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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 15**

Multiple-Choice Problems

1. To double both the pressure and the volume of a fixed amount of an ideal gas, you would multiply its absolute temperature by

A. 1 (i.e., keep the temperature the same). B. 2. C. 4. D. 16

2. Oxygen molecules are 16 times more massive than hydrogen molecules. If samples of these two gases are at the same temperature, what must be true about the motion of the molecules?

A. The rms molecular speed is the same for both gases.

B. The average kinetic energy is the same for both gases.

C. The rms speed of the hydrogen molecules is 4 times greater than that of the oxygen molecules.

D. The rms speed of the hydrogen molecules is 16 times greater than that of the oxygen molecules.

3. An ideal gas in a cubical box having sides of length L exerts a pressure p on the walls of the box. If all of this gas is put into a box having sides of length 2L without changing its temperature, the pressure it exerts on the walls of the larger box will be

A. 4p B. p/2 C. p/4 D. p/8

4. If you mix different amounts of two ideal gases that are originally at different temperatures, what must be true of the final state after the temperature stabilizes?

A. Both gases will reach the same final temperature.

B. The final rms molecular speed will be the same for both gases.

C. The final average kinetic energy of a molecule will be the same for both gases.

5. If you double the rms speed of the molecules of an ideal gas, which of the following statements is or are true about the gas?

A. Its absolute temperature is doubled.

B. Its Celsius temperature is doubled.

C. Its absolute temperature is quadrupled.

D. Its Celsius temperature is quadrupled.

6. In an ideal gas, which of the following quantities can be determined by measuring just the temperature of the gas?

A. The average kinetic energy of the molecules.

B. The total kinetic energy of the molecules.

C. The pressure of the gas.

7. You add equal amounts of heat to two identical cylinders containing equal amounts of the same ideal gas. Cylinder A is allowed to expand, while cylinder B is not. How do the temperature changes of the two cylinders compare?

A. The two cylinders will experience the same temperature change.

B. Cylinder A will experience a greater temperature change.

C. Cylinder B will experience a greater temperature change.

8. When ice melts at 0 °C its volume decreases. Compared to the amount of heat added, the change in internal energy is

A. greater B. less C. the same

9. The formula for the change in the internal energy of a fixed amount of an ideal gas is valid

A. only for constant-volume processes

B. only for adiabatic processes

C. only for isobaric processes

D. for any process involving that fixed amount of the gas.

10. For the process shown in the pV diagram in Figure 15.29, the total work in going from a to d along the path shown is

A. 15 × 105 J B. 9 × 105 J C. 6 × 105 J D. 1 × 105 J

11. You have two boxes, one containing some hot gas and the other containing some cold gas. This is all you know about these boxes. What can you validly conclude about the characteristics of the gas in the boxes?

A. The molecules in the hot gas are moving faster, on average, than those in the cold gas.

B. The molecules of the hot gas have greater average kinetic energy than those of the cold gas.

C. The molecules of the hot gas have more total kinetic energy than those of the cold gas.

D. The pressure of the hot gas is greater than that of the cold gas.

12. The gas shown in Figure 15.30 is in a completely insulated rigid container. Weight is added to the frictionless piston, compressing the gas. As this is done,

A. the temperature of the gas stays the same because the container is insulated.

B. the temperature of the gas increases because heat is added to the gas.

C. the temperature of the gas increases because work is done on the gas.

D. the pressure of the gas stays the same because the temperature of the gas is constant.

13. Which of the following must be true about an ideal gas that undergoes an isothermal expansion?

A. No heat enters the gas.

B. The pressure of the gas decreases.

C. The internal energy of the gas does not change.

D. The gas does positive work.

14. An ideal gas is initially confined to one side of a perfectly insulated rigid chamber by a movable frictionless piston, as shown in Figure 15.31. (The other side of the chamber is evacuated.) What is true of this gas after the piston is suddenly pulled to the right end of the chamber?

A. The expansion causes the temperature of the gas to decrease.

B. The pressure of the gas decreases.

C. The temperature of the gas does not change.

D. The expansion causes the internal energy of the gas to decrease.

15. Suppose that, in the previous problem, the piston is not pulled but instead is allowed to move slowly to the right as it is hit by the gas molecules. What is true about this gas just as the piston reaches the right end of the chamber?

A. The expansion causes the temperature of the gas to decrease.

B. The pressure of the gas decreases.

C. The temperature of the gas does not change.

D. The expansion causes the internal energy of the gas to decrease.

**Problems**

1. A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 m3 of air at a pressure of 3.40 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 m3. If the temperature remains constant, what is the final value of the pressure?

2. Helium gas with a volume of 2.60 L under a pressure of 1.30 atm and at a temperature of 41.0 °C is warmed until both the pressure and volume of the gas are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is 4.00 g/mol.

3. A 3.00 L tank contains air at 3.00 atm and 20.0 °C. The tank is sealed and cooled until the pressure is 1.00 atm. (a) What is the temperature then in degrees Celsius, assuming that the volume of the tank is constant? (b) If the temperature is kept at the value found in part (a) and the gas is compressed, what is

the volume when the pressure again becomes 3.00 atm?

4. A 20.0 L tank contains 0.225 kg of helium at 18.0°C. The molar mass of helium is 4.00 g/mol (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

5. A room with dimensions 7.00 m by 8.00 m by 2.50 m is filled with pure oxygen at 22.0 °C and 1.00 atm. The molar mass of oxygen is 32.0 g/mol (a) How many moles of oxygen are required? (b) What is the mass of this oxygen, in kilograms?

6. Three moles of an ideal gas are in a rigid cubical box with sides of length 0.200 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0 °C (b) What is the force when the temperature of the gas is increased to 100.0 °C

7. A large cylindrical tank contains 0.750 m3 of nitrogen gas at 27 °C and 1.5 × 105 (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to 0.480 m3 and the temperature is increased to 157 °C?

8. Planetary atmospheres. (a) Calculate the density of the atmosphere at the surface of Mars (where the pressure is 650 Pa and the temperature is typically 253 K, with a CO2 atmosphere), Venus (with an average temperature of 730 K and pressure of 92 atm, with a CO2 atmosphere), and Saturn’s moon Titan (where the pressure is 1.5 atm and the temperature is –178 °C with a N2 atmosphere). (b) Compare each of these densities with that of the earth’s atmosphere, as determined in Example 15.4.

9. The gas inside a balloon will always have a pressure nearly equal to atmospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas) to a volume of 0.600 L at a temperature of 19.0 °C. What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen (77.3 K)?

10. Lung volume. The total lung volume for a typical person is 6.00 L. A person fills her lungs with air at an absolute pressure of 1.00 atm. Then, holding her breath, she compresses her chest cavity, decreasing her lung volume to 5.70 L. What is the pressure of the air in her compressed lungs, assuming that the temperature of the air remains constant?

11. A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm3 of air at atmospheric pressure (1.01×105 Pa) and a temperature of 27.0 °C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm3 and the gauge pressure has increased to 2.72×106 Pa. Compute the final temperature.

12. A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm) to the surface (where the pressure is 1.00 atm). The temperature at the bottom is 4.0 °C and the temperature at the surface is 23.0 °C. (a) What is the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? (b) Would it be safe for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not?

13. At an altitude of 11,000 m (a typical cruising altitude for a jet airliner), the air temperature is –56.5 °C and the air density is 0.364 kg/m3. What is the pressure of the atmosphere at that altitude? The molar mass of air is 28.8 g/mol.

14. If a certain amount of ideal gas occupies a volume V at STP on earth, what would be its volume (in terms of V) on Venus, where the temperature is 1003 °C and the pressure is 92 atm?

15. Calculate the volume of 1.00 mol of liquid water at a temperature of 20 °C (at which its density is 998 kg/m3 and compare this volume with the volume occupied by 1.00 mol of water at the critical point, which is 56 × 10–6 m3. Water has a molar mass of 18.0 g/mol.

16. Solid water (ice) is slowly warmed from a very low temperature. (a) What minimum external pressure p1 must be applied to the solid if a melting phase transition is to be observed? Describe the sequence of phase transitions that occur if the applied pressure p is such that p < p1 (b) Above a certain maximum pressure p2 no boiling transition is observed. What is this pressure? Describe the sequence of phase transitions that occur if p1 < p < p2?

17. The atmosphere of the planet Mars is 95.3% carbon dioxide (CO2) and about 0.03% water vapor. The atmospheric pressure is only about 600 Pa, and the surface temperature varies from -30 °C to -100 °C. The polar ice caps contain both CO2 ice and water ice. Could there be liquid CO2 on the surface of Mars? Could there be liquid water? Why or why not?

18. Find the mass of a single sulfur (S) atom and an ammonia molecule. Use the periodic table in Appendix C to find the molar masses.

19. How many water molecules are there in a 1.00 L bottle of water? The molar mass of water is 18.0 g/mol.

20. In the air we breathe at 72 °F and 1.0 atm pressure, how many molecules does a typical cubic centimeter contain, assuming that the air is all N2?

