Figure 19-28 shows a hypothetical speed distribution for

Find the mass in kilograms of 7.50 ' 1024 atoms of arsenic, which has a molar mass of 74.9 g/mol.

Gold has a molar mass of 197 g/mol. (a) How many moles of gold are in a 2.50 g sample of pure gold? (b) How many atoms are in the sample?

Oxygen gas having a volume of 1000 cm3 SSM at 40.0 C and - 1.01 ' 105 Pa expands until its volume is 1500 cm3 and its pressure is 1.06 ' 105 Pa. Find (a) the number of moles of oxygen present and (b) the final temperature of the sample.

A quantity of ideal gas at 10.0-C and 100 kPa occupies a volume of 2.50 m3 . (a) How many moles of the gas are present? (b) If the pressure is now raised to 300 kPa and the temperature is raised to 30.0-C, how much volume does the gas occupy? Assume no leaks.

The best laboratory vacuum has a pressure of about 1.00 ' 10$18 atm, or 1.01 ' 10$13 Pa. How many gas molecules are there per cubic centimeter in such a vacuum at 293 K?

Water bottle in a hot car. In the American Southwest, the temperature in a closed car parked in sunlight during the summer can be high enough to burn flesh. Suppose a bottle of water at a refrigerator temperature of 5.00-C is opened, then closed, and then left in a closed car with an internal temperature of 75.0-C. Neglecting the thermal expansion of the water and the bottle, find the pressure in the air pocket trapped in the bottle. (The pressure can be enough to push the bottle cap past the threads that are intended to keep the bottle closed.)

Suppose 1.80 mol of an ideal gas is taken from a volume of 3.00 m3 to a volume of 1.50 m3 via an isothermal compression at 30-C. (a) How much energy is transferred as heat during the compression, and (b) is the transfer to or from the gas?

Compute (a) the number of moles and (b) the number of molecules in 1.00 cm3 of an ideal gas at a pressure of 100 Pa and a temperature of 220 K.

An automobile tire has a volume of 1.64 ' 10$2 m3 and contains air at a gauge pressure (pressure above atmospheric pressure) of 165 kPa when the temperature is 0.00-C.What is the gauge pressure of the air in the tires when its temperature rises to 27.0-C and its volume increases to 1.67 ' 10$2 m3 ? Assume atmospheric pressure is 1.01 ' 105 Pa.

A container encloses 2 mol of an ideal gas that has molar mass M1 and 0.5 mol of a second ideal gas that has molar mass M2 # 3M1. What fraction of the total pressure on the container wall is attributable to the second gas? (The kinetic theory explanation of pressure leads to the experimentally discovered law of partial pressures for a mixture of gases that do not react chemically: The total pressure exerted by the mixture is equal to the sum of the pressures that the several gases would exert separately if each were to occupy the vessel alone. The moleculevessel collisions of one type would not be altered by the presence of another type.)

Air that initially occupies 0.140 m3 at a gauge pressure of 103.0 kPa is expanded isothermally to a pressure of 101.3 kPa and then cooled at constant pressure until it reaches its initial volume. Compute the work done by the air. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.)

Submarine rescue. When the U.S. submarine Squalus became disabled at a depth of 80 m, a cylindrical chamber was lowered from a ship to rescue the crew. The chamber had a radius of 1.00 m and a height of 4.00 m, was open at the bottom, and held two rescuers. It slid along a guide cable that a diver had attached to a hatch on the submarine. Once the chamber reached the hatch and clamped to the hull, the crew could escape into the chamber. During the descent, air was released from tanks to prevent water from flooding the chamber. Assume that the interior air pressure matched the water pressure at depth h as given by p0 " rgh, where p0 # 1.000 atm is the surface pressure and r # 1024 kg/m3 is the density of seawater. Assume a surface temperature of 20.0-C and a submerged water temperature of $30.0-C. (a) What is the air volume in the chamber at the surface? (b) If air had not been released from the tanks, what would have been the air volume in the chamber at depth h # 80.0 m? (c) How many moles of air were needed to be released to maintain the original air volume in the chamber?

