University Physics with Modern Physics

Hugh D. Young

Chapter 19

The First Law of Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

Two moles of an ideal gas are heated at constant pressure from $T$ = 27$^\circ$C to $T$ = 107$^\circ$C. (a) Draw a $pV$-diagram for this process. (b) Calculate the work done by the gas.

Supratim Pal

Supratim Pal

Numerade Educator

Problem 2

Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is 27.0$^\circ$C and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done 2.40 $\times$ 10$^3$ J of work.

Katie Mcalpine

Numerade Educator

Problem 3

Two moles of an ideal gas are compressed in a cylinder at a constant temperature of 65.0$^\circ$C until the original pressure has tripled. (a) Sketch a $pV$-diagram for this process. (b) Calculate the amount of work done.

Ze-Han Lee

Numerade Educator

Problem 4

The graph in Fig. E19.4 shows a $pV$-diagram of the air in a human lung when a person is inhaling and then exhaling a deep breath. Such graphs, obtained in clinical practice, are normally somewhat curved, but we have modeled one as a set of straight lines of the same general shape. ($Important:$ The pressure shown is the gauge pressure, not the absolute pressure.) (a) How many joules of net work does this person's lung do during one complete breath? (b) The process illustrated here is somewhat different from those we have been studying, because the pressure change is due to changes in the amount of gas in the lung, not to temperature changes. (Think of your own breathing. Your lungs do not expand because they've gotten hot.) If the temperature of the air in the lung remains a reasonable 20$^\circ$C, what is the maximum number of moles in this person's lung during a breath?

Salamat Ali

Numerade Educator

Problem 5

During the time 0.305 mol of an ideal gas undergoes an isothermal compression at 22.0$^\circ$C, 392 J of work is done on it by the surroundings. (a) If the final pressure is 1.76 atm, what was the initial pressure? (b) Sketch a $pV$-diagram for the process.

Ze-Han Lee

Numerade Educator

Problem 6

A gas undergoes two processes. In the first, the volume remains constant at 0.200 m$^3$ and the pressure increases from 2.00 $\times$ 10$^5$ Pa to 5.00 $\times$ 10$^5$ Pa. The second process is a compression to a volume of 0.120 m$^3$ at a constant pressure of 5.00 $\times$ 10$^5$ Pa. (a) In a pV-diagram, show both processes. (b) Find the total work done by the gas during both processes.

Katie Mcalpine

Numerade Educator

Problem 7

(a) In Fig. 19.7a, consider the closed loop $1\rightarrow 3\rightarrow 2\rightarrow 4\rightarrow 1$. This is a $cyclic$ process in which the initial and final states are the same. Find the total work done by the system in this cyclic process, and show that it is equal to the area enclosed by the loop. (b) How is the work done for the process in part (a) related to the work done if the loop is traversed in the opposite direction, $1\rightarrow 4\rightarrow 2\rightarrow3\rightarrow 1$? Explain.

Ze-Han Lee

Numerade Educator

Problem 8

$\textbf{Figure E19.8}$ shows a $pV$-diagram for an ideal gas in which its absolute temperature at $b$ is one-fourth of its absolute temperature at $a$. (a) What volume does this gas occupy at point $b$? (b) How many joules of work was done by or on the gas in this process? Was it done by or on the gas? (c) Did the internal energy of the gas increase or decrease from $a$ to $b$? How do you know? (d) Did heat enter or leave the gas from $a$ to $b$? How do you know?

Katie Mcalpine

Numerade Educator

Problem 9

A gas in a cylinder expands from a volume of 0.110 m$^3$ to 0.320 m$^3$. Heat flows into the gas just rapidly enough to keep the pressure constant at 1.65 $\times$ 10$^5$ Pa during the expansion. The total heat added is 1.15 $\times$ 10$^5$ J. (a) Find the work done by the gas. (b) Find the change in internal energy of the gas. (c) Does it matter whether the gas is ideal? Why or why not?

Ze-Han Lee

Numerade Educator

Problem 10

Five moles of an ideal monatomic gas with an initial temperature of 127$^\circ$C expand and, in the process, absorb 1500 J of heat and do 2100 J of work. What is the final temperature of the gas?

