Chapter 19 the molar mass M of the molecules in the sample

Chapter: Chapter 19

Learning Objectives

LO 19.1.0 Solve problems related to Avogadro’s number.

LO 19.1.1 Identify Avogadro’s number NA.

LO 19.1.2 Apply the relationship between the number of moles n, the number of molecules N,

and

Avogadro’s number NA.

LO 19.1.3 Apply the relationships between the mass m of a sample, the molar mass M of the

molecules in the sample, the number of moles n in the sample, and Avogadro’s number NA.

LO 19.2.0 Solve problems related to ideal gases.

LO 19.2.1 Identify why an ideal gas is said to be ideal.

LO 19.2.2 Apply either of the two forms of the ideal gas law, written in terms of the number of

moles n or the number of molecules N.

LO 19.2.3 Relate the ideal gas constant R and the Boltzmann constant k.

LO 19.2.4 Identify that the temperature in the ideal gas law must be in kelvins.

LO 19.2.5 Sketch p-V diagrams for a constant-temperature expansion of a gas and a

constant-temperature contraction.

LO 19.2.6 Identify the term isotherm.

LO 19.2.7 Calculate the work done by a gas, including the algebraic sign, for an expansion and a

contraction along an isotherm.

LO 19.2.8 For an isothermal process, identify that the change in internal energy ΔE is zero and

that the energy Q transferred as heat is equal to the work W done.

LO 19.2.9 On a p-V diagram, sketch a constant-volume process and identify the amount of work

done in terms of area on the diagram.

LO 19.2.10 On a p-V diagram, sketch a constant-pressure process and determine the work done

in terms of area on the diagram.

LO 19.3.0 Solve problems related to pressure, temperature, and RMS speed.

LO 19.3.1 Identify that the pressure on the interior walls of a gas container is due to the

molecular collisions with the walls.

LO 19.3.2 Relate the pressure on a container wall to the momentum of the gas molecules and the

time intervals between their collisions with the wall.

LO 19.3.3 For the molecules of an ideal gas, relate the root-mean-square speed vrms and the

average speed vavg.

LO 19.3.4 Relate the pressure of an ideal gas to the rms speed vrms of the molecules.

LO 19.3.5 For an ideal gas, apply the relationship between the gas temperature T and the rms

speed vrms and molar mass M of the molecules.

LO 19.4.0 Solve problems related to translational kinetic energy.

LO 19.4.1 For an ideal gas, relate the average kinetic energy of the molecules to their rms speed.

LO 19.4.2 Apply the relationship between the average kinetic energy and the temperature of

the gas.

LO 19.4.3 Identify that a measurement of a gas temperature is effectively a measurement of the

average kinetic energy of the gas molecules.

LO 19.5.0 Solve problems related to mean free path.

LO 19.5.1 Identify what is meant by mean free path.

LO 19.5.2 Apply the relationship between the mean free path, the diameter of the molecules,

and the number of molecules per unit volume.

LO 19.6.0 Solve problems related to the distribution of molecular speeds.

LO 19.6.1 Explain how Maxwell’s speed distribution law is used to find the fraction of

molecules with speeds in a certain speed range.

LO 19.6.2 Sketch a graph of Maxwell’s speed distribution, showing the probability distribution

versus speed and indicating the relative positions of the average speed vavg, the most probable

speed vP, and the rms speed vrms.

LO 19.6.3 Explain how Maxwell’s speed distribution is used to find the average speed, the rms

speed, and the most probable speed.

LO 19.6.4 For a given temperature T and molar mass M, calculate the average speed vavg, the

most probable speed vP, and the rms speed vrms.

LO 19.7.0 Solve problems related to the molar specific heats of an ideal gas.

LO 19.7.1 Identify that the internal energy of an ideal monatomic gas is the sum of the

translational kinetic energies of its atoms.

LO 19.7.2 Apply the relationship between the internal energy Eint of a monatomic ideal gas, the

number of moles n, and the gas temperature T.

LO 19.7.3 Distinguish between monatomic, diatomic, and polyatomic ideal gases.

LO 19.7.4 For monatomic, diatomic, and polyatomic ideal gases, evaluate the molar specific

heats for a constant-volume process and a constant-pressure process.

LO 19.7.5 Calculate a molar specific heat at constant pressure Cp by adding R to the molar

specific heat at constant volume CV, and explain why (physically) Cp is greater.

LO 19.7.6 Identify that the energy transferred to an ideal gas as heat in a constant-volume

process goes entirely into the internal energy (the random translational motion) but that in a

constant-pressure process energy also goes into the work done to expand the gas.

LO 19.7.7 Identify that for a given change in temperature, the change in the internal energy of

an ideal gas is the same for any process and is most easily calculated by assuming a

constant-volume process.