A sample of an ideal gas is taken through the cyclic process abca shown in Fig. 19-20. The scale of the vertical axis is set by pb # 7.5 kPa and pac # 2.5 kPa. At point a, T # 200 K. (a) How many moles of gas are in the sample? What are (b) the temperature of the gas at point b, (c) the temperature of the gas at point c, and (d) the net energy added to the gas as heat during the cycle? pb pac 1.0 3.0 Volume (m3) Pressure (kPa) b a c. A B Figure 19-20 Problem 13

In the temperature range 310 K to 330 K, the pressure p of a certain nonideal gas is related to volume V and temperature T by p # (24.9 J/K) T V $ (0.00662 J/K2 ) T2 V How much work is done by the gas if its temperature is raised from 315 K to 325 K while the pressure is held constant?

Suppose 0.825 mol of an ideal gas undergoes an isothermal expansion as energy is added to it as heat Q. If Fig. 19-21 shows the final volume Vf versus Q, what is the gas temperature? The scale of the vertical axis is set by Vfs # 0.30 m3 , and the scale of the horizontal axis is set by Qs # 1200 J. Figure 19-21 Problem 15. Vfs 0 Q (J) Qs Vf (m3)

An air bubble of volume 20 cm3 is at the bottom of a lake 40 m deep, where the temperature is 4.0-C. The bubble rises to the surface, which is at a temperature of 20-C.Take the temperature of the bubbles air to be the same as that of the surrounding water. Just as the bubble reaches the surface, what is its volume?

Container A in Fig. 19-22 holds an ideal gas at a pressure of 5.0 ' 105 Pa and a temperature of 300 K. It is connected by a thin tube (and a closed valve) to container B, with four times the volume of A. Container B holds the same ideal gas at a pressure of 1.0 ' 105 Pa and a temperature of 400 K. The valve is opened to allow the pressures to equalize, but the temperature of each container is maintained.What then is the pressure? A B Figure 19-22 Problem 17.

The temperature and pressure in the Suns atmosphere are 2.00 ' 106 K and 0.0300 Pa. Calculate the rms speed of free electrons (mass 9.11 ' 10$31 kg) there, assuming they are an ideal gas.

(a) Compute the rms speed of a nitrogen molecule at 20.0-C. The molar mass of nitrogen molecules (N2) is given in Table 19-1. At what temperatures will the rms speed be (b) half that value and (c) twice that value?

Calculate the rms speed of helium atoms at 1000 K. See Appendix F for the molar mass of helium atoms.

The lowest possible temperature in outer space is 2.7 K. What is the rms speed of hydrogen molecules at this temperature? (The molar mass is given in Table 19-1.)

Find the rms speed of argon atoms at 313 K. See Appendix F for the molar mass of argon atoms.

A beam of hydrogen molecules (H2) is directed toward a wall, at an angle of 55- with the normal to the wall. Each molecule in the beam has a speed of 1.0 km/s and a mass of 3.3 ' 10$24 g. The beam strikes the wall over an area of 2.0 cm2 , at the rate of 1023 molecules per second.What is the beams pressure on the wall?

At 273 K and 1.00 ' 10$2 atm, the density of a gas is 1.24 ' 10$5 g/cm3 . (a) Find vrms for the gas molecules. (b) Find the molar mass of the gas and (c) identify the gas. See Table 19-1.

Determine the average value of the translational kinetic energy of the molecules of an ideal gas at temperatures (a) 0.00-C and (b) 100-C. What is the translational kinetic energy per mole of an ideal gas at (c) 0.00-C and (d) 100-C?

What is the average translational kinetic energy of nitrogen molecules at 1600 K?

Water standing in the open at 32.0-C evaporates because of the escape of some of the surface molecules. The heat of vaporization (539 cal/g) is approximately equal to n, where is the average energy of the escaping molecules and n is the number of molecules per gram. (a) Find . (b) What is the ratio of to the average kinetic energy of H2O molecules, assuming the latter is related to temperature in the same way as it is for gases?

At what frequency would the wavelength of sound in air be equal to the mean free path of oxygen molecules at 1.0 atm pressure and 0.00-C? The molecular diameter is 3.0 ' 10$8 cm.

The atmospheric density at an altitude of 2500 km is about 1 molecule/cm3 . (a) Assuming the molecular diameter of 2.0 ' 10$8 cm, find the mean free path predicted by Eq. 19-25. (b) Explain whether the predicted value is meaningful.

The mean free path of nitrogen molecules at 0.0-C and 1.0 atm is 0.80 ' 10$5 cm. At this temperature and pressure there are 2.7 ' 1019 molecules/cm3 .What is the molecular diameter?