21. We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at a temperature of 50 °C while the gas in box B is at 10 °C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? (a) The pressure in A is higher than in B. (b) There are more molecules in A than in B. (c) A and B cannot contain the same type of gas. (d) The molecules in A have more average kinetic energy per molecule than those in B. (e) The molecules in A are moving faster than those in B. Explain the reasoning behind your answers.

22. (a) A deuteron, H, is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300 million K. What is the rms speed of the deuterons? Is this a significant fraction of the speed of light (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c?

23. Oxygen (O2) has a molar mass of 32.0g/mol. (a) What is the root-mean-square speed of an oxygen molecule at a temperature of 300 K? (b) What is its average translational kinetic energy at that speed?

24. Suppose some insects have speeds of 10.0 m/s, 8.00 m/s, 7.00 m/s and 2.00 m/s. Find (a) the rms speed of these critters and (b) their average speed.

25. In a gas at standard temperature and pressure, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about 6 billion people at present)?

26. At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at 20.0 °C?

27. Where is the hydrogen? The average temperature of the atmosphere near the surface of the earth is about 20 °C. (a) What is the root-mean-square speed of hydrogen molecules, H2, at this temperature? (b) The escape speed from the earth is about 11 km/s. Is the average molecule moving fast enough to escape? (c) Compare the rms speeds of oxygen (O2) and nitrogen (N2) with that of H2 (d) So why has the hydrogen been able to escape the earth’s gravity, but the heavier gases (such as O2 and N2 have not, even though none of these gases has an rms speed equal to the escape speed of the earth?

28. A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms; (b) their root-mean-square speeds.

29. STP. The conditions of standard temperature and pressure (STP) are a temperature of 0.00 °C and a pressure of 1.00 atm. (a) How many liters does 1.00 mol of any ideal gas occupy at STP? (b) For a scientist on Venus, an absolute pressure of 1 Venusian-atmosphere is 92 Earth-atmospheres. Of course

she would use the Venusian-atmosphere to define STP. Assuming she kept the same temperature, how many liters would 1 mole of ideal gas occupy on Venus?

30. Breathing at high altitudes. If you have ever hiked or climbed to high altitudes in the mountains, you surely have noticed how short of breath you get. This occurs because the air is thinner, so each breath contains fewer O2 molecules than at sea level. At the top of Mt. Everest, the pressure is only 1/3 atm. Air contains 21% O2 and 78% N2 and an average human breath is 0.50 L of air. At the top of Mt. Everest, (a) how many O2 molecules does each breath contain when the temperature is -15 °F and (b) what percent is this of the number of O2 molecules you would get from a breath at sea level at -15 °F?

31. How often do we need to breathe? A resting person requires 14.5 L of O2 per hour to maintain metabolic activities. Such a person breathes in 0.50 L of air at approximately 20 °C with each breath. The inhaled air is 20.9% O2, while the exhaled air is 16.3% O2. (a) How many breaths per minute does a resting person need to take to provide the necessary oxygen? (b) How many O2 molecules does a resting person inhale per breath?

32. (a) How much heat does it take to increase the temperature of 2.50 moles of an ideal monatomic gas from 25.0 °C to 55.0 °C if the gas is held at constant volume? (b) How much heat is needed if the gas is diatomic rather than monatomic? (c) Sketch a pV diagram for these processes.

33. (a) If you supply 1850 J of heat to 2.25 moles of an ideal diatomic gas initially at 10.0 °C in a perfectly rigid container, what will be the final temperature of the gas? (b) Suppose the gas in the container were an ideal monatomic gas instead. How much heat would you need to add to produce the same temperature change? (c) Sketch a pV diagram of these processes.

34. Compute the specific heat capacity (in J/(kg K) at constant volume of nitrogen (N2) gas, and compare it with the specific heat capacity of liquid water. Use Appendix C to determine the molar mass of N2.

35. Perfectly rigid containers each hold n moles of ideal gas, one being hydrogen (H2) and other being neon (Ne). If it takes 100 J of heat to increase the temperature of the hydrogen by 2.50 °C, by how many degrees will the same amount of heat raise the temperature of the neon?

36. Assume that the gases in this problem can be treated as ideal over the temperature ranges involved, and consult Appendix C to determine the necessary molar masses. (a) How much heat is needed to raise the temperature of 75.0 g of N2 from 12.1 °C to 49.5 °C at constant volume? (b) If instead you want to produce the same temperature change in 75.0 g of O2 at constant volume, how much heat do you need?

37. A metal cylinder with rigid walls contains 2.50 mol of oxygen gas. The gas is cooled until the pressure decreases to 30.0% of its original value. You can ignore the thermal contraction of the cylinder. (a) Draw a pV diagram of this process. (b) Calculate the work done by the gas.

38. A gas under a constant pressure of 1.5 × 105 Pa and with an initial volume of 0.0900 m3 is cooled until its volume becomes 0.0600 m3 (a) Draw a pV diagram of this process. (b) Calculate the work done by the gas.

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

40. 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 × 10-2 m3 to 4.50 × 10–2 m3 (a) Calculate the initial and final temperatures of the gas. (b) Calculate the amount of work the gas does in expanding. (c) Calculate the amount of heat added to the gas. (d) Calculate the change in internal energy of the gas.

41. Work done in a cyclic process. In Figure 15.32, consider the closed loop 1→2→3→4→1. This is a cyclic process in which the initial and final states are the same. (a) Find the total work done by the system in this process, and show that it is equal to the area enclosed by the loop. (b) How is the work done during the process in part (a) related to the work done if the loop is traversed in the opposite direction, 1→4→3→2→1. Explain. (c) How much work is done in the cycle 3→4→2→3?

42. Work done by the lungs. The graph in Figure 15.33 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. (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. If the temperature of the air in the lungs remains a reasonable 20 °C, what is the maximum number of moles in this person’s lungs during a breath?

43. In a certain chemical process, a lab technician supplies 254 J of heat to a system. At the same time, 73 J of work are done on the system by its surroundings. What is the increase in the internal energy of the system?

44. A gas in a cylinder expands from a volume of 0.110 m3 to 0.320 m3. Heat flows into the gas just rapidly enough to keep the pressure constant at 1.80 × 105 Pa during the expansion. The total heat added is 1.15 × 105 J (a) Find the work done by the gas. (b) Find the change in internal energy of the gas.

45. A gas in a cylinder is held at a constant pressure of 2.30 × 105 Pa and is cooled and compressed from 1.70 m3 to 1.20 m3. The internal energy of the gas decreases by 1.40 × 105 J (a) Find the work done by the gas. (b) Find the amount of the heat that flowed into or out of the gas, and state the direction (inward or outward) of the flow.

46. Five moles of an ideal monatomic gas with an initial temperature of 127 °C expand and, in the process, absorb 1200 J of heat and do 2100 J of work. What is the final temperature of the gas?

47. When a system is taken from state a to state b in Figure 15.34 along the 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 Ua = 0 and Ud = 8.0 J find the heat absorbed in the processes ad and db.

48. An ideal gas expands while the pressure is kept constant. During this process, does heat ﬂow into the gas or out of the gas? Justify your answer.

49. You are keeping 1.75 moles of an ideal gas in a container surrounded by a large ice-water bath that maintains the temperature of the gas at 0.00°C. (a) How many joules of work would have to be done on this gas to compress its volume from 4.20 L to 1.35 L? (b) How much heat came into (or out of) the

gas during this process? Was it into or out of?

50. Suppose you do 457 J of work on 1.18 moles of ideal He gas in a perfectly insulated container. By how much does the internal energy of this gas change? Does it increase or decrease?

51. A cylinder with a movable piston contains 3.00 mol of N2 gas (assumed to behave like an ideal gas). (a) The N2 is heated at constant volume until 1557 J of heat have been added. Calculate the change in temperature. (b) Suppose the same amount of heat is added to the N2, but this time the gas is allowed to expand while remaining at constant pressure. Calculate the temperature change. (c) In which case, (a) or (b), is the final internal energy of the N2 higher? How do you know? What accounts for the difference between the two cases?

52. Figure 15.35 shows a pV diagram for an ideal gas in which its pressure tripled from a to b when 534 J of heat was put into the gas. (a) How much work was done on or by the gas between a and b? (b)Without doing any calculations, decide whether the temperature of this gas increased, decreased, or remained the same between a and b. Explain your reasoning. (c) By how much did the internal energy of the gas change between a and b? Did it increase or decrease? (d) What is the temperature of the gas at point b in terms of its temperature at a, Ta?

53. Figure 15.36 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?

54. The pV diagram in Figure 15.37 shows a process abc involving 0.450 mole 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 put in during the process to increase the internal energy of the gas by 15,000 J?

55. 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 the path abc in Fig. 15.38. (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 the 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?

56. In the process illustrated by the pV diagram in Figure 15.39, the temperature of the ideal gas remains constant at 85 °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?

57. A cylinder contains 0.250 mol of carbon dioxide gas at a temperature of 27.0 °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 °C. Assume that CO2 the may be treated as an ideal gas. (a) Draw a pV diagram of this process. (b) How much work is done by the gas in the 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?