Katie Mcalpine

Numerade Educator

Problem 11

The process abc shown in the $pV$-diagram in $Fig. E19.11$ involves 0.0175 mol of an ideal gas. (a) What was the lowest temperature the gas reached in this process? Where did it occur? (b) How much work was done by or on the gas from $a$ to $b$? From $b$ to $c$? (c) If 215 $J$ of heat was put into the gas during $abc$, how many of those joules went into internal energy?

Ze-Han Lee

Numerade Educator

Problem 12

A gas in a cylinder is held at a constant pressure of 1.80 $\times$ 10$^5$ $Pa$ and is cooled and compressed from 1.70 m$^3$ to 1.20 m$^3$. The internal energy of the gas decreases by 1.40 $\times$ 10$^5$ J. (a) Find the work done by the gas. (b) Find the absolute value of the heat flow, [$Q$] , into or out of the gas, and state the direction of the heat flow. (c) Does it matter whether the gas is ideal? Why or why not?

Katie Mcalpine

Numerade Educator

Problem 13

The $pV$-diagram in Fig. E19.13 shows a process $abc$ involving 0.450 mol of an ideal gas. (a) What was the temperature of this gas at points $a$, $b$, and $c$? (b) How much work was done by or on the gas in this process? (c) How much heat had to be added during the process to increase the internal energy of the gas by 15,000 J?

Ze-Han Lee

Numerade Educator

Problem 14

When water is boiled at a pressure of 2.00 atm, the heat of vaporization is 2.20 $\times$ 10$^6$ J/kg and the boiling point is 120$^\circ$C. At this pressure, 1.00 kg of water has a volume of 1.00 $\times$ 10$^{-3}$ m$^3$, and 1.00 kg of steam has a volume of 0.824 m$^3$. (a) Compute the work done when 1.00 kg of steam is formed at this temperature. (b) Compute the increase in internal energy of the water.

Katie Mcalpine

Numerade Educator

Problem 15

An ideal gas is taken from a to b on the $pV$-diagram shown in $Fig. E19.15$. During this process, 700 J of heat is added and the pressure doubles. (a) How much work is done by or on the gas? Explain. (b) How does the temperature of the gas at $a$ compare to its temperature at $b$? Be specific. (c) How does the internal energy of the gas at $a$ compare to the internal energy at $b$? Be specific and explain.

Ze-Han Lee

Numerade Educator

Problem 16

During an isothermal compression of an ideal gas, 410 J of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

Katie Mcalpine

Numerade Educator

Problem 17

A cylinder contains 0.250 mol of carbon dioxide ($CO_2$) gas at a temperature of 27.0$^\circ$C. The cylinder is provided with a frictionless piston, which maintains a constant pressure of 1.00 atm
on the gas. The gas is heated until its temperature increases to 127.0$^\circ$C. Assume that the CO$_2$ may be treated as an ideal gas. (a) Draw a $pV$-diagram for this process. (b) How much work is done by the gas in this process? (c) On what is this work done? (d) What is the change in internal energy of the gas? (e) How much heat was supplied to the gas? (f) How much work would have been done if the pressure had been 0.50 atm?

Ze-Han Lee

Numerade Educator

Problem 18

A cylinder contains 0.0100 mol of helium at $T$ = 27.0$^\circ$C. (a) How much heat is needed to raise the temperature to 67.0$^\circ$C while keeping the volume constant? Draw a $pV$-diagram for this process. (b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from 27.0$^\circ$C to 67.0$^\circ$C? Draw a $pV$-diagram for this process. (c) What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat? (d) If the gas is ideal, what is the change in its internal energy in part (a)? In part (b)? How do the two answers compare? Why?

Katie Mcalpine

Numerade Educator

Problem 19

In an experiment to simulate conditions inside an automobile engine, 0.185 mol of air at 780 K and 3.00 $\times$ 10$^6$ Pa is contained in a cylinder of volume 40.0 cm$^3$. Then 645 J of heat is transferred to the cylinder. (a) If the volume of the cylinder is constant while the heat is added, what is the final temperature of the air? Assume that the air is essentially nitrogen gas, and use the data in Table 19.1 even though the pressure is not low. Draw a $pV$-diagram for this process. (b) If instead the volume of the cylinder is allowed to increase while the pressure remains constant, repeat part (a).

Ze-Han Lee

Numerade Educator

Problem 20

When a quantity of monatomic ideal gas expands at a constant pressure of 4.00 $\times$ 10$^4$ Pa, the volume of the gas increases from 2.00 $\times$ 10$^{-3}$ m$^3$ to 8.00 $\times$ 10$^{-3}$ m$^3$. What is the change in the internal energy of the gas?