In a certain particle accelerator, protons travel around a circular path of diameter 23.0 m in an evacuated chamber, whose residual gas is at 295 K and 1.00 ' 10$6 torr pressure. (a) Calculate the number of gas molecules per cubic centimeter at this pressure. (b) What is the mean free path of the gas molecules if the molecular diameter is 2.00 ' 10$8 cm?

At 20-C and 750 torr pressure, the mean free paths for argon gas (Ar) and nitrogen gas (N2) are lAr # 9.9 ' 10$6 cm and lN2 # 27.5 ' 10$6 cm. (a) Find the ratio of the diameter of an Ar atom to that of an N2 molecule. What is the mean free path of argon at (b) 20-C and 150 torr, and (c) $40-C and 750 torr?

The speeds of 10 molecules are 2.0, 3.0, 4.0, . . . , 11 km/s. What are their (a) average speed and (b) rms speed?

The speeds of 22 particles are as follows (Ni represents the number of particles that have speed vi ): Ni 24682 vi (cm/s) 1.0 2.0 3.0 4.0 5.0 What are (a) vavg, (b) vrms, and (c) vP?

Ten particles are moving with the following speeds: four at 200 m/s, two at 500 m/s, and four at 600 m/s. Calculate their (a) average and (b) rms speeds. (c) Is vrms 4 vavg?

The most probable speed of the molecules in a gas at temperature T2 is equal to the rms speed of the molecules at temperature T1. Find T2 /T1.

Figure 19-23 shows a hypothetical speed distribution for a sample of N gas particles (note that P(v) # 0 for speed v 4 2v0).What are the values of (a) av0, (b) vavg/v0, and (c) vrms/v0? (d) What fraction of the particles has a speed between 1.5v0 and 2.0v0? SSM WWW SSM SSM v0 Speed 2v 0 0 a P(v) Figure 19-23 Problem 37.

Figure 19-24 gives the probability distribution for nitrogen gas.The scale of the horizontal axis is set by vs # 1200 m/s.What are the (a) gas temperature and (b) rms speed of the molecules? 0 vs v (m/s) P(v) Figure 19-24 Problem 38

At what temperature does the rms speed of (a) H2 (molecular hydrogen) and (b) O2 (molecular oxygen) equal the escape speed from Earth (Table 13-2)? At what temperature does the rms speed of (c) H2 and (d) O2 equal the escape speed from the Moon (where the gravitational acceleration at the surface has magnitude 0.16g)? Considering the answers to parts (a) and (b), should there be much (e) hydrogen and (f) oxygen high in Earths upper atmosphere, where the temperature is about 1000 K?

Two containers are at the same temperature. The first contains gas with pressure p1, molecular mass m1, and rms speed vrms1. The second contains gas with pressure 2.0p1, molecular mass m2, and average speed vavg2 # 2.0vrms1. Find the mass ratio m1/m2.

A hydrogen molecule (diameter 1.0 ' 10$8 cm), traveling at the rms speed, escapes from a 4000 K furnace into a chamber containing cold argon atoms (diameter 3.0 ' 10$8 cm) at a density of 4.0 ' 1019 atoms/cm3 . (a) What is the speed of the hydrogen molecule? (b) If it collides with an argon atom, what is the closest their centers can be, considering each as spherical? (c) What is the initial number of collisions per second experienced by the hydrogen molecule? (Hint:Assume that the argon atoms are stationary.Then the mean free path of the hydrogen molecule is given by Eq. 19-26 and not Eq. 19-25.)

What is the internal energy of 1.0 mol of an ideal monatomic gas at 273 K?

The temperature of 3.00 mol of an ideal diatomic gas is increased by 40.0 C without the pressure of the gas changing. The molecules in the gas rotate but do not oscillate. (a) How much energy is transferred to the gas as heat? (b) What is the change in the internal energy of the gas? (c) How much work is done by the gas? (d) By how much does the rotational kinetic energy of the gas increase?

One mole of an ideal diatomic gas goes from a to c along the diagonal path in Fig. 19-25. The scale of the vertical axis is set by pab # 5.0 kPa and pc # 2.0 kPa, and the scale of the horizontal axis is set by Vbc # 4.0 m3 and Va # 2.0 m3 . During the transition, (a) what is the change in internal energy of the gas, and (b) how - pab pc Va Vbc Volume (m3) Pressure (kPa) a b c Figure 19-25 Problem 44. much energy is added to the gas as heat? (c) How much heat is required if the gas goes from a to c along the indirect path abc?