58. Heating air in the lungs. Human lung capacity varies from about 4 L to 6 L, so we shall use an average of 5.0 L. The air enters at the ambient temperature of the atmosphere and must be heated to internal body temperature at an approximately constant pressure of 1.0 atm in our model. Suppose you are outside on a winter day when the temperature is -10 °F (a) How many moles of air does your lung hold if the 5.0 L is at the internal body temperature of 37 °C? (b) How much heat must your body have supplied to get the 5.0 L of air up to internal body temperature, assuming that the atmosphere is all N2? (c) Suppose instead that you manage to inhale the full 5.0 L of air in one breath and hold it in your lungs without expanding (or contracting) them. How much heat would your body have had to supply in that case to raise the air up to internal body temperature?

59. The graph in Figure 15.40 shows a pV diagram for 1.10 moles of ideal oxygen, O2 (a) Find the temperature at points a, b, c, and d. (b) How many joules of heat enters (or leaves) the oxygen in segment (i) ab, (ii) bc, (iii) cd, (iv) da? (c) In each of the preceding segments, does the heat enter or leave the gas? How do you know?

60. An ideal gas at 4.00 atm and 350 K is permitted to expand adiabatically to 1.50 times its initial volume. Find the final pressure and temperature if the gas is (a) monatomic with Cp/CV = 5/3 (b) diatomic with Cp/CV = 7/5.

61. An experimenter adds 970 J of heat to 1.75 mol of an ideal gas to heat it from 10.0 °C to 25.0 °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 γ for the gas.

62. 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?

63. A player bounces a basketball on the floor, compressing it to 80.0% of its original volume. The air (assume it is essentially N2 gas) inside the ball is originally at a temperature of 20.2 °C and a pressure of 2.00 atm. The ball’s diameter is 23.9 cm. (a) What temperature does the air in the ball reach at its maximum compression? (b) By how much does the internal energy of the air change between the ball’s original state and its maximum compression?

64. In the pV diagram shown in Figure 15.41, 85.0 J of work was done by 0.0650 mole of ideal gas during an adiabatic process. (a) How much heat entered or left this gas from a to b? (b) By how many joules did the internal energy of the gas change? (c) What is the temperature of the gas at b?

65. Modern vacuum pumps make it easy to attain pressures on the order of 10-13 atm in the laboratory. At a pressure of 9.00 × 10-14 atm and an ordinary temperature of 300 K, how many molecules are present in 1.00 cm3 of gas?

66. How many atoms are you? Estimate the number of atoms in the body of a 65 kg physics student. Note that the human body is mostly water, which has molar mass 18.0 g/mol and that each water molecule contains three atoms.

67. The effect of altitude on the lungs. (a) Calculate the change in air pressure you will experience if you climb a 1000 m mountain, assuming that the temperature and air density do not change over this distance and that they were 22 °C and 1.2 kg/m3, respectively, at the bottom of the mountain. (b) If you took a 0.50 L breath at the foot of the mountain and managed to hold it until you reached the top, what would be the volume of this breath when you exhaled it there?

68. (a) Calculate the mass of nitrogen present in a volume of 3000 cm3 if the temperature of the gas is 22.0 °C and the absolute pressure is 2.00 × 10-13 atm, a partial vacuum easily obtained in laboratories. The molar mass of nitrogen (N2) is 28.0 g/mol (b) What is the density (in kg/m3) of the N2?

69. An automobile tire has a volume of 0.0150 m3 on a cold day when the temperature of the air in the tire is 5.0 °C and atmospheric pressure is 1.02 atm. Under these conditions, the gauge pressure is measured to be 1.70 atm. After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to 45.0 °C and the volume to 0.0159 m3. What is the gauge pressure at that time?

70. A cylinder 1.00 m tall with inside diameter 0.120 m is used to hold propane gas (molar mass 44.1 g/mol) for use in a barbecue. It is initially filled with gas until the gauge pressure is 1.30 × 106 Pa and the temperature is 22.0 °C. The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is 2.50 × 106 Pa. Calculate the mass of propane that has been used.

71. The surface of the sun. The surface of the sun has a temperature of about 5800 K and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is 1.67 × 10-31 kg) (b) What would be the mass of an atom that had half the rms speed of hydrogen?

72. Atmosphere of Titan. Titan, the largest satellite of Saturn, has a thick nitrogen atmosphere. At its surface, the pressure is 1.5 Earth-atmospheres and the temperature is 94 K. (a) What is the surface temperature in °C (b) Calculate the surface density in Titan’s atmosphere in molecules per cubic meter. (c) Compare the density of Titan’s surface atmosphere to the density of Earth’s atmosphere at 22 °C. Which body has a denser atmosphere?

73. Helium gas expands slowly to twice its original volume, doing 300 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.

74. A cylinder with a piston contains 0.250 mol of ideal oxygen at a pressure of 2.40 × 105 Pa and a temperature of 355 K. 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. (b) Compute the temperature during the isothermal compression. (c) Compute the maximum pressure. (d) Compute the total work done by the piston on the gas during the series of processes.

75. You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0 °C. Assume that all the gas is N2 of molar mass 28.0 g/mol. (a) Find the mass of a single N2 molecule. (b) How much translational kinetic energy does an average N2 molecule have? (c) How many N2 molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?

76. (a) One-third of a mole of He gas is taken along the path abc shown as the solid line in Figure 15.42. 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 from state a to state c along the horizontal dashed line in Fig. 15.42, how much heat would be transferred into or out of the gas? (c) How does Q in part (b) compare against Q in part (a)? Explain.

77. A bicyclist uses a tire pump whose cylinder is initially full of air at an absolute pressure of 1.01 × 105 Pa. The length of stroke of the pump (the length of the cylinder) is 36.0 cm. At what part of the stroke (i.e., what length of the air column) does air begin to enter a tire in which the gauge pressure is 2.76 × 105 Pa. Assume that the temperature remains constant during the compression.

78. The bends. If deep-sea divers rise to the surface too quickly, nitrogen bubbles in their blood can expand and prove fatal. This phenomenon is known as the bends. If a scuba diver rises quickly from a depth of 25 m in Lake Michigan (which is fresh water), what will be the volume at the surface of N2 an bubble that occupied 1.00 mm3 in his blood at the lower depth? Does it seem that this difference is large enough to be a problem?

79. Figure 15.43 shows a pV diagram for 0.0040 mole of ideal H2 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 of these segments.

80. The graph in Figure 15.44 shows a pV diagram for 3.25 moles 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.

81. A flask with a volume of 1.50 L, provided with a stopcock, contains ethane gas (C2H6) at 300 K and atmospheric pressure (1.013 × 105 Pa). The molar mass of ethane is 30.0 g/mol. The system is warmed to a temperature of 380 K, with the stopcock open to the atmosphere. The stopcock is then closed, and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?

82. Diesel ignition. Conventional engines ignite their fuel by using the spark from a spark plug. But in a diesel engine, the air enters the chamber at the temperature of the atmosphere and is compressed by the piston until it reaches 550 °C, at which time the fuel is injected into the chamber and ignited by the hot air. There is no spark plug and no heat is put into the air. (One of the drawbacks of diesel engines is that they are hard to start in cold weather, as we shall shortly see.) Suppose that a certain chamber has a maximum volume of 0.50 L and uses 0.050 mole of air. We can model the air as all ideal N2 and use the appropriate values from Table 15.4. (a) If the air temperature is 20 °C, what is the volume of the air (which started at 0.50 L) when it has been compressed enough so that its temperature has risen to 550 °C? (b) What is the change in the internal energy of the air during this compression? (c) How much work did the piston do on this gas while compressing it? (d) Suppose it is a cold winter morning, with the air temperature 10 °F. If the piston compressed the air by the same amount as before, what will be the highest temperature the gas will reach in this case? (e) Do you now see why a diesel engine is hard to start in cold weather? Can you suggest any reasonable technological solutions to help start a diesel engine on a cold day?

83. Initially at a temperature of 80.0 °C, 0.28 m3 of air expands at a constant gauge pressure of 1.38 × 105 Pa to a volume of 1.42 m3 and then expands further adiabatically to a final volume of 2.27 m3 and a final gauge pressure of 2.29 × 105 Pa. Draw a pV diagram for this sequence of processes, and compute the total work done by the air. Cv for air is 20.8 J/(mol K).

84. In a cylinder, 4.00 mol of helium initially at 1.00 × 106 Pa and 300 K expands until its volume doubles. Compute the work done by the gas if the expansion is (a) isobaric, (b) adiabatic. (c) Show each process on a pV diagram. In which case is the magnitude of the work done by the gas the greatest? (d) In which case is the magnitude of the heat transfer greatest? (e) In which case is the magnitude of the change in internal energy greatest?

85. Starting with 2.50 mol of N2 gas (assumed to be ideal) in a cylinder at 1.00 atm and 20.0 °C, a chemist first heats the gas at constant volume, adding 1.52 × 104 J of heat, then continues heating and allows the gas to expand at constant pressure to twice its original volume. (a) Calculate the final temperature of the gas. (b) Calculate the amount of work done by the gas. (c) Calculate the amount of heat added to the gas while it was expanding. (d) Calculate the change in internal energy of the gas for the whole process.