Katie Mcalpine

Numerade Educator

Problem 21

Heat $Q$ flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?

Ze-Han Lee

Numerade Educator

Problem 22

Three moles of an ideal monatomic gas expands at a constant pressure of 2.50 atm; the volume of the gas changes from 3.20 $\times$ 10$^{-2}$ m$^3$ to 4.50 $\times$ 10$^{-2}$ m$^3$. Calculate (a) the initial and final temperatures of the gas; (b) the amount of work the gas does in expanding; (c) the amount of heat added to the gas; (d) the change in internal energy of the gas.

Katie Mcalpine

Numerade Educator

Problem 23

An experimenter adds 970 J of heat to 1.75 mol of an ideal gas to heat it from 10.0$^\circ$C to 25.0$^\circ$C at constant pressure. The gas does +223 J of work during the expansion. (a) Calculate the change in internal energy of the gas. (b) Calculate $_\Upsilon$ for the gas.

Ze-Han Lee

Numerade Educator

Problem 24

Propane gas ($C_3H_8$) behaves like an ideal gas with $_\Upsilon$ = 1.127. Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

Katie Mcalpine

Numerade Educator

Problem 25

The temperature of 0.150 mol of an ideal gas is held constant at 77.0$^\circ$C while its volume is reduced to 25.0% of its initial volume. The initial pressure of the gas is 1.25 atm. (a) Determine the work done by the gas. (b) What is the change in its internal energy? (c) Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?

Ze-Han Lee

Numerade Educator

Problem 26

Five moles of monatomic ideal gas have initial pressure 2.50 $\times$ 10$^3$ Pa and initial volume 2.10 m$^3$. While undergoing an adiabatic expansion, the gas does 1480 J of work. What is the final pressure of the gas after the expansion?

Katie Mcalpine

Numerade Educator

Problem 27

A monatomic ideal gas that is initially at 1.50 $\times$ 10$^5$ Pa and has a volume of 0.0800 m$^3$ is compressed adiabatically to a volume of 0.0400 m$^3$. (a) What is the final pressure? (b) How much work is done by the gas? (c) What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

Ze-Han Lee

Numerade Educator

Problem 28

The engine of a Ferrari F355 F1 sports car takes in air at 20.0$^\circ$C and 1.00 atm and compresses it adiabatically to 0.0900 times the original volume. The air may be treated as an ideal gas with $_\Upsilon$ = 1.40. (a) Draw a $pV$-diagram for this process. (b) Find the final temperature and pressure.

Katie Mcalpine

Numerade Educator

Problem 29

During an adiabatic expansion the temperature of 0.450 mol of argon (Ar) drops from 66.0$^\circ$C to 10.0$^\circ$C. The argon may be treated as an ideal gas. (a) Draw a $pV$-diagram for this process. (b) How much work does the gas do? (c) What is the change in internal energy of the gas?

Ze-Han Lee

Numerade Educator

Problem 30

A player bounces a basketball on the floor, compressing it to 80.0% of its original volume. The air (assume it is essentially N$_2$ gas) inside the ball is originally at 20.0$^\circ$C and 2.00 atm. The ball's inside diameter is 23.9 cm. (a) What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal. (b) By how much does the internal energy of the air change between the ball's original state and its maximum compression?

Katie Mcalpine

Numerade Educator

Problem 31

On a warm summer day, a large mass of air (atmospheric pressure 1.01 $\times$ 10$^5$ Pa) is heated by the ground to 26.0$^\circ$C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only 0.850 $\times$ 10$^5$ Pa. Assume that air is an ideal gas, with $\Upsilon$ = 1.40. (This rate of cooling for dry, rising air, corresponding to roughly 1 C$^\circ$ per 100 m of altitude, is called the dry $adiabatic$ $lapse$ $rate$.)

Ze-Han Lee

Numerade Educator

Problem 32

A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at 1.00 $\times$ 10$^5 $Pa and occupies a volume of 2.50 $\times$ 10$^{-3}$ m$^3$. (a) Find the initial temperature of the gas in kelvins. (b) If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.