The mass of a gas molecule can be computed from its specific heat at constant volume cV. (Note that this is not CV.) Take cV # 0.075 cal/g 9C- for argon and calculate (a) the mass of an argon atom and (b) the molar mass of argon.

Under constant pressure, the temperature of 2.00 mol of an ideal monatomic gas is raised 15.0 K. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change %Eint in the internal energy of the gas, and (d) the change %K in the average kinetic energy per atom?

The temperature of 2.00 mol of an ideal monatomic gas is raised 15.0 K at constant volume.What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change %Eint in the internal energy of the gas, and (d) the change %K in the average kinetic energy per atom?

When 20.9 J was added as heat to a particular ideal gas, the volume of the gas changed from 50.0 cm3 to 100 cm3 while the pressure remained at 1.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas present was 2.00 ' 10$3 mol, find (b) Cp and (c) CV.

A container holds a mixture of three nonreacting gases: 2.40 mol of gas 1 with CV1 12.0 J/mol9K, 1.50 mol of gas 2 with CV2 # 12.8 J/mol9K, and 3.20 mol of gas 3 with CV3 # 20.0 J/mol9K. What is CV of the mixture?

We give 70 J as heat to a diatomic gas, which then expands at constant pressure.The gas molecules rotate but do not oscillate. By how much does the internal energy of the gas increase?

When 1.0 mol of oxygen (O2) gas is heated at constant pressure starting at 0 C, how much energy must be added to the gas as heat to double its volume? (The molecules rotate but do not oscillate.)

Suppose 12.0 g of oxygen (O2) gas is heated at constant atmospheric pressure from 25.0 C to 125 C. (a) How many moles of oxygen are present? (See Table 19-1 for the molar mass.) (b) How much energy is transferred to the oxygen as heat? (The -- - ILW # molecules rotate but do not oscillate.) (c) What fraction of the heat is used to raise the internal energy of the oxygen?

Suppose 4.00 mol of an ideal diatomic gas, with molecular rotation but not oscillation, experienced a temperature increase of 60.0 K under constant-pressure conditions. What are (a) the energy transferred as heat Q, (b) the change %Eint in internal energy of the gas, (c) the work W done by the gas, and (d) the change %K in the total translational kinetic energy of the gas?

We know that for an adiabatic process pVg # a constant. Evaluate a constant for an adiabatic process involving exactly 2.0 mol of an ideal gas passing through the state having exactly p # 1.0 atm and T # 300 K. Assume a diatomic gas whose molecules rotate but do not oscillate.

A certain gas occupies a volume of 4.3 L at a pressure of 1.2 atm and a temperature of 310 K. It is compressed adiabatically to a volume of 0.76 L. Determine (a) the final pressure and (b) the final temperature, assuming the gas to be an ideal gas for which g # 1.4.

Suppose 1.00 L of a gas with g # 1.30, initially at 273 K and 1.00 atm, is suddenly compressed adiabatically to half its initial volume. Find its final (a) pressure and (b) temperature. (c) If the gas is then cooled to 273 K at constant pressure, what is its final volume?

The volume of an ideal gas is adiabatically reduced from 200 L to 74.3 L. The initial pressure and temperature are 1.00 atm and 300 K. The final pressure is 4.00 atm. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is the final temperature? (c) How many moles are in the gas?

Opening champagne. In a bottle of champagne, the pocket of gas (primarily carbon dioxide) between the liquid and the cork is at pressure of pi # 5.00 atm. When the cork is pulled from the bottle, the gas undergoes an adiabatic expansion until its pressure matches the ambient air pressure of 1.00 atm. Assume that the ratio of the molar specific heats is . If the gas has initial temperature Ti # 5.00 C, what is its temperature at the end of the adiabatic expansion?