86. A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500.0 m3 and the surrounding air is at 15.0 °C, what must the temperature of the air in the balloon be for it to lift a total load of 290 kg (in addition to the mass of the hot air)? The density of air at 15.0 °C and atmospheric pressure is 1.23 kg/m3.

**Chapter 16**

Multiple-Choice Problems

1. An insulated box has a barrier that confines a gas to only one side of the box. The barrier springs a leak, allowing the gas to low and occupy both sides of the box. Which statement best describes the entropy of this system?

A. The entropy is greater in the first state, with all the gas on one side of the box.

B. The entropy is greater in the second state, with the gas on both sides of the box.

C. The entropy is the same in both states, since no heat was added to the gas and its temperature did not change.

2. Suppose you put a hot object in contact with a cold object and observe (with some wonder) that heat flows from the cold object to the hot object, making the cold one colder and the hot one hotter. This process would violate

A. only the first law of thermodynamics.

B. only the second law of thermodynamics.

C. both the first and the second laws of thermodynamics.

3. Carnot engine A operates between temperatures of 500 °C and 300 °C. Carnot engine B operates between temperatures of 900 °C and 700 °C. How do the efficiencies of the two engines compare?

A. Engine A has higher efficiency.

B. Engine B has higher efficiency.

C. The efficiencies of the two engines are the same.

4. The thermal efficiency formula e = W/QH is valid for which of the following heat engines?

A. Carnot engine. B. Otto engine. C. Diesel engine. D. Any other type of heat engine.

5. The thermal efficiency formula e = 1 – Tc/TH is valid for which of the following heat engines?

A. Carnot engine. B. Otto engine. C. Diesel engine. D. Any other type of heat engine.

6. A refrigerator consumes a certain amount of electrical energy in order to operate. In terms of the thermodynamic cycle of this refrigerator, this electrical energy is equal to

A. the work done on the gas.

B. the heat extracted from the food in the refrigerator.

C. the heat expelled into the room.

D. the sum of all three quantities in choices A, B, and C.

7. Which of the following statements is a consequence of the second law of thermodynamics?

A. The efficiency of a heat engine can never be greater than unity.

B. Heat can never flow from a cold object to a hot object unless work is done.

C. Energy is conserved in all thermodynamics processes.

D. Even a perfect heat engine cannot convert all of the heat put into it to work.

8. You want to increase the efficiency of a Carnot heat engine by changing the temperature of either its hot or its cold reservoir by the same number of degrees ΔT. Which of the following choices will produce the greater increase in its thermodynamic efficiency?

A. Increase the temperature of the hot reservoir by ΔT

B. Decrease the temperature of the cold reservoir by ΔT

C. It would not matter. Both choices would produce the same increase in efficiency, since ΔT is the same for both.

9. Two Carnot heat engines operate between the same hot and cold reservoirs, but one has a greater compression ratio than the other. Which statement is true about their thermal efficiencies?

A. The one with the greater compression ratio will have the greater efficiency.

B. The one with the greater compression ratio will have the lesser efficiency.

C. Both will have the same efficiency.

10. You put 250 g of water into the freezer and make ice cubes. Which of the following statements is true about this process?

A. The entropy of the water increases.

B. The entropy of the room increases.

C. The amount of heat removed from the water is equal to the amount of heat ejected into the room.

D. More heat is ejected into the room than was removed from the water.

E. The work done by the refrigerator is equal to the heat removed from the water.

11. If you mix cold milk with hot coffee in an insulated Styrofoam™ cup, which of the following things happen?

A. The entropy of the milk increases.

B. The entropy of the coffee decreases by the same amount that the entropy of the milk increased.

C. The net entropy of the coffee–milk mixture does not change, because no heat was added to this system.

D. The entropy of the coffee–milk mixture increases.

12. Suppose a Carnot refrigerator is used to cool the interior of a food storage locker from room temperature to 5 °C. As the temperature of the locker drops, the performance coefficient of the refrigerator

A. increases. B. decreases. C. is constant.

13. A glass of water left outside on a cold night freezes into solid ice, and its entropy decreases. Which statement is true of this process:

A. The entropy of the surrounding air does not change.

B. The entropy of the water changes more than the entropy of the surrounding air.

C. The entropy of the surrounding air changes more than the entropy of the water.

D. The entropy of the water and the entropy of the surrounding air change by equal amounts.

14. If we run an ideal Carnot heat engine in reverse, which of the following statements about it must be true?

A. Heat enters the gas at the cold reservoir and goes out of the gas at the hot reservoir.

B. The amount of heat transferred at the hot reservoir is equal to the amount of heat transferred at the cold reservoir.

C. It is able to perform a net amount of useful work, such as pumping water from a well, during each cycle.

D. It can transfer heat from a cold object to a hot object.

15. Which of the following processes would be a violation of the second law of thermodynamics?

A. All the kinetic energy of an object is transformed into heat.

B. All the heat put into the operating gas of a heat engine during one cycle is transformed into work.

C. A refrigerator removes 100 cal of heat from milk while using only 75 cal of electrical energy to operate.

D. A heat engine does 25 J of work while expelling only 10 J of heat to the cold reservoir.

**Problems**

1. A coal-fired power plant that operates at an efficiency of 38% generates 750 MW of electric power. How much heat does the plant discharge to the environment in one day?

2. Each cycle, a certain heat engine expels 250 J of heat when you put in 325 J of heat. Find the efficiency of this engine and the amount of work you get out of the 325 J heat input.

3. A diesel engine performs 2200 J of mechanical work and discards 4300 J of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

4. An aircraft engine takes in 9000 J of heat and discards 6400 J each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

5. A gasoline engine. A gasoline engine takes in of heat and delivers 3700 J of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of 4.60 ×104 J (a) What is the thermal efficiency of the engine? (b) How much heat is discarded in each cycle? (c) What mass of fuel is burned in each cycle? (d) If the engine goes through 60.0 cycles per second, what is its power output in kilowatts? In horsepower?

6. A gasoline engine has a power output of 180 kW (about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

7. A certain nuclear power plant has a mechanical power output (used to drive an electric generator) of 330 MW. Its rate of heat input from the nuclear reactor is 1300 MW. (a) What is the thermal efficiency of the system? (b) At what rate is heat discarded by the system?

8. Figure 16.15 shows a pV diagram for a heat engine that uses 1.40 moles of an ideal diatomic gas. (a) How much heat goes into this gas per cycle, and where in the cycle does it occur? (b) How much heat is ejected by the gas per cycle, and where does it occur? (c) How much work does this engine do each cycle? (d) What is the thermal efficiency of the engine?

9. The pV diagram in Figure 16.16 shows a cycle of a heat engine that uses 0.250 mole of an ideal gas having The curved part ab of the cycle is adiabatic. (a) Find the pressure of the gas at point a. (b) How much heat enters this gas per cycle, and where does it happen? (c) How much heat leaves this gas in a cycle, and where does it occur? (d) How much work does this engine do in a cycle? (e) What is the thermal efficiency of the engine?

10. An Otto engine uses a gas having γ = 1.40 and has a compression ratio of 8.50. (a) Out of every gallon of fuel this engine burns, what fraction is wasted (i.e., produces energy that is wasted)? (b) If you could change the compression ratio to 10.0 instead, what percentage of the fuel would be wasted?

11. What compression ratio r must an Otto cycle have to achieve an ideal efficiency of 65.0% if the gas used in the chamber has γ = 1.40.

12. For an Otto engine with a compression ratio of 7.50, you have your choice of using an ideal monatomic or ideal diatomic gas. Which one would give you greater efficiency? Calculate the efficiency in both cases to find out.

13. (a) Calculate the theoretical efficiency for an Otto cycle engine with γ = 1.40 and r = 9.50 (b) If this engine takes in 10,000 J of heat from burning its fuel, how much heat does it discard to the outside air?

14. In one cycle, a freezer uses 785 J of electrical energy in order to remove 1750 J of heat from its freezer compartment at 10 °F. (a) What is the coefficient of performance of this freezer? (b) How much heat does it expel into the room during this cycle?

15. A refrigerator has a coefficient of performance of 2.10. Each cycle, it absorbs 3.40 × 104 J of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

16. A window air-conditioner unit absorbs 9.80 × 104 J of heat per minute from the room being cooled and in the same period deposits 1.44 × 105 J of heat into the outside air. What is the power consumption of the unit in watts?

17. A freezer has a coefficient of performance of 2.40. The freezer is to convert 1.80 kg of water at 25.0 °C to 1.80 kg of ice at -5.0 °C in 1 hour. (a) What amount of heat must be removed from the water at 25.0 °C to convert it to ice at -5.0 °C (b) How much electrical energy is consumed by the freezer during this hour? c) How much wasted heat is rejected to the room in which the freezer sits?

18. A cooling unit for chilling the water of an aquarium gives specifications of 1/10 hp and 1270 Btu/h. Assuming the unit produces its 1/10 hp at 70.0% efficiency, calculate its performance coefficient.

