passes through the cycle shown in Fig. 18-37. Determine the energy transferred by the system as heat during constant-pressure process $C A$ if the energy added as heat $O_{A B}$ during constant-volume process $A B$ is $25.0 \mathrm{~J}$, no energy is transferred as heat during adiabatic process $B C$, and the net work done during the cycle is $15.0 \mathrm{~J}$.

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Problem 41

An aluminum cup of $200 \mathrm{~cm}^{3} \mathrm{ca}-$ Figure 18-37 Problem 40. pacity is completely filled with glycerin at $22^{\circ} \mathrm{C}$. How much glycerin, if any, will spill out of the cup if the temperature of both the cup and the glycerin is increased to $28^{\circ} \mathrm{C} ?$ (The coefficient of volume expansion of glycerin is $5.1 \times 10^{-4} / \mathrm{C}^{9}$.)

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Problem 42

A $0.600 \mathrm{~kg}$ sample is placed in a cooling apparatus that removes energy as heat at a constant rate. Figure 18-38 gives the temperature $T$ of the sample versus time $t$; the horizontal scale is set by $t_{s}=80.0$ min. The sample freezes during the energy removal. The specific heat of the sample in its initial liquid phase is $3000 \mathrm{~J} / \mathrm{kg}-\mathrm{K}$. What are (a) the sample's heat of fusion and (b) its specific $\quad 0 \quad t_{x}$ of fusion and (b) its specific heat in the frozen phase?
Figure 18-38 Problem 42.

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Problem 43

A cylindrical copper rod of
length $0.60 \mathrm{~m}$ and cross-
sectional area $6.0 \mathrm{~cm}^{2}$ is insulated along its side. The ends are held at a temperature difference of $100 \mathrm{C}^{\circ}$ by having one end in a water-ice mixture and the other in a mixture of boiling water and steam. At what rate (a) is energy conducted by the rod and (b) does the ice melt?

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Problem 44

A $0.485 \mathrm{~kg}$ sample of liquid water and a sample of ioe are placed in a thermally insulated container. The container also contains a device that transfers energy as heat from the liquid water to the ice at a constant rate $P$, until thermal equilibrium is reached. The temperatures $T$ of the liquid water and the ice are given in Fig. 18-39 as functions of time $t$; the horizontal scale is set by $t_{s}=80.0$ min. (a) What is rate $P$ ? (b) What is the initial mass of the ice in the container? (c) When thermal equilibrium is reached, what is the mass of the ice produced in this process?
Figure 18-39 Problem 44.

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Problem 45

At what temperature is the Fahrenheit scale reading equal to (a) three times that of the Celsius scale and (b) one-third that of the Celsius scale?

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Problem 46

At $20^{\circ} \mathrm{C}$, a brass cube has edge length $25 \mathrm{~cm}$. What is the increase in the surface area when it is heated from $20^{\circ} \mathrm{C}$ to $75^{\circ} \mathrm{C} ?$

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Problem 47

Figure $18-40$ represents a closed cy- $p$ cle for a gas (the figure is not drawn to scale). The change in the internal energy of the gas as it moves from $a$ to $c$ along the path $a b c$ is $-50 \mathrm{~J}$. As it moves from $c$ to $d, 45 \mathrm{~J}$ must be transferred to it as heat. An additional transfer of $20 \mathrm{~J}$ to it as heat is needed as it moves from $d$ to
as heat is needed as it moves from $d$ to $v$
a. How much work is done on the gas as Figure 18-40 Problem $47 .$ it moves from $c$ to $d$ ?

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Problem 48

At $20^{\circ} \mathrm{C}$, a rod is exactly $20.05 \mathrm{~cm}$ long on a steel ruler. Both are placed in an oven at $250^{\circ} \mathrm{C}$, where the rod now measures $20.11$ $\mathrm{cm}$ on the same ruler. What is the coefficient of linear expansion for the material of which the rod is made?

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Problem 49

In Fig. 18-41a, two identical rectangular rods of metal are welded cnd $T_{1}$ to end, with a temperature of $T_{1}=0^{\circ} \mathrm{C}$ on the left side and a temperature of $T_{2}=100^{\circ} \mathrm{C}$ on the right side. In $6.0 \mathrm{~min}, 43 \mathrm{~J}$ is conducted at a constant rate from the right side to the left side. How much time would be required to conduct $43 \mathrm{~J}$ if the rods were welded

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Problem 50

An aluminum-alloy rod has a length of $10.000 \mathrm{~cm}$ at $20.000^{\circ} \mathrm{C}$ and a length of $10.020 \mathrm{~cm}$ at the boiling point of water. (a) What is the length of the rod at the freezing point of water? (b) What is the temperature if the length of the rod is $10.009 \mathrm{~cm}$ ?

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Problem 51

In a solar water heater, energy from the Sun is gathered by water that circulates through tubes in a rooftop collector. The solar radiation enters the collector through a transparent cover and warms the water in the tubes; this water is pumped into a holding tank. Assume that the efficiency of the overall system is $25 \%$ (that is, $80 \%$ of the incident solar energy is lost from the system). What collector area is necessary to raise the temperature of 200 L of water in the tank from $20^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ in $1.0 \mathrm{~h}$ when the intensity of incident sunlight is $750 \mathrm{~W} / \mathrm{m}^{2} ?$

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Problem 52

An aluminum flagpole is $30 \mathrm{~m}$ high. By how much does its length increase as the temperature increases by $15 \mathrm{C}^{\circ}$ ?