Katie Mcalpine

Numerade Educator

Problem 33

A quantity of air is taken from state $a$ to state $b$ along a path that is a straight line in the $pV$-diagram ($Fig. P19.33$). (a) In this process, does the temperature of the gas increase, decrease, or stay the same? Explain. (b) If $V_a$ = 0.0700 m$^3$, $V_b$ = 0.1100 m$^3$, $\rho _a$ = 1.00 $\times$ 10$^5$ Pa, and $\rho _b$ = 1.40 $\times$ 10$^5$ Pa, what is the work $W$ done by the gas in this process? Assume that the gas may be treated as ideal.

Ze-Han Lee

Numerade Educator

Problem 34

One-half mole of an ideal gas is taken from state a to state c as shown in $Fig. P19.34$. (a) Calculate the final temperature of the gas. (b) Calculate the work done on (or by) the gas as it moves from state $a$ to state $c$. (c) Does heat leave the system or enter the system during this process? How much heat? Explain.

Katie Mcalpine

Numerade Educator

Problem 35

$Figure P19.35$ shows the $pV$-diagram for a process in which the temperature of the ideal gas remains constant at 85$^\circ$C. (a) How many moles of gas are involved? (b) What volume does this gas occupy at $a$? (c) How much work was done by or on the gas from $a$ to $b$? (d) By how much did the internal energy of the gas change during this process?

Ze-Han Lee

Numerade Educator

Problem 36

The graph in $Fig. P19.36$ shows a $pV$-diagram for 3.25 mol of ideal helium (He) gas. Part $ca$ of this process is isothermal. (a) Find the pressure of the He at point $a$. (b) Find the temperature of the He at points $a$, $b$, and $c$. (c) How much heat entered or left the He during segments $ab$, $bc$, and $ca$? In each segment, did the heat enter or leave? (d) By how much did the internal energy of the He change from a to $b$, from $b$ to $c$, and from $c$ to $a$? Indicate whether this energy increased or decreased.

Katie Mcalpine

Numerade Educator

Problem 37

When a system is taken from state $a$ to state $b$ in $Fig. P19.37$ along path $acb$, 90.0 J of heat flows into the system and 60.0 J of work is done by the system. (a) How much heat flows into the system along path $adb$ if the work done by the system is 15.0 J? (b) When the system is returned from $b$ to $a$along the curved path, the absolute value of the work done by the system is 35.0 J. Does the system absorb or liberate heat? How much heat? (c) If $U_a$ = 0 and $U_d$ = 8.0 J, find the heat absorbed in the processes $ad$ and $db$.

Ze-Han Lee

Numerade Educator

Problem 38

A thermodynamic system is taken from state $a$ to state $c$ in $Fig. P19.38$ along either path $abc$ or path $adc$. Along path $abc$, the work $W$ done by the system is 450 J. Along path $adc$, $W$ is 120 J. The internal energies of each of the four states shown in the figure are $U_a$ = 150 J, $U_b$ = 240 J, $U_c$ = 680 J, and $U_d$ = 330 J. Calculate the heat flow $Q$ for each of the four processes $ab$, $bc$, $ad$, and $dc$. In each process, does the system absorb or liberate heat?

Katie Mcalpine

Numerade Educator

Problem 39

A volume of air (assumed to be an ideal gas) is first cooled without changing its volume and then expanded without changing its pressure, as shown by path $abc$ in $Fig. P19.39$. (a) How does the final temperature of the gas compare with its initial temperature? (b) How much heat does the air exchange with its surroundings during process $abc$? Does the air absorb heat or release heat during this process? Explain. (c) If the air instead expands from state $a$ to state $c$ by the straight-line path shown, how much heat does it exchange with its surroundings?

Ze-Han Lee

Numerade Educator

Problem 40

Three moles of argon gas (assumed to be an ideal gas) originally at 1.50 $\times$ 10$^4$ Pa and a volume of 0.0280 m$^3$ are first heated and expanded at constant pressure to a volume of 0.0435 m$^3$, then heated at constant volume until the pressure reaches 3.50 $\times$ 10$^4$ Pa, then cooled and compressed at constant pressure until the volume is again 0.0280 m$^3$, and finally cooled at constant volume until the pressure drops to its original value of 1.50 $\times$ 10$^4$ Pa. (a) Draw the $pV$-diagram for this cycle. (b) Calculate the total work done by (or on) the gas during the cycle. (c) Calculate the net heat exchanged with the surroundings. Does the gas gain or lose heat overall?