19. A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 J to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir? (c) What is the thermal efficiency of the cycle?

20. A heat engine is to be built to extract energy from the temperature gradient in the ocean. If the surface and deepwater temperatures are 25 °C and 8 °C, respectively, what is the maximum efficiency such an engine can have?

21. A Carnot engine is operated between two heat reservoirs at temperatures of 520 K and 300 K. (a) If the engine receives 6.45 kJ of heat energy from the reservoir at 520 K in each cycle, how many joules per cycle does it reject to the reservoir at 300 K? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

22. A Carnot engine has an efficiency of 59% and performs 2.5 × 104 J of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature (20.0 °C). What is the temperature of its heat source?

23. An ice-making machine operates in a Carnot cycle. It takes heat from water at 0.0 °C and rejects heat to a room at 24.0 °C. Suppose that 85.0 kg of water at 0.0 °C are converted to ice at 0.0 °C. (a) How much heat is rejected to the room? (b) How much energy must be supplied to the device?

24. A Carnot freezer that runs on electricity removes heat from the freezer compartment, which is at and expels it into the room at You put an ice-cube tray containing 375 g of water at 18°C into the freezer. (a) What is the coefficient of performance of this freezer? (b) How much energy is needed to freeze this water? (c) How much electrical energy must be supplied to the freezer to freeze the water? (d) How much heat does the freezer expel into the room while freezing the ice?

25. The pV diagram in Figure 16.17 shows a general Carnot cycle for an engine, with the hot and cold thermal reservoir segments identified. Segments ab and cd are isothermal, while the other two are adiabatic. (a) If this engine is used as a heat engine, what is the direction of the cycle, clockwise or counterclockwise? In which segments does heat enter the gas, and in which ones does it leave the gas? (b) If the engine is used as a refrigerator, what is the direction of the cycle? In which segments does heat enter the gas, and in which ones does it leave the gas? Also, which segments take place at the inside of the refrigerator, and which ones occur in the air of the room in which the refrigerator is operating? (c) If the engine is used as a heat pump–air conditioner, what is the direction of the cycle,

and in which segments does heat enter the gas and in which ones does it leave the gas? Which segments take place inside of the house and which ones occur outside? (d) If the engine is used as a heat pump–house heater, what is the direction of the cycle? In which segments does heat enter the gas and in which ones does it leave the gas? Which segments take place inside of the house and which ones occur outside?

26. A sophomore with nothing better to do adds heat to 0.350 kg of ice at 0.00 °C until it is all melted. (a) What is the change in entropy of the water? (b) The source of the heat is a very massive body at a temperature of 25.0 °C. What is the change in entropy of this body? (c) What is the total change in entropy of the water and the heat source?

27. A 4.50 kg block of ice at 0.00 °C falls into the ocean and melts. The average temperature of the ocean is 3.50 °C, including all the deep water. By how much does the melting of this ice change the entropy of the world? Does it make it larger or smaller?

28. A large factory furnace maintained at 175 °C at its outer surface is wrapped in an insulating blanket of thermal conductivity 0.055 W/(m K) which is thick enough that the outer surface of the insulation is at 42 °C while heat escapes from the furnace at a steady rate of 125 W for each square meter of surface area. By how much does each square meter of the furnace change the entropy of the factory every second?

29. You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with 270 kg of water and attempt to warm it further by pouring in 5.00 kg of boiling water from the stove. (a) Is this a reversible or an irreversible process? Use physical reasoning to explain. (b) Calculate the final temperature of the bath water. (c) Calculate the net change in entropy of the system (bath water + boiling water), assuming no heat exchange with the air or the tub itself.

30. If 25.0 g of the metal gallium melts in your hand (see Fig. 14.14), what is the change in entropy of the gallium in this process? What about the change in entropy of your hand? Is it positive or negative? Is its magnitude greater or less than that of the change in entropy of the gallium? The melting temperature of gallium is 29.8 °C and its heat of fusion is 8.04 × 104 J/kg.

31. Three moles of an ideal gas undergo a reversible isothermal compression at 20.0°C. During this compression, 1850 J of work is done on the gas. What is the change in entropy of the gas?

32. Entropy change due to driving. Premium gasoline produces 1.23 × 108 J of heat per gallon when it is burned at a temperature of approximately 400°C (although the amount can vary with the fuel mixture). If the car’s engine is 25% efficient, three-fourths of that heat is expelled into the air, typically at 20 °C. If your car gets 35 miles per gallon of gas, by how much does the car’s engine change the entropy of the world when you drive 1.0 mile? Does it decrease or increase it?

33. Entropy of metabolism. An average sleeping person metabolizes at a rate of about 80 W by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all of this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37 °C, and the skin is typically 7 C° cooler. By how much does the person’s entropy change per second due to this heat transfer?

34. Entropy change from digesting fat. Digesting fat produces 9.3 food calories per gram of fat, and typically 80% of this energy goes to heat when metabolized. The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally 37 °C, and the surface is usually about 30 °C. By how much does the digestion and metabolism of a 2.50 g pat of butter change your body’s entropy? Does it increase or decrease?

35. Solar collectors. A well-insulated house of moderate size in a temperate climate requires an average heat input rate of 20.0 kW. If this heat is to be supplied by a solar collector with an average (night and day) energy input of 300 W/m2 and a collection efficiency of 60.0%, what area of solar collector is required?

36. Solar power. A solar power plant is to be built with an average power output capacity of 2500 MW in a location where the average power from the sun’s radiation is 200 W/m2 at the earth’s surface. What land area (in km2 and mi2) must the solar collectors occupy if they are (a) photocells with 42% efﬁciency, (b) mirrors that generate steam for a turbine-generator unit with an overall efficiency of 21%?

37. An experimental power plant at the Natural Energy Laboratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water temperatures are 27 °C and 6 °C, respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 kW of power, at what rate must heat be extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of 10 °C. What must be the flow rate of cold water through the system? Give your answer in kg/h and L/h.

38. Solar water heater. A solar water heater for domestic hot water supply uses solar collecting panels with a collection efficiency of 50% in a location where the average solar-energy input is 200 W/m2. If the water comes into the house at 15.0 °C and is to be heated to 60.0 °C, what volume of water can be heated per hour if the area of the collector is 30.0 m2.

39. You are designing a Carnot engine that has 2 mol of CO2 as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of 527 °C and a maximum pressure of 5.00 atm. With a heat input of 400 J per cycle, you want 300 J of useful work. (a) Find the temperature of the cold reservoir. (b) For how many cycles must this engine run to melt completely a 10.0-kg block of ice originally at 0.0 °C, using only the heat rejected by the engine?

40. A heat engine takes 0.350 mol of an ideal diatomic gas around the cycle shown in the pV diagram of Figure 16.18. Process 1 → 2 is at constant volume, process 2 → 3 is adiabatic, and process is 3 → 1 at a constant pressure of 1.00 atm. The value of γ for this gas is 1.40. (a) Find the pressure and volume at points 1, 2, and 3. (b) Calculate Q, W, and ΔU for each of the three processes. (c) Find the net work done by the gas in the cycle. (d) Find the net heat flow into the engine in one cycle. (e) What is the thermal efficiency of the engine? How does this efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximum temperatures T1 and T2.

41. As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 moles of He gas as its working substance and that operates from a high-temperature reservoir at 500 °C. The engine is to lift a 15.0 kg weight 2.00 m per cycle, using 500 J of heat input. The gas in the engine chamber can have a minimum volume of 5.00 L during the cycle. (a) Draw a pV diagram for this cycle. In your diagram, show where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

42. The Kwik-Freez Appliance Co. wants you to design a food freezer that will keep the freezing compartment at -5 °C and will operate in a room at 20.0 °C. The freezer is to make 5.00 kg of ice at 0.0 °C, starting with water at 20.0 °C. Find the least possible amount of electrical energy needed to make this ice and the smallest possible amount of heat expelled into the room.

43. A Carnot engine operates between two heat reservoirs at temperatures TH and TC. An inventor proposes to increase the efficiency by running one engine between TH and an intermediate temperature T′ and a second engine between T′ and TC using as input the heat expelled by the first engine. Compute

the efficiency of this composite system, and compare it to that of the original engine.

44. A cylinder contains oxygen gas (O2) at a pressure of 2.00 atm. The volume is 4.00 L, and the temperature is 300 K. Assume that the oxygen may be treated as an ideal gas. The oxygen is carried through the following processes:

(i) Heated at constant pressure from the initial state (state 1) to state 2, which has T = 450 K

(ii) Cooled at constant volume to 250 K (state 3).

(iii) Compressed at constant temperature to a volume of 4.00 L (state 4).

(iv) Heated at constant volume to 300 K, which takes the system back to state 1.

(a) Show these four processes in a pV diagram, giving the numerical values of p and V in each of the four states. (b) Calculate Q and W for each of the four processes. (c) Calculate the net work done by the oxygen. (d) What is the efficiency of this device as a heat engine? How does this efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximum temperatures of 250 K and 450 K?