LO 19.7.8 For an ideal gas, apply the relationship between heat Q, number of moles n, and

temperature change ΔT, using the appropriate molar specific heat.

LO 19.7.9 Between two isotherms on a p-V diagram, sketch a constant-volume process and a

constant-pressure process, and for each identify the work done in terms of area on the graph.

LO 19.7.10 Calculate the work done by an ideal gas for a constant-pressure process.

LO 19.7.11 Identify that the work done by a gas is zero for a constant-volume process.

LO 19.8.0 Solve problems related to degrees of freedom and molar specific heats.

LO 19.8.1 Identify that a degree of freedom is associated with each way a gas can store energy

(translation, rotation, and oscillation).

LO 19.8.2 Identify that an energy of 1/2kT per molecule is associated with each degree of

freedom.

LO 19.8.3 Identify that a monatomic gas can have an internal energy consisting of only

translational motion.

LO 19.8.4 Identify that at low temperatures a diatomic gas has energy in only translational

motion, at higher temperatures it also energy in molecular rotation, and at even higher

temperatures it can also have energy in molecular oscillations.

LO 19.8.5 Calculate the molar specific heat for monatomic and diatomic ideal gases in a

constant-volume process and a constant-pressure process.

LO 19.9.0 Solve problems related to the adiabatic expansion of an ideal gas.

LO 19.9.1 On a p-V diagram, sketch an adiabatic expansion (or contraction) and identify that

there is no heat exchange Q with the environment.

LO 19.9.2 Identify that in an adiabatic expansion, the gas does work on the environment,

decreasing the gas’s internal energy, and that in an adiabatic contraction, work is done on the

gas, increasing the internal energy.

LO 19.9.3 In an adiabatic expansion or contraction, relate the initial pressure and volume to the

final pressure and volume.

LO 19.9.4 In an adiabatic expansion or contraction, relate the initial temperature and volume to

the final temperature and volume.

LO 19.9.5 Calculate the work done in an adiabatic process by integrating the pressure with

respect to volume.

LO 19.9.6 Identify that a free expansion of a gas into a vacuum is adiabatic but no work is done

and thus, by the first law of thermodynamics, the internal energy and temperature of the gas do

not change.

Multiple Choice

1. Avogadro’s number is:

A) 6.02 x 1018 mol-1

B) 6.02 x 1020 mol-1

C) 6.02 x 1022 mol-1

D) 6.02 x 1023 mol-1

E) 6.02 x 1024 mol-1

2. How many molecules are in a sample of 10-3 mol?

A) 6.02 x 1018

B) 6.02 x 1020

C) 6.02 x 1022

D) 6.02 x 1023

E) cannot tell without knowing the molecular mass

3. Platinum has a molar mass of 195 g/mol. If you have a ring that contains 2.3 g of platinum,

how many moles does it contain?

A) 0.012 mol

B) 85 mol

C) 450 mol

D) 7.2 x 1021 mol

E) 1.4 x 1024 mol

4. Evidence that a gas consists mostly of empty space is the fact that:

A) the density of a gas becomes much greater when it is liquefied

B) gases exert pressure on the walls of their containers

C) gases are transparent

D) heating a gas increases the molecular motion

E) nature abhors a vacuum

5. Air enters a hot-air furnace at 7C and leaves at 77C. If the pressure does not change each

entering cubic meter of air expands to:

A) 0.80 m3

B) 1.25 m3

C) 1.9 m3

D) 7.0 m3

E) 11 m3

6. 273 cm3 of an ideal gas is at 0C. It is heated at constant pressure to 10C. It will now

occupy:

A) 263 cm3

B) 273 cm3

C) 278 cm3

D) 283 cm3

E) 293 cm3

7. Two identical rooms in a house are connected by an open doorway. The temperatures in the

two rooms are maintained at different values. Which room contains more air?

A) the room with higher temperature

B) the room with lower temperature

C) the room with higher pressure

D) neither because both have the same pressure

E) neither because both have the same volume

8. It is known that 28 grams of a certain ideal gas occupy 22.4 liters at standard conditions

(0C, 1 atm). The volume occupied by 42 grams of this gas at standard conditions is:

A) 14.9 liters

B) 22.4 liters

C) 33.6 liters

D) 42 liters

E) more data are needed

9. An automobile tire is pumped up to a gauge pressure of 2.0  105 Pa when the temperature is

27C. What is its gauge pressure after the car has been running on a hot day so that the tire

temperature is 77C? Assume that the volume remains fixed and take atmospheric pressure to be

1.013  105 Pa.