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Problem 53

A person makes a quantity of iced tea by mixing $250 \mathrm{~g}$ of hot ten (essentially water) with an equal mass of ice at its melting point. Assume the mixture has negligible energy exchanges with its environment. If the tea's initial temperature is $T_{i}=90^{\circ} \mathrm{C}$, when thermal equilibrium is reached what are (a) the mixture's temperature $T_{f}$ and (b) the remaining mass $m_{f}$ of ice? If $T_{i}=70^{\circ} \mathrm{C}$, when thermal equilibrium is reached what are (c) $T_{f}$ and (d) $m_{f}$ ?

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Problem 54

A $22.0 \mathrm{~g}$ copper ring at $0.000^{\circ}$
$C$ has an inner diameter of $D=2.54000 \mathrm{~cm} .$ An aluminum sphere at $100.0^{\circ} \mathrm{C}$ has a diameter of $d=2.54508 \mathrm{~cm}$. The sphere is put on top of the ring (Fig. 18-42), and the two are allowed to come to thermal equilibrium, with no heat lost to the surroundings. The sphere just passes through the ring at the equilibrium tempera-
ture. What is the mass of the Figure 18-42 Problem $54 .$ sphere?

Prachita Kush

Numerade Educator

Problem 55

A steel rod is $3.000 \mathrm{~cm}$ in diameter at $-10.00^{\circ} \mathrm{C}$. A brass ring has an interior diameter of $2.992 \mathrm{~cm}$ at $-10.00^{\circ} \mathrm{C}$ At what common temperature will the ring just slide onto the rod?

Prachita Kush

Numerade Educator

Problem 56

A $150 \mathrm{~g}$ copper bowl contains $260 \mathrm{~g}$ of water, both at $20.0^{\circ} \mathrm{C}$ A very hot $300 \mathrm{~g}$ copper cylinder is dropped into the water, causing the water to boil, with $5.00 \mathrm{~g}$ being converted to steam. The final temperature of the system is $100^{\circ} \mathrm{C}$. Neglect energy transfers with the environment. (a) How much energy (in calories) is transferred to the water as heat? (b) How much to the bowl? (c) What is the original temperature of the cylinder?

Prachita Kush

Numerade Educator

Problem 57

Suppose that on a linear temperature scale $X$, water boils at $-72.0^{\circ} \mathrm{X}$ and freezes at $-123.0^{\circ} \mathrm{X}$. What is a temperature of $59.0 \mathrm{~K}$ on the X scale? (Approximate water's boiling point as $373 \mathrm{~K}$.)

Prachita Kush

Numerade Educator

Problem 58

How much water remains unfrozen after $50.2 \mathrm{~kJ}$ is transferred as heat from $240 \mathrm{~g}$ of liquid water initially at its freezing point?

Prachita Kush

Numerade Educator

Problem 59

What is the volume of a lead ball at $20.00^{\circ} \mathrm{C}$ if the ball's volume at $60.00^{\circ} \mathrm{C}$ is $33.58 \mathrm{~cm}^{3}$ ?

Prachita Kush

Numerade Educator

Problem 60

Calculate the specific heat of a metal from the following data. A container made of the metal has a mass of $3.6 \mathrm{~kg}$ and contains $15 \mathrm{~kg}$ of water. A $1.8 \mathrm{~kg}$ piece of the metal initially at a temperature of $180^{\circ} \mathrm{C}$ is dropped into the water. The container and water initially have a temperature of $16.0^{\circ} \mathrm{C}$, and the final temperature of the entire (insulated) system is $18.0^{\circ} \mathrm{C}$.

Prachita Kush

Numerade Educator

Problem 61

A circular hole in an aluminum plate is $3.115 \mathrm{~cm}$ in diameter at $0.000^{\circ} \mathrm{C}$. What is its diameter when the temperature of the plate is raised to $180.0^{\circ} \mathrm{C}$ ?

Prachita Kush

Numerade Educator

Problem 62

When the temperature of a copper coin is raised by $100 \mathrm{C}^{\circ}$, its diameter increases by $0.20 \%$. To two significant figures, give the percent increase in (a) the area of a face, (b) the thickness, (c) the volume, and (d) the mass of the coin. (c) Calculate the coefficient of linear expansion of the coin.

Prachita Kush

Numerade Educator

Problem 63

A gas thermometer is constructed of two gas-containing bulbs, each in a water bath, as shown in Fig. 18-43. The pressure difference between the two bulbs is measured by a mercury manometer as shown. Appropriate reservoirs, not shown in Figure 18-43 Problem 63. the diagram, maintain constant gas
Figure 18-43 Problem 63.
volume in the two bulbs. There is no difference in pressure when both baths are at the triple point of water. The pressure difference is 120 torr when one bath is at the triple point and the other is at the boiling point of water. It is $80.0$ torr when one bath is at the triple point and the other is at an unknown temperature to be measured. What is the unknown temperature?

Prachita Kush

Numerade Educator

Problem 64

When the temperature of a metal cylinder is raised from $47.4^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$, its length increases by $0.10 \%$. (a) Find the percent change in density. (b) What is the metal? Use Table 18-2.

Prachita Kush

Numerade Educator

Problem 65

Suppose the temperature of a gas is $372.90 \mathrm{~K}$ when it is at the boiling point of water. What then is the limiting value of the ratio of the pressure of the gas at that boiling point to its pressure at the triple point of water? (Assume the volume of the gas is the same at both temperatures.)

Prachita Kush

Numerade Educator

Problem 66

In a certain experiment, a small radioactive source must move at selected, extremely slow speeds. This motion is accomplished by fastening the source to one end of an aluminum rod and heating the central section of the rod in a controlled way. If the effective heated section of the rod in Fig. 18-44 has length $d=2.00 \mathrm{~cm}$, at what constant rate must the temperature of the rod be changed if the source is to move at a constant speed of $150 \mathrm{~nm} / \mathrm{s}$ ?
Figure 18-44 Problem 66 .

Prachita Kush

Numerade Educator