45. Human entropy. A person having skin of surface area and temperature 30.0 °C is resting in an insulated room where the ambient air temperature is 20.0 °C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second?

46. A typical coal-fired power plant generates 1000 MW of usable power at an overall thermal efficiency of 40%. (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of 2.65 × 107 J/kg. How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river’s temperature is 18.0 °C before it reaches the power plant and after 18.5 °C it has received the plant’s waste heat. Calculate the river’s flow rate, in cubic meters per second. (e) By how much does the river’s entropy increase each second?

47. A human engine. You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is 37.0 °C) and the other end at the surface of your skin, at 30.0 °C. (a) What is the maximum efficiency of such a heat engine? Would it be a very useful engine? (b) Suppose you want to use this human engine to lift a 2.50 kg box from the floor to a tabletop 1.20 m above the floor. How much must you increase the gravitational potential energy and how much heat input is needed to accomplish this? (c) How many 350 calorie (those are food calories, remember) candy bars must you eat to lift the box in this way? Recall that 80% of the food energy goes into heat.

48. What is the thermal efficiency of an engine that operates by taking n moles of diatomic ideal gas through the cycle 1 → 2 → 3 → 4 → 1 shown in Fig. 16.19?

49. One end of a copper rod is immersed in boiling water at 100 °C and the other end in an ice–water mixture at 0 °C. The sides of the rod are insulated. After steady-state conditions have been achieved in the rod, 0.160 kg of ice melts in a certain time interval. For this time interval, find (a) the entropy change of the boiling water, (b) the entropy change of the ice–water mixture, (c) the entropy change of the copper rod, (d) the total entropy change of the entire system.

50. The pV diagram in Figure 16.20 shows a heat engine operating on 0.850 mol of H2 (See Table 15.4.) Segment ca is isothermal. (a) Without doing any calculations, identify the segments during which heat enters the gas and those during which it leaves the gas. Explain your reasoning. (b) Find the thermal efficiency of this heat engine, treating the gas as ideal.

51. Calculate the coefficient of performance of the engine in the previous problem if it is run in reverse, as a refrigerator.

**Chapter 17**

Multiple-Choice Problems

1. Just after two identical point charges are released when they are a distance D apart in outer space, they have an acceleration a. If you release them from a distance D/2 instead, their acceleration will be

A. a/4 B. a/2 C. 2a D. 4a

2. If the electric field is E at a distance d from a point charge, its magnitude will be 2E at a distance

A. d/4 B. d/2 C. d/√2 D. d√2 E. 2d

3. Two unequal point charges are separated as shown in Figure 17.36. The electric field due to this combination of charges can be zero

A. only in region 1.

B. only in region 2.

C. only in region 3.

D. in both regions 1 and 3.

4. Two protons close to each other are released from rest and are completely free to move. After being released,

A. their speeds gradually decrease to zero as they move apart.

B. their speeds gradually increase as they move apart.

C. their accelerations gradually decrease to zero as they move apart.

D. their accelerations gradually increase as they move apart.

5. A spherical balloon contains a charge +Q uniformly distributed over its surface. When it has a diameter D, the electric field at its surface has magnitude E. If the balloon is now blown up to twice this diameter without changing the charge, the electric field at its surface is

A. 4E B. 2E C. E D. E/2 E. E/4

6. Two microscopic bags each contain two protons. When they are separated by a distance d, the electrical force on each bag due to the other bag is F. You now transfer a proton from one bag to another without changing anything else. The electrical force on each bag is now

A. F B. (3/4)F C. (1/2)F D. (1/4)F

7. An electron is moving horizontally in a laboratory when a uniform electric field is suddenly turned on. This field points vertically downward. Which of the paths shown will the electron follow, assuming that gravity can be neglected?

8. Point P in Figure 17.37 is equidistant from two point charges ±Q of equal magnitude. If a negative point charge is placed at P without moving the original charges, the net electrical force the charges ±Q will exert on it is

A. directly upward.

B. directly downward.

C. zero.

D. directly to the right.

E. directly to the left.

9. A charge +Q is suspended by a silk thread inside of a neutral metal box without touching the metal. What is true about the charge on the inner and outer surfaces of the box?

A. The charge on both the inner and the outer surfaces is zero.

B. The charge is -Q on the inner surface and +Q on the outer surface.

C. The charge is +Q on the inner surface and -Q on the outer surface.

D. The charge on both the inner and the outer surfaces is +Q

10. A charge Q and a charge 3Q are released in a uniform electric field. If the force this field exerts on 3Q is F, the force it will exert on Q is

A. F B. F/3 C. F/9

11. Three equal point charges are held in place as shown in Figure 17.38. If F1 is the force on q due to Q1 and F2 is the force on q due to Q2 how do F1 and F2 compare?

A. F1 = 2F2 B. F1 = 3F2 C. F1 = 4F2 D. F1 = 9F2

12. An electric field of magnitude E is measured at a distance R from a point charge Q. If the charge is doubled to 2Q and the electric field is now measured at a distance of 2R from the charge, the new measured value of the field will be:

A. 2E B. E C. E/2 D. E/4

13. A very small ball containing a charge –Q hangs from a light string between two vertical charged plates, as shown in Figure 17.39. When released from rest, the ball will

A. swing to the right.

B. swing to the left.

C. remain hanging vertically.

14. A point charge Q at the center of a sphere of radius R produces an electric flux of ΦE coming out of the sphere. If the charge remains the same but the radius of the sphere is doubled, the electric flux coming out of it will be:

A. ΦE/2 B. ΦE/4 C. 2ΦE D. 4ΦE E. ΦE

15. Two charged small spheres are a distance R apart and exert an electrostatic force F on each other. If the distance is halved to R/2 the force exerted on each sphere will be

A. 4F B. 2F C. F/2 D. F/4

**Problems**

1. A positively charged glass rod is brought close to a neutral sphere that is supported on a nonconducting plastic stand as shown in Figure 17.40. Sketch the distribution of charges on the sphere if it is made of (a) aluminum, (b) nonconducting plastic.

2. A positively charged rubber rod is moved close to a neutral copper ball that is resting on a nonconducting sheet of plastic. (a) Sketch the distribution of charges on the ball. (b) With the rod still close to the ball, a metal wire is briefly connected from the ball to the earth and then removed. After the rubber rod is also removed, sketch the distribution of charges (if any) on the copper ball.

3. Two iron spheres contain excess charge, one positive and the other negative. (a) Show how the charges are arranged on these spheres if they are very far from each other. (b) If the spheres are now brought close to each other, but do not touch, sketch how the charges will be distributed on their surfaces. (c) In part (b), show how the charges would be distributed if both spheres were negative.

4. Electrical storms. During an electrical storm, clouds can build up very large amounts of charge, and this charge can induce charges on the earth’s surface. Sketch the distribution of charges at the earth’s surface in the vicinity of a cloud if the cloud is positively charged and the earth behaves like a conductor.

5. In ordinary laboratory circuits, charges in the μC and nC range are common. How many excess electrons must you add to an object to give it a charge of (a) -2.50 μC (b) -2.50 nC

6. Signal propagation in neurons. Neurons are components of the nervous system of the body that transmit signals as electrical impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neutron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately 5.6 × 1011 Na+ (sodium ions) per meter, each with charge +e, enter the axon. How many coulombs of charge enter a 1.5 cm length of the axon during this process?

7. Particles in a gold ring. You have a pure (24-karat) gold ring with mass 17.7 g. Gold has an atomic mass of 197 g mol and an atomic number of 79. (a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

8. Two equal point charges of 3.00 × 10-6 C are placed 0.200 m apart. What are the magnitude and direction of the force each charge exerts on the other?

9. At what distance would the repulsive force between two electrons have a magnitude of 2.00 N? Between two protons?

10. A negative charge of -0.500 μC exerts an upward 0.200 N force on an unknown charge 0.300 m directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the -0.500 μC charge?

11. Forces in an atom. The particles in the nucleus of an atom are approximately 10-15 m apart, while the electrons in an atom are about 10-10 m from the nucleus. (a) Calculate the electrical repulsion between two protons in a nucleus if they are 1.00 × 10-15 m apart. If you were holding these protons, do you think you could feel the effect of this force? How many pounds would the force be? (b) Calculate the electrical attraction that a proton in a nucleus exerts on an orbiting electron if the two particles are 1.00 × 10-10 m apart. If you were holding the electron, do you think you could feel the effect of this force?

12. (a) What is the total negative charge, in coulombs, of all the electrons in a small 1.00 g sphere of carbon? One mole of C is 12.0 g, and each atom contains 6 protons and 6 electrons. (b) Suppose you could take out all the electrons and hold them in one hand, while in the other hand you hold what is left of the original sphere. If you hold your hands 1.50 m apart at arms length, what force will each of them feel? Will it be attractive or repulsive?

13. As you walk across a synthetic-fiber rug on a cold, dry winter day, you pick up an excess charge of -55 μC (a) How many excess electrons did you pick up? (b) What is the charge on the rug as a result of your walking across it?