Figure 19-26 shows two paths that may be taken by a gas from an initial point i to a final point f. Path 1 consists of an isothermal expansion (work is 50 J in magnitude), an adiabatic expansion (work is 40 J in magnitude), an isothermal compression (work is 30 J in magnitude), and then an adiabatic compression (work is 25 J in magnitude). What is the change in the internal energy of the gas if the gas goes from point i to point f along path 2? PV Path 1 Isothermal Isothermal Adiabatic Path 2 f i Figure 19-26 Problem 59

Adiabatic wind. The normal airflow over the Rocky Mountains is west to east.The air loses much of its moisture content and is chilled as it climbs the western side of the mountains.When it descends on the eastern side, the increase in pressure toward lower altitudes causes the temperature to increase. The flow, then called a chinook wind, can rapidly raise the air temperature at the base of the mountains. Assume that the air pressure p depends on altitude y according to p p0 exp ($ay), where p0 # 1.00 atm and a 1.16 10$4 m$1 . Also assume that the ratio of the molar specific heats is . A parcel of air with an initial temperature of .00 C descends adiabatically from y1 4267 m to y 1567 m.What is its temperature at the end of the descent?

A gas is to be expanded from initial state i to final state f along either path 1 or path 2 on a p-V diagram. Path 1 consists of three steps: an isothermal expansion (work is 40 J in magnitude), an adiabatic expansion (work is 20 J in magnitude), and another isothermal expansion (work is 30 J in magnitude). Path 2 consists of two steps: a pressure reduction at constant volume and an expansion at constant pressure.What is the change in the internal energy of the gas along path 2?

An ideal diatomic gas, with rotation but no oscillation, undergoes an adiabatic compression. Its initial pressure and volume are 1.20 atm and 0.200 m3 . Its final pressure is 2.40 atm. How much work is done by the gas?

Figure 19-27 shows a cycle undergone by 1.00 mol of an ideal monatomic gas. The temperatures are T1 # 300 K, T2 # 600 K, and T3 # 455 K. For 1 : 2, what are (a) heat Q, (b) the change in internal energy %Eint, and (c) the work done W? For 2 : 3, what are (d) Q, (e) %Eint, and (f) W? For 3 : 1, what are (g) Q, (h) %Eint, and (i) W? For the full cycle, what are (j) Q, (k) %Eint, and (l) W? The initial pressure at point 1 is 1.00 atm (# 1.013 ' 105 Pa).What are the (m) volume and (n) pressure at point 2 and the (o) volume and (p) pressure at point 3? Pressure 2 Adiabatic 1 3 Volume Figure 19-27 Problem 63.

Calculate the work done by an external agent during an isothermal compression of 1.00 mol of oxygen from a volume of 22.4 L at 0-C and 1.00 atm to a volume of 16.8 L.

An ideal gas undergoes an adiabatic compression from p # 1.0 atm, V # 1.0 ' 106 L, T # 0.0-C to p # 1.0 ' 105 atm, V # 1.0 ' 103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? What is the total translational kinetic energy per mole (d) before and (e) after the compression? (f) What is the ratio of the squares of the rms speeds before and after the compression?

An ideal gas consists of 1.50 mol of diatomic molecules that rotate but do not oscillate. The molecular diameter is 250 pm. The gas is expanded at a constant pressure of 1.50 ' 105 Pa,with a transfer of 200 J as heat.What is the change in the mean free path of the molecules?

An ideal monatomic gas initially has a temperature of 330 K and a pressure of 6.00 atm. It is to expand from volume 500 cm3 to volume 1500 cm3 . If the expansion is isothermal, what are (a) the final pressure and (b) the work done by the gas? If, instead, the expansion is adiabatic, what are (c) the final pressure and (d) the work done by the gas?

In an interstellar gas cloud at 50.0 K, the pressure is 1.00 ' 10$8 Pa. Assuming that the molecular diameters of the gases in the cloud are all 20.0 nm, what is their mean free path?

The envelope and basket of a hot-air balloon have a combined weight of 2.45 kN, and the envelope has a capacity (volume) of 2.18 ' 103 m3 . When it is fully inflated, what should be the temperature of the enclosed air to give the balloon a lifting capacity (force) of 2.67 kN (in addition to the balloons weight)? Assume that the surrounding air, at 20.0-C, has a weight per unit volume of 11.9 N/m3 and a molecular mass of 0.028 kg/mol, and is at a pressure of 1.0 atm.

An ideal gas, at initial temperature T1 and initial volume 2.0 m3 , is expanded adiabatically to a volume of 4.0 m3 , then expanded isothermally to a volume of 10 m3 , and then compressed adiabatically back to T1.What is its final volume?

The temperature of 2.00 mol of an ideal monatomic gas is raised 15.0 K in an adiabatic process. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change %Eint in internal energy of the gas, and (d) the change %K in the average kinetic energy per atom?