Katie Mcalpine

Numerade Educator

Problem 41

Two moles of an ideal monatomic gas go through the cycle $abc$. For the complete cycle, 800 J of heat flows out of the gas. Process $ab$ is at constant pressure, and process $bc$ is at constant volume. States $a$ and $b$ have temperatures $T_a$ = 200 K and $T_b$ = 300 K. (a) Sketch the $pV$-diagram for the cycle. (b) What is the work $W$ for the process $ca$?

Ze-Han Lee

Numerade Educator

Problem 42

Three moles of an ideal gas are taken around cycle $acb$ shown in $Fig. P19.42$. For this gas, $C_p$ = 29.1 J/mol $\cdot$ K. Process $ac$ is at constant pressure, process $ba$ is at constant volume, and process $cb$ is adiabatic. The temperatures of the gas in states $a$, $c$, and $b$ are $T_a$ = 300 K, $T_c$ = 492 K, and $T_b$ = 600 K. Calculate the total work $W$ for the cycle.

Katie Mcalpine

Numerade Educator

Problem 43

$Figure P19.43$ shows a $pV$-diagram for 0.0040 mol of ideal H$_2$ gas. The temperature of the gas does not change during segment $bc$. (a) What volume does this gas occupy at point $c$? (b) Find the temperature of the gas at points $a$, $b$, and $c$. (c) How much heat went into or out of the gas during segments $ab$, $ca$, and $bc$? Indicate whether the heat has gone into or out of the gas. (d) Find the change in the internal energy of this hydrogen during segments $ab$, $bc$, and $ca$. Indicate whether the internal energy increased or decreased during each segment.

Ze-Han Lee

Numerade Educator

Problem 44

(a) One-third of a mole of He gas is taken along the path $abc$ shown in $Fig. P19.44$. Assume that the gas may be treated as ideal. How much heat is transferred into or out of the gas? (b) If the gas instead went directly from state a to state $c$ along the horizontal dashed line in Fig. P19.44, how much heat would be transferred into or out of the gas? (c) How does $Q$ in part (b) compare with $Q$ in part (a)? Explain.

Katie Mcalpine

Numerade Educator

Problem 45

Starting with 2.50 mol of N$_2$ gas (assumed to be ideal) in a cylinder at 1.00 atm and 20.0$^\circ$C, a chemist first heats the gas at constant volume, adding 1.36 $\times$ 10$^4$ J of heat, then continues heating and allows the gas to expand at constant pressure to twice its original volume. Calculate (a) the final temperature of the gas; (b) the amount of work done by the gas; (c) the amount of heat added to the gas while it was expanding; (d) the change in internal energy of the gas for the whole process.

Ze-Han Lee

Numerade Educator

Problem 46

Nitrogen gas in an expandable container is cooled from 50.0$^\circ$C to 10.0$^\circ$C with the pressure held constant at 3.00 $\times$ 10$^5$ Pa. The total heat liberated by the gas is 2.50 $\times$ 10$^4$ J. Assume that the gas may be treated as ideal. Find (a) the number of moles of gas; (b) the change in internal energy of the gas; (c) the work done by the gas. (d) How much heat would be liberated by the gas for the same temperature change if the volume were constant?

Katie Mcalpine

Numerade Educator

Problem 47

A cylinder with a frictionless, movable piston like that shown in Fig. 19.5 contains a quantity of helium gas. Initially the gas is at 1.00 $\times$ 10$^5$ Pa and 300 K and occupies a volume of 1.50 L. The gas then undergoes two processes. In the first, the gas is heated and the piston is allowed to move to keep the temperature at 300 K. This continues until the pressure reaches 2.50 $\times$ 10$^4$ Pa. In the second process, the gas is compressed at constant pressure until it returns to its original volume of 1.50 L. Assume that the gas may be treated as ideal. (a) In a $pV$-diagram, show both processes. (b) Find the volume of the gas at the end of the first process, and the pressure and temperature at the end of the second process. (c) Find the total work done by the gas during both processes. (d) What would you have to do to the gas to return it to its original pressure and temperature?