14. Two small plastic spheres are given positive electrical charges. When they are 15.0 cm apart, the repulsive force between them has magnitude 0.220 N. What is the charge on each sphere (a) if the two charges are equal? (b) if one sphere has four times the charge of the other?

15. Two small aluminum spheres, each having mass 0.0250 kg, are separated by 80.0 cm. (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 g/mol, and its atomic number is 13.) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude 1.00 × 104 N (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

16. Two small spheres spaced 20.0 cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 4.57 × 10-21 N?

17. An average human weighs about 650 N. If two such generic humans each carried 1.0 coulomb of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction between them to equal their 650-N weight?

18. If a proton and an electron are released when they are 2.0 × 10-10 m apart (typical atomic distances), find the initial acceleration of each of them.

19. Three point charges are arranged on a line. Charge q3 = + 5.00 nC and is at the origin. Charge q2 = -3.00 nC and is at x = +4.00 cm. Charge q1 is at x = + 2.00 cm. What is q1 (magnitude and sign) if the net force on q3 is zero?

20. If two electrons are each 1.5 × 10-10 m from a proton, as shown in Figure 17.41, find the magnitude and direction of the net electrical force they will exert on the proton.

21. Two point charges are located on the y axis as follows: charge q1 = -1.50 nC at y = -0.600 m and charge q2 = +3.20 nC at the origin (y = 0). What is the net force (magnitude and direction) exerted by these two charges on a third charge q3 = +5.00 nC located at y = -0.400 m?

22. Two point charges are placed on the x axis as follows: Charge q1 = 4.00 nC is located at x = 0.200 m and charge q2 = 5.00 nC is at x = -3.00 m. What are the magnitude and direction of the net force exerted by these two charges on a negative point charge q3 = -0.600 nC placed at the origin?

23. Three charges are at the corners of an isosceles triangle as shown in Figure 17.42. The ± 5.00 μC charges form a dipole. (a) Find the magnitude and direction of the net force that the -10.0 μC charge exerts on the dipole. (b) For an axis perpendicular to the line connecting the two charges of the dipole at its midpoint and perpendicular to the plane of the paper, find the magnitude and direction of the torque exerted on the dipole by the -10.0 μC charge.

24. Base pairing in DNA, I. The two sides of the DNA double helix are connected by pairs of bases (adenine, thymine, cytosine, and guanine). Because of the geometric shape of these molecules, adenine bonds with thymine and cytosine bonds with guanine. Figure 17.43 shows the thymine–adenine bond. Each charge shown is ±e, and the H – N distance is 0.110 nm. (a) Calculate the net force that thymine exerts on adenine. Is it attractive or repulsive? To keep the calculations fairly simple, yet reasonable, consider only the forces due to the O – H – N and the N – H – N combinations, assuming that these two combinations are parallel to each other. Remember, however, that in the O – H – N set, the O- exerts a force on both the H+ and the N- and likewise along the O – H – N set. (b) Calculate the force on the electron in the hydrogen atom, which is 0.0529 nm from the proton. Then compare the strength of the bonding force of the electron in hydrogen with the bonding force of the adenine–thymine molecules.

25. Base pairing in DNA, II. Refer to the previous problem. Figure 17.44 shows the bonding of the cytosine and guanine molecules. The O – H and H – N distances are each 0.110 nm. In this case, assume that the bonding is due only to the forces along the O – H – O, N – H – N and O – H – N combinations, and assume also that these three combinations are parallel to each other. Calculate the net force that cytosine exerts on guanine due to the preceding three combinations. Is this force attractive or repulsive?

26. Surface tension. Surface tension is the force that causes the surface of water (and other liquids) to form a “skin” that resists penetration. Because of this force, water forms into beads, and insects such as water spiders can walk on water. As we shall see, the force is electrical in nature. The surface of a polar liquid, such as water, can be viewed as a series of dipoles strung together in the stable arrangement in which the dipole moment vectors are parallel to the surface, all pointing in the same direction. Suppose now that something presses inward on the surface, distorting the dipoles as shown in Figure 17.45. Show that the two slanted dipoles exert a net upward force on the dipole between them and hence oppose the downward external force. Show also that the dipoles attract each other and thus resist being separated. Notice that the force between dipoles opposes penetration of the liquid’s surface and is a simple model for surface tension.

27. If the central charge shown in Figure 17.46 is displaced 0.350 nm to the right while the other charges are held in place, find the magnitude and direction of the net force that the other two charges exert on it.

28. Two unequal charges repel each other with a force F. If both charges are doubled in magnitude, what will be the new force in terms of F?

29. In an experiment in space, one proton is held fixed and another proton is released from rest a distance of 2.50 mm away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numbers!) acceleration–time and velocity–time graphs of the released proton’s motion.

30. A charge +Q is located at the origin and a second charge, +4Q, is at distance d on the x-axis. Where should a third charge, q, be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

31. A small object carrying a charge of -8.00 nC is acted upon by a downward force of 20.0 nN when placed at a certain point in an electric field. (a) What are the magnitude and direction of the electric field at the point in question? (b) What would be the magnitude and direction of the force acting on a proton placed at this same point in the electric field?

32. (a) What must the charge (sign and magnitude) of a 1.45 g particle be for it to remain balanced against gravity when placed in a downward-directed electric field of magnitude 650 N/C? (b) What is the magnitude of an electric field in which the electric force it exerts on a proton is equal in magnitude to the proton’s weight?

33. A uniform electric field exists in the region between two oppositely charged plane parallel plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, 3.20 cm distant from the first, in a time interval of 1.5 × 10-8 s (a) Find the magnitude of this electric field. (b) Find the speed of the electron when it strikes the second plate.

34. A particle has a charge of -3.00 nC (a) Find the magnitude and direction of the electric field due to this particle at a point 0.250 m directly above it. (b) At what distance from the particle does its electric field have a magnitude of 12.0 N/C.

35. The electric field caused by a certain point charge has a magnitude of 6.50 × 103 N/C at a distance of 0.100 m from the charge. What is the magnitude of the charge?

36. At what distance from a particle with a charge of 5.00 nC does the electric field of that charge have a magnitude of 4.00 N/C?

37. Electric fields in the atom. (a) Within the nucleus. What strength of electric field does a proton produce at the distance of another proton, about 5.0 × 10-15 m away? (b) At the electrons. What strength of electric field does this proton produce at the distance of the electrons, approximately away?

38. A proton is traveling horizontally to the right at 4.50 × 106 m/s (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of 3.20 cm. (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

39. Electric field of axons. A nerve signal is transmitted through a neuron when an excess of Na+ ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately 10.0 μm in diameter, and measurements show that about 5.6 × 1011 Na+ ions per meter (each of charge +e) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of nearly point charges moving along the axon. Let us look at a 0.10 mm length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a 0.10 mm length of the axon? (b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is 5.00 cm below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0 μN/C. How far from this segment of axon could a shark be and still detect its electric field?

40. Two point charges are separated by 25.0 cm (see Figure 17.47). Find the net electric field these charges produce at (a) point A, (b) point B. (c) What would be the magnitude and direction of the electric force this combination of charges would produce on a proton at A?

41. A point charge of -4.00 nC is at the origin, and a second point charge of +6.00 nC is on the x axis at x = 0.800 m. Find the magnitude and direction of the electric field at each of the following points on the x-axis: (a) x = 20.0 cm (b) x = 1.20 m (c) x = -20.0 cm

42. In a rectangular coordinate system, a positive point charge q = 6.00 nC is placed at the point x = +0.150 m, y = 0, and an identical point charge is placed at x = -0.150 m, y = 0. Find the x and y components and the magnitude and direction of the electric field at the following points: (a) the origin; (b) x = 0.350 m, y = 0 (c) x = 0.150 m, y = -0.400 m (d) x = 0, y = 0.200 m

43. Two particles having charges of +0.500 nC and +8.00 nC are separated by a distance of 1.20 m. (a) At what point along the line connecting the two charges is the net electric field due to the two charges equal to zero? (b) Where would the net electric field be zero if one of the charges were negative?

44. Three negative point charges lie along a line as shown in Figure 17.48. Find the magnitude and direction of the electric field this combination of charges produces at point P, which lies 6.00 cm from the -2.00 μC charge measured perpendicular to the line connecting the three charges.

45. Torque and force on a dipole. An electric dipole is in a uniform external electric field E as shown in Figure 17.49. (a) What is the net force this field exerts on the dipole? (b) Find the orientations of the dipole for which the torque on it about an axis through its center perpendicular to the plane of the figure is zero. (c) Which of the orientations in part (b) is stable, and which is unstable?

46. (a) An electron is moving east in a uniform electric field of 1.50 N/C directed to the west. At point A, the velocity of the electron is 4.50 × 105 m/s toward the east. What is the speed of the electron when it reaches point B, 0.375 m east of point A? (b) A proton is moving in the uniform electric field of part (a). At point A, the velocity of the proton is 1.90 × 104 m/s east. What is the speed of the proton at point B?

47. The electric field due to a certain point charge has a magnitude E at a distance of 1.0 cm from the charge. (a) What will be the magnitude of this field (in terms of E) if we move 1.0 cm farther away from the charge? (b) What will be the magnitude of the field (in terms of E) if we move an additional 1.0 cm farther away than in part (a)?