At what temperature do atoms of helium gas have the same rms speed as molecules of hydrogen gas at 20.0-C? (The molar masses are given in Table 19-1.)

At what frequency do molecules (diameter 290 pm) collide in (an ideal) oxygen gas (O2) at temperature 400 K and pressure 2.00 atm?

(a) What is the number of molecules per cubic meter in air at 20-C and at a pressure of 1.0 atm (# 1.01 ' 105 Pa)? (b) What is the mass of 1.0 m3 of this air? Assume that 75% of the molecules are nitrogen (N2) and 25% are oxygen (O2).

The temperature of 3.00 mol of a gas with CV # 6.00 cal/mol9K is to be raised 50.0 K. If the process is at constant volume, what are (a) the energy transferred as heat Q, (b) the work W done by the gas, (c) the change %Eint in internal energy of the gas, and (d) the change %K in the total translational kinetic energy? If the process is at constant pressure, what are (e) Q, (f) W, (g) Eint, and (h) %K? If the process is adiabatic, what are (i) Q, (j) W, (k) %Eint, and (l) %K?

During a compression at a constant pressure of 250 Pa, the volume of an ideal gas decreases from 0.80 m3 to 0.20 m3 . The initial temperature is 360 K, and the gas loses 210 J as heat. What are (a) the change in the internal energy of the gas and (b) the final temperature of the gas?

Figure 19-28 shows a hypothetical speed distribution for particles of a certain gas: P(v) # Cv2 for 0 8 v 2 v0 and P(v) # 0 for v 4 v0. Find (a) an expression for C in terms of v0, (b) the average speed of the particles, and (c) their rms speed. P(v) Speed v 0 0 Figure 19-28 Problem 77

(a) An ideal gas initially at pressure p0 undergoes a free expansion until its volume is 3.00 times its initial volume. What then is the ratio of its pressure to p0? (b) The gas is next slowly and adiabatically compressed back to its original volume. The pressure after compression is (3.00)1/3p0. Is the gas monatomic, diatomic, or polyatomic? (c) What is the ratio of the average kinetic energy per molecule in this final state to that in the initial state?

An ideal gas undergoes isothermal compression from an initial volume of 4.00 m3 to a final volume of 3.00 m3 . There is 3.50 mol of the gas, and its temperature is 10.0-C. (a) How much work is done by the gas? (b) How much energy is transferred as heat between the gas and its environment?

Oxygen (O2) gas at 273 K and 1.0 atm is confined to a cubical container 10 cm on a side. Calculate %Ug/Kavg, where %Ug is the change in the gravitational potential energy of an oxygen molecule falling the height of the box and Kavg is the molecules average translational kinetic energy.

An ideal gas is taken through a complete cycle in three steps: adiabatic expansion with work equal to 125 J, isothermal contraction at 325 K, and increase in pressure at constant volume. (a) Draw a p-V diagram for the three steps. (b) How much energy is transferred as heat in step 3, and (c) is it transferred to orfrom the gas?

(a) What is the volume occupied by 1.00 mol of an ideal gas at standard conditions that is, 1.00 atm (# 1.01 ' 105 Pa) and 273 K? (b) Show that the number of molecules per cubic centimeter (the Loschmidt number) at standard conditions is 2.69 ' 109

A sample of ideal gas expands from an initial pressure and volume of 32 atm and 1.0 L to a final volume of 4.0 L. The initial temperature is 300 K. If the gas is monatomic and the expansion isothermal, what are the (a) final pressure pf, (b) final temperature Tf, and (c) work W done by the gas? If the gas is monatomic and the expansion adiabatic, what are (d) pf, (e) Tf, and (f) W? If the gas is diatomic and the expansion adiabatic, what are (g) pf, (h) Tf, and (i) W?

An ideal gas with 3.00 mol is initially in state 1 with pressure p1 # 20.0 atm and volume V1 # 1500 cm3 . First it is taken to state 2 with pressure p2 # 1.50p1 and volume V2 # 2.00V1. Then it is taken to state 3 with pressure p3 2.00p1 and volume V3 0.500V1. What is the temperature of the gas in (a) state 1 and (b) state 2? (c) What is the net change in internal energy from state 1 to state 3?