Ze-Han Lee

Numerade Educator

Problem 48

A cube of copper 2.00 cm on a side is suspended by a string. (The physical properties of copper are given in Tables 14.1, 17.2, and 17.3.) The cube is heated with a burner from 20.0$^\circ$C to 90.0$^\circ$C. The air surrounding the cube is at atmospheric pressure (1.01 $\times$ 10$^5$ Pa). Find (a) the increase in volume of the cube; (b) the mechanical work done by the cube to expand against the pressure of the surrounding air; (c) the amount of heat added to the cube; (d) the change in internal energy of the cube. (e) Based on your results, explain whether there is any substantial difference between the specific heats $c_p$ (at constant pressure) and $c_V$ (at constant volume) for copper under these conditions.

Katie Mcalpine

Numerade Educator

Problem 49

During certain seasons strong winds called chinooks blow from the west across the eastern slopes of the Rockies and downhill into Denver and nearby areas. Although the mountains are cool, the wind in Denver is very hot; within a few minutes after the chinook wind arrives, the temperature can climb 20 C$^\circ$ ("chinook" refers to a Native American people of the Pacific Northwest). Similar winds occur in the Alps (called foehns) and in southern California (called Santa Anas). (a) Explain why the temperature of the chinook wind rises as it descends the slopes.
Why is it important that the wind be fast moving? (b) Suppose a strong wind is blowing toward Denver (elevation 1630 m) from Grays Peak (80 km west of Denver, at an elevation of 4350 m),
where the air pressure is 5.60 $\times$ 10$^4$ Pa and the air temperature is -15.0$^\circ$C. The temperature and pressure in Denver before the wind arrives are 2.0$^\circ$C and 8.12 $\times$ 10$^4$ Pa. By how many Celsius degrees will the temperature in Denver rise when the chinook arrives?

Ze-Han Lee

Numerade Educator

Problem 50

A large research balloon containing $2.00 \times 10^{3} \mathrm{~m}^{3}$ of helium gas at 1.00 atm and a temperature of $15.0^{\circ} \mathrm{C}$ rises rapidly from ground level to an altitude at which the atmospheric pressure is only 0.900 atm (Fig. $\mathbf{P} 19.50$ ). Assume the helium behaves like an ideal gas and the balloon's ascent is too rapid to permit much heat exchange with the surrounding air.
(a) Calculate the volume of the gas at the higher altitude.
(b) Calculate the temperature of the gas at the higher altitude.
(c) What is the change in internal energy of the helium as the balloon rises to the higher altitude?

Katie Mcalpine

Numerade Educator

Problem 51

An air pump has a cylinder 0.250 m long with a movable piston. The pump is used to compress air from the atmosphere (at absolute pressure 1.01 $\times$ 10$^5$ Pa) into a very large tank at 3.80 $\times$ 10$^5$ Pa gauge pressure. (For air, $C_V$ = 20.8 J/mol $\cdot$ K.) (a) The piston begins the compression stroke at the open end of the cylinder. How far down the length of the cylinder has the piston moved when air first begins to flow from the cylinder into the tank? Assume that the compression is adiabatic. (b) If the air is taken into the pump at 27.0$^\circ$C, what is the temperature of the compressed air? (c) How much work does the pump do in putting 20.0 mol of air into the tank?

Ze-Han Lee

Numerade Educator

Problem 52

A certain ideal gas has molar heat capacity at constant volume $C_V$ . A sample of this gas initially occupies a volume $V_0$ at pressure $p_0$ and absolute temperature $T_0$ . The gas expands isobarically to a volume $2V_0$ and then expands further adiabatically to a final volume $4V_0$ . (a) Draw a $pV$-diagram for this sequence of processes. (b) Compute the total work done by the gas for this sequence of processes. (c) Find the final temperature of the gas. (d) Find the absolute value of the total heat flow $Q$ into or out of the gas for this sequence of processes, and state the direction of heat flow.

Katie Mcalpine

Numerade Educator

Problem 53

A monatomic ideal gas expands slowly to twice its original volume, doing 450 J of work in the process. Find the heat added to the gas and the change in internal energy of the gas if the process is (a) isothermal; (b) adiabatic; (c) isobaric.

Ze-Han Lee

Numerade Educator

Problem 54

A cylinder with a piston contains 0.250 mol of oxygen at 2.40 $\times$ 10$^5$ Pa and 355 K. The oxygen may be treated as an ideal gas. The gas first expands isobarically to twice its original volume. It is then compressed isothermally back to its original volume, and finally it is cooled isochorically to its original pressure. (a) Show the series of processes on a $pV$-diagram. Compute (b) the temperature during the isothermal compression; (c) the maximum pressure; (d) the total work done by the piston on the gas during the series of processes.