48. For the dipole shown in Figure 17.50, show that the electric field at points on the x axis points vertically downward and has magnitude kq(2a)/(a2 + x2)3/2. What does this expression reduce to when the distance between the two charges is much less than x?

49. Figure 17.51 shows some of the electric field lines due to three point charges arranged along the vertical axis. All three charges have the same magnitude. (a) What are the signs of the three charges? Explain your reasoning. (b) At what point(s) is the magnitude of the electric field the smallest? Explain your reasoning. Explain how the fields produced by each individual point charge combine to give a small net field at this point or points.

50. A proton and an electron are separated as shown in Figure 17.52. Points A, B, and C lie on the perpendicular bisector of the line connecting these two charges. Sketch the direction of the net electric field due to the two charges at (a) point A, (b) point B, and (c) point C.

51. Sketch electric field lines in the vicinity of two charges, Q and -4Q located a small distance apart on the x-axis.

52. Two point charges Q and +q (where q is positive) produce the net electric field shown at point P in Figure 17.53. The field points parallel to the line connecting the two charges. (a) What can you conclude about the sign and magnitude of Q? Explain your reasoning. (b) If the lower charge were negative instead, would it be possible for the field to have the direction shown in the figure? Explain your reasoning.

53. Two very large parallel sheets of the same size carry equal magnitudes of charge spread uniformly over them, as shown in Figure 17.54. In each of the cases that follow, sketch the net pattern of electric field lines in the region between the sheets, but far from their edges. (a) The top sheet is positive and the bottom sheet is negative, as shown, (b) both sheets are positive, (c) both sheets are negative.

54. (a) A closed surface encloses a net charge of 2.50 μC. What is the net electric flux through the surface? (b) If the electric flux through a closed surface is determined to be 1.40 N·m2/C, how much charge is enclosed by the surface?

55. Figure 17.55 shows cross sections of five closed surfaces S1, S2, etc. Find the net electric flux passing through each of these surfaces.

56. A point charge 8.00 nC is at the center of a cube with sides of length 0.200 m. What is the electric flux through (a) the surface of the cube, (b) one of the six faces of the cube?

57. A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of -15.0 μC. Find the electric field (a) just inside the paint layer, (b) just outside the paint layer, and (c) 5.00 cm outside the surface of the paint layer.

58. On a humid day, an electric field of 2.00 × 104 N/C is enough to produce sparks about an inch long. Suppose that in your physics class, a van de Graaff generator with a sphere radius of 15.0 cm is producing sparks 6 inches long. (a) Use Gauss’s law to calculate the amount of charge stored on the surface of the sphere before you bravely discharge it with your hand. (b) Assume all the charge is concentrated at the center of the sphere, and use Coulomb’s law to calculate the electric field at the surface of the sphere.

59. (a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere 30.0 cm in diameter to produce an electric field of 1150 N/C just outside the surface of the sphere? (b) What is the electric field at a point 10.0 cm outside the surface of the sphere?

60. In a certain region of space, the electric field E is uniform; i.e., neither its direction nor its magnitude changes in the region. (a) Use Gauss’s law to prove that this region of space must be electrically neutral; that is, there must be no charge in this region. (b) Is the converse true? That is, in a region of space where there is no charge, must E be uniform? Explain.

61. A total charge of magnitude Q is distributed uniformly within a thick spherical shell of inner radius a and outer radius b. (a) Use Gauss’s law to find the electric field within the cavity (b) Use Gauss’s law to prove that the electric field outside the shell is exactly the same as if all the charge were concentrated as a point charge Q at the center of the sphere. (c) Explain why the result in part (a) for a thick shell is the same as that found in Example 17.10 for a thin shell.

62. During a violent electrical storm, a car is struck by a falling high-voltage wire that puts an excess charge of -850 μC on the metal car. (a) How much of this charge is on the inner surface of the car? (b) How much is on the outer surface?

63. A neutral conductor completely encloses a hole inside of it. You observe that the outer surface of this conductor carries a charge of -12 μC (a) Can you conclude that there is a charge inside the hole? If so, what is this charge? (b) How much charge is on the inner surface of the conductor?

64. An irregular neutral conductor has a hollow cavity inside of it and is insulated from its surroundings. An excess charge of +16 nC is sprayed onto this conductor. (a) Find the charge on the inner and outer surfaces of the conductor. (b) Without touching the conductor, a charge of -11 nC is inserted into the cavity through a small hole in the conductor. Find the charge on the inner and outer surfaces of the conductor in this case.

65. Three point charges are arranged along the x axis. Charge q1 = -4.50 nC is located at x = 0.200 m, and charge q2 = +2.50 nC is at x = -0.300 m. A positive point charge is located at the origin. (a) What must the value of q3 be for the net force on this point charge to have magnitude 4.00 μN (b) What is the direction of the net force on q3? (c) Where along the x axis can q3 be placed and the net force on it be zero, other than the trivial answers of x = +∞ and x = -∞?

66. An electron is released from rest in a uniform electric field. The electron accelerates vertically upward, traveling 4.50 m in the first 3.00 μs after it is released. (a) What are the magnitude and direction of the electric field? (b) Are we justified in ignoring the effects of gravity? Justify your answer quantitatively.

67. A charge q1 = +5.00 nC is placed at the origin of an xy-coordinate system, and a charge q2 = -2.00 nC is placed on the positive x axis at x = 4.00 cm (a) If a third charge q3 = +6.00 nC is now placed at the point x = 4.00 cm, y = 3.00 cm, find the x and y components of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

68. A charge of -3.00 nC is placed at the origin of an xy-coordinate system, and a charge of 2.00 nC is placed on the y axis at (a) If a third charge, of 5.00 nC, is now placed at the point find the x and y components of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

69. Point charges of 3.00 nC are situated at each of three corners of a square whose side is 0.200 m. What are the magnitude and direction of the resultant force on a point charge of -1.00 μC if it is placed (a) at the center of the square, (b) at the vacant corner of the square?

70. An electron is projected with an initial speed v0 = 5.00 × 106 m/s into the uniform field between the parallel plates in Figure 17.57. The direction of the field is vertically downward, and the field is zero except in the space between the two plates. The electron enters the field at a point midway between the plates. If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field.

71. A small 12.3 g plastic ball is tied to a very light 28.6 cm string that is attached to the vertical wall of a room. A uniform horizontal electric field exists in this room. When the ball has been given an excess charge of -1.11 μC you observe that it remains suspended, with the string making an angle of 17.4° with the wall. Find the magnitude and direction of the electric field in the room.

72. A -5.00 nC point charge is on the x axis at x = 1.20 m. A second point charge Q is on the x axis at -0.600 m. What must be the sign and magnitude of Q for the resultant electric field at the origin to be (a) 45.0 N/C in the +x direction, (b) 45.0 N/C in the –x direction?

73. The earth has a downward-directed electric field near its surface of about 150 N/m. If a raindrop with a diameter of 0.020 mm is suspended, motionless, in this field, how many excess electrons must it have on its surface?

74. A 9.60 μC point charge is at the center of a cube with sides of length 0.500 m. (a) What is the electric ﬂux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were 0.250 m long? Explain.

75. Two point charges q1 and q2 are held 4.00 cm apart. An electron released at a point 4.00 cm that is equidistant from both charges (see Figure 17.59) undergoes an initial acceleration of 8.25 × 1018 m/s2 directly upward in the figure, parallel to the line connecting q1 and q2. Find the magnitude and sign of q1 and q2.

76. Electrophoresis. Electrophoresis is a process used by biologists to separate different biological molecules (such as proteins) from each other according to their ratio of charge to size. The materials to be separated are in a viscous solution that produces a drag force FD proportional to the size and speed of the molecule. We can express this relationship as FD = KRv, where R is the radius of the molecule (modeled as being spherical), is its speed, and K is a constant that depends on the viscosity of the solution. The solution is placed in an external electric field E so that the electric force on a particle of charge q is F = qE. (a) Show that when the electric field is adjusted so that the two forces (electrical and viscous drag) just balance, the ratio of q to R is Kv/E (b) Show that if we leave the electric field on for a time T, the distance x that the molecule moves during that time is x = (ET/k)(q/R) (c) Suppose you have a sample containing three different biological molecules for which the molecular ratio q/R for material 2 is twice that of material 1 and the ratio for material 3 is three times that of material 1. Show that the distances migrated by these molecules after the same amount of time are x2 = 2x1 and x3 = 3x1. In other words, material 2 travels twice as far as material 1, and material 3 travels three times as far as material 1. Therefore, we have separated these molecules according to their ratio of charge to size. In practice, this process can be carried out in a special gel or paper, along which the biological molecules migrate. The process can be rather slow, requiring several hours for separations of just a centimeter or so.

77. An early model of the hydrogen atom viewed it as an electron orbiting a proton in a circular path with a radius of 5.29 × 10-11 m. What would be the speed of the electron in this model? You’ll need some information from Appendix E, and may need to review Chapter 6 on circular motion.

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