A steel tank contains 300 g of ammonia gas (NH3) at a pressure of 1.35 ' 106 Pa and a temperature of 77-C. (a) What is the volume of the tank in liters? (b) Later the temperature is 22-C and the pressure is 8.7 ' 105 Pa. How many grams of gas have leaked out of the tank?

In an industrial process the volume of 25.0 mol of a monatomic ideal gas is reduced at a uniform rate from 0.616 m3 to 0.308 m3 in 2.00 h while its temperature is increased at a uniform rate from 27.0-C to 450-C.nThroughout the process, the gas passes through thermodynamic equilibrium states. What are (a) the cumulative work done on the gas, (b) the cumulative energy absorbed by the gas as heat, and (c) the molar specific heat for the process? (Hint: To evaluate the integral for the work, you might use " a " bx A " Bx dx # bx B " aB $ bA B2 ln(A " Bx) an indefinite integral.) Suppose the process is replaced with a twostep process that reaches the same final state. In step 1, the gas volume is reduced at constant temperature, and in step 2 the temperature is increased at constant volume. For this process, what are (d) the cumulative work done on the gas, (e) the cumulative energy absorbed by the gas as heat,and (f) the molar specific heat for the process?

Figure 19-29 shows a cycle consisting of five paths: AB is isothermal at 300 K, BC is adiabatic with work # 5.0 J, CD is at a constant pressure of 5 atm, DE is isothermal, and EA is adiabatic with a change in internal energy of 8.0 J. What is the change in internal energy of the gas along path CD? A E D C B V p Figure 19-29 Problem 87

An ideal gas initially at 300 K is compressed at a constant pressure of 25 N/m2 from a volume of 3.0 m3 to a volume of 1.8 m3 . In the process, 75 J is lost by the gas as heat.What are (a) the change in internal energy of the gas and (b) the final temperature of the gas?

A pipe of length L # 25.0 m that is open at one end contains air at atmospheric pressure. It is thrust vertically into a freshwater lake until the water rises halfway up in the pipe (Fig. 19-30). What is the depth h of the lower end of the pipe? Assume that the temperature is the same everywhere and does not change.. h L/2 L/ Figure 19-30 Problem 89

In a motorcycle engine, a piston is forced down toward the crankshaft when the fuel in the top of the pistons cylinder undergoes combustion. The mixture of gaseous combustion products then expands adiabatically as the piston descends. Find the average power in (a) watts and (b) horsepower that is involved in this expansion when the engine is running at 4000 rpm, assuming that the gauge pressure immediately after combustion is 15 atm, the initial volume is 50 cm3 , and the volume of the mixture at the bottom of the stroke is 250 cm3 . Assume that the gases are diatomic and that the time involved in the expansion is one-half that of the total cycle.

For adiabatic processes in an ideal gas, show that (a) the bulk modulus is given by B # $V dp dV # gp, where g # Cp/CV. (See Eq. 17-2.) (b) Then show that the speed of sound in the gas is vs # A gp r # A gRT M ,where r is the density, T is the temperature, and M is the molar mass. (See Eq. 17-3.)

Air at 0.000-C and 1.00 atm pressure has a density of 1.29 ' 10$3 g/cm3 , and the speed of sound is 331 m/s at that temperature. Compute the ratio g of the molar specific heats of air. (Hint: See Problem 91.)

The speed of sound in different gases at a certain temperature T depends on the molar mass of the gases. Show that v1 v2 # A M2 M1 ,where v1 is the speed of sound in a gas of molar mass M1 and v2 is the speed of sound in a gas of molar mass M2.(Hint: See Problem 91.)

From the knowledge that CV, the molar specific heat at constant volume, for a gas in a container is 5.0R, calculate the ratio of the speed of sound in that gas to the rms speed of the molecules, for gas temperature T. (Hint: See Problem 91.)

The molar mass of iodine is 127 g/mol. When sound at frequency 1000 Hz is introduced to a tube of iodine gas at 400 K, an internal acoustic standing wave is set up with nodes separated by 9.57 cm.What is g for the gas? (Hint: See Problem 91.)

For air near 0-C, by how much does the speed of sound increase for each increase in air temperature by 1 C-? (Hint: See Problem 91.)

Two containers are at the same temperature. The gas in the first container is at pressure p1 and has molecules with mass m1 and root-mean-square speed vrms1.The gas in the second is at pressure 2p1 and has molecules with mass m2 and average speed vavg2 # 2vrms1. Find the ratio m1/m2 of the masses of their molecules.