Katie Mcalpine

Numerade Educator

Problem 55

Use the conditions and processes of Problem 19.54 to compute (a) the work done by the gas, the heat added to it, and its internal energy change during the initial expansion; (b) the work done, the heat added, and the internal energy change during the final cooling; (c) the internal energy change during the isothermal compression.

Ze-Han Lee

Numerade Educator

Problem 56

A cylinder with a piston contains 0.150 mol of nitrogen at 1.80 $\times$ 10$^5$ Pa and 300 K. The nitrogen may be treated as an ideal gas. The gas is first compressed isobarically to half its original volume. It then expands adiabatically back to its original volume, and finally it is heated isochorically to its original pressure. (a) Show the series of processes in a $pV$-diagram. (b) Compute the temperatures at the beginning and end of the adiabatic expansion. (c) Compute the minimum pressure.

Katie Mcalpine

Numerade Educator

Problem 57

Use the conditions and processes of Problem 19.56 to compute (a) the work done by the gas, the heat added to it, and its internal energy change during the initial compression; (b) the work done by the gas, the heat added to it, and its internal energy change during the adiabatic expansion; (c) the work done, the heat added, and the internal energy change during the final heating.

Ze-Han Lee

Numerade Educator

Problem 58

In a cylinder, 1.20 mol of an ideal monatomic gas, initially at 3.60 $\times$ 10$^5$ Pa and 300 K, expands until its volume triples. Compute the work done by the gas if the expansion is (a) isothermal; (b) adiabatic; (c) isobaric. (d) Show each process in a $pV$-diagram. In which case is the absolute value of the work done by the gas greatest? Least? (e) In which case is the absolute value of the heat transfer greatest? Least? (f) In which case is the absolute value of the change in internal energy of the gas greatest? Least?

Katie Mcalpine

Numerade Educator

Problem 59

You have recorded measurements of the heat flow $Q$ into 0.300 mol of a gas that starts at $T_1$ = 20.0$^\circ$C and ends at a temperature $T_2$. You measured $Q$ for three processes: one isobaric, one isochoric, and one adiabatic. In each case, $T_2$ was the same. $Figure$ $P19.59$ summarizes your results. But you lost a page from your lab notebook and don't have a record of the value of $T_2$; you also don't know which process was isobaric, isochoric, or adiabatic. Each process was done at a sufficiently low pressure for the gas to be treated as ideal. (a) Identify each process $a$, $b$, or $c$ as isobaric, isochoric, or adiabatic. (b) What is the value of $T_2$? (c) How much work is done by the gas in each process? (d) For which process is the magnitude of the volume change the greatest? (e) For each process, does the volume of the gas increase, decrease, or stay the same?

Ze-Han Lee

Numerade Educator

Problem 60

You compress a gas in an insulated cylinder-no heat flows into or out of the gas. The gas pressure is fairly low, so treating the gas as ideal is a good approximation. When you measure the pressure as a function of the volume of the gas, you obtain these results:

(a) Graph log ($p$) versus log ($V$), with $p$ in Pa and $V$ in m$^3$. Explain why the data points fall close to a straight line. (b) Use your graph to calculate $\gamma$ for the gas. Is the gas monatomic, diatomic, or polyatomic? (c) When $p$ = 0.101 atm and $V$ = 2.50 L, the temperature is 22.0$^\circ$C. Apply the ideal-gas equation and calculate the temperature for each of the other pairs of $p$ and $V$ values. In this compression, does the temperature of the gas increase, decrease, or stay constant?

Lainey Roebuck

Numerade Educator

Problem 61

You place a quantity of gas into a metal cylinder that has a movable piston at one end. No gas leaks out of the cylinder as the piston moves. The external force applied to the piston can be varied to change the gas pressure as you move the piston to change the volume of the gas. A pressure gauge attached to the interior wall of the cylinder measures the gas pressure, and you can calculate the volume of the gas from a measurement of the piston's position in the cylinder. You start with a pressure of 1.0 atm and a gas volume of 3.0 L. Holding the pressure constant, you increase the volume to 5.0 L. Then, keeping the volume constant at 5.0 L, you increase the pressure to 3.0 atm. Next you decrease the pressure linearly as a function of volume until the volume is 3.0 L and the pressure is 2.0 atm. Finally, you keep the volume constant at 3.0 L and decrease the pressure to 1.0 atm, returning the gas to its initial pressure and volume. The walls of the cylinder are good conductors of heat, and you provide the required heat sources and heat sinks so that the necessary heat flows can occur. At these relatively high pressures, you suspect that the ideal-gas equation will not apply with much accuracy. You don't know what gas is in the cylinder or whether it is monatomic, diatomic, or polyatomic. (a) Plot the cycle in the $pV$-plane. (b) What is the net heat flow for the gas during this cycle? Is there net heat flow into or out of the gas?

Ze-Han Lee

Numerade Educator

Problem 62

The power output of an automobile engine is directly proportional to the mass of air that can be forced into the volume of the engine's cylinders to react chemically with gasoline. Many cars have a $turbocharger$, which compresses the air before it enters the engine, giving a greater mass of air per volume. This rapid, essentially adiabatic compression also heats the air. To compress it further, the air then passes through an $intercooler$ in which the air exchanges heat with its surroundings at essentially constant pressure. The air is then drawn into the cylinders. In a typical installation, air is taken into the turbocharger at atmospheric pressure (1.01 $\times$ 10$^5$ Pa), density $\rho$ = 1.23 kg/m$^3$, and temperature 15.0$^\circ$C. It is compressed adiabatically to 1.45 $\times$ 10$^5$ Pa. In the intercooler, the air is cooled to the original temperature of 15.0$^\circ$C at a constant pressure of 1.45 $\times$ 10$^5$ Pa. (a) Draw a $pV$-diagram for this sequence of processes. (b) If the volume of one of the engine's cylinders is 575 cm$^3$, what mass of air exiting from the intercooler will fill the cylinder at 1.45 $\times$ 10$^5$ Pa? Compared to the power output of an engine that takes in air at 1.01 $\times$ 10$^5$ Pa at 15.0$^\circ$C, what percentage increase in power is obtained by using the turbocharger and intercooler? (c) If the intercooler is not used, what mass of air exiting from the turbocharger will fill the cylinder at 1.45 $\times$ 10$^5$ Pa? Compared to the power output of an engine that takes in air at 1.01 $\times$ 10$^5$ Pa at 15.0$^\circ$C, what percentage increase in power is obtained by using the turbocharger alone?

Katie Mcalpine

Numerade Educator

Problem 63

In a test of the effects of low temperatures on the gas mixture, a cylinder filled at 20.0$^\circ$C to 2000 psi (gauge pressure) is cooled slowly and the pressure is monitored. What is the expected pressure at -5.00$^\circ$C if the gas remains a homogeneous mixture? (a) 500 psi; (b) 1500 psi; (c) 1830 psi; (d) 1920 psi.

Ze-Han Lee

Numerade Educator

Problem 64

In another test, the valve of a 500-L cylinder full of the gas mixture at 2000 psi (gauge pressure) is opened wide so that the gas rushes out of the cylinder very rapidly. Why might some $N_2O$ condense during this process? (a) This is an isochoric process in which the pressure decreases, so the temperature also decreases. (b) Because of the rapid expansion, heat is removed from the system, so the internal energy and temperature of the gas decrease. (c) This is an isobaric process, so as the volume increases, the temperature decreases proportionally. (d) With the rapid expansion, the expanding gas does work with no heat input, so the internal energy and temperature of the gas decrease.

Katie Mcalpine

Numerade Educator

Problem 65

You have a cylinder that contains 500 L of the gas mixture pressurized to 2000 psi (gauge pressure). A regulator sets the gas flow to deliver 8.2 $L$/min at atmospheric pressure. Assume that this flow is slow enough that the expansion is isothermal and the gases remain mixed. How much time will it take to empty the cylinder? (a) 1 h; (b) 33 h; (c) 57 h; (d) 140 h.

Ze-Han Lee

Numerade Educator

Problem 66

In a hospital, pure oxygen may be delivered at 50 psi (gauge pressure) and then mixed with $N_2O$. What volume of oxygen at 20$^\circ$C and 50 psi (gauge pressure) should be mixed with 1.7 kg of $N_2O$ to get a 50%/50% mixture by volume at 20$^\circ$C? (a) 0.21 m$^3$; (b) 0.27 m$^3$; (c) 1.9 m$^3$; (d) 100 m$^3$.

Katie Mcalpine

Numerade Educator