A) 1.6  105 Pa

B) 2.3  105 Pa

C) 2.5  105 Pa

D) 3.6  105 Pa

E) 8.6  105 Pa

10. A sample of an ideal gas is compressed by a piston from 10 m3 to 5 m3 and simultaneously

cooled from 273C to 0C. As a result there is:

A) an increase in pressure

B) a decrease in pressure

C) a decrease in density

D) no change in volume

E) an increase in density

11. A 2.0-m3 weather balloon is loosely filled with helium at 1 atm (76 cm Hg) and at 27C. At

an elevation of 20,000 ft, the atmospheric pressure is down to 38 cm Hg and the helium has

expanded, being under no constraint from the confining bag. If the temperature at this elevation

is –48C, the gas volume is:

A) 0.75 m3

B) 1.3 m3

C) 3.0 m3

D) 4.0 m3

E) 5.3 m3

12. Oxygen (molar mass = 32 g) occupies a volume of 12 liters when its temperature is 20C

and its pressure is 1 atm. Using R = 0.082 literatm/moleK, calculate the mass of the oxygen:

A) 6.4 g

B) 11 g

C) 16 g

D) 32 g

E) 64 g

13. An ideal gas occupies 12 liters at 293 K and 1 atm (76 cm Hg). Its temperature is now

raised to 373 K and its pressure increased to 215 cm Hg. The new volume is:

A) 3.3 liters

B) 5.4 liters

C) 27 liters

D) 46 liters

E) none of these

14. Use R = 8.2  10–5 m3  atm/mol  K and NA = 6.02  1023 mol–1. The approximate number

of air molecules in a 1 m3 volume at room temperature (300 K) and atmospheric pressure is:

A) 41

B) 450

C) 2.4  1025

D) 2.7  1026

E) 5.4  1026

15. An air bubble doubles in volume as it rises from the bottom of a lake (1000 kg/m3).

Ignoring any temperature changes, the depth of the lake is:

A) 21 m

B) 0.76 m

C) 4.9 m

D) 10 m

E) 0.99 m

16. An ideal gas undergoes an isothermal process starting with a pressure of 2  105 Pa and a

volume of 6 cm3. Which of the following might be the pressure and volume of the final state?

A) 1  105 Pa and 10cm3

B) 3  105 Pa and 6 cm3

C) 4  105 Pa and 4 cm3

D) 6  105 Pa and 2 cm3

E) 8  105 Pa and 2 cm3

17. The pressures p and volumes V of the five ideal gases, with the same number of molecules,

are given below. Which has the highest temperature?

A) p = 1  105 Pa and V = 10cm3

B) p = 3  105 Pa and V = 6 cm3

C) p = 4  105 Pa and V = 4 cm3

D) p = 6  105 Pa and V = 2 cm3

E) p = 8  105 Pa and V = 2 cm3

18. A given mass of gas is enclosed in a suitable container so that it may be maintained at

constant volume. Under these conditions, there can be no change in what property of the gas?

A) Pressure

B) Density

C) Molecular kinetic energy

D) Internal energy

E) Temperature

19. In order that a single process be both isothermal and occur at constant pressure:

A) one must use an ideal gas

B) such a process is impossible

C) a change of phase is essential

D) one may use any real gas such as N2

E) one must use a solid

20. Over 1 cycle of a cyclic process in which a system does net work on its environment:

A) the change in the pressure of the system cannot be zero

B) the change in the volume of the system cannot be zero

C) the change in the temperature of the system cannot be zero

D) the change in the internal energy of the system cannot be zero

E) none of the above

21. What is the relationship between the ideal gas constant R and the Boltzmann constant k?

A) R = nk/N

B) R = Nk/n

C) R = n/Nk

D) R = N/nk

E) depends on the molar specific heat of the gas

22. In using the ideal gas law, the temperature T must be measured in:

A) Celsius

B) Kelvin

C) Fahrenheit

D) either Celsius or Kelvin

E) any units as long as the correct values of k or R are used

23. An isothermal process for an ideal gas is represented on a p-V diagram by:

A) a horizontal line

B) a vertical line

C) a portion of an ellipse

D) a portion of a parabola

E) a hyperbola

24. A real gas undergoes a process which can be represented as a curve on a p-V diagram. This

curve is an isotherm if:

A) the volume of the gas does not change

B) the temperature of the gas does not change

C) the pressure of the gas does not change

D) the gas does no work on its environment

E) the gas exchanges no heat with its environment

25. A real gas undergoes a process which can be represented as a curve on a p-V diagram. The

work done by the gas during this process is:

A) pV

B) p(V2 – V1)

C) (p2 – p1)V

D)  p dV

E) V dp

26. The energy absorbed as heat by an ideal gas for an isothermal process equals:

A) the work done by the gas

B) the work done on the gas

C) the change in the internal energy of the gas

D) the negative of the change in internal energy of the gas

E) zero since the process is isothermal

27. When an ideal gas undergoes a slow isothermal expansion:

A) the work done by the gas is the same as the energy absorbed as heat

B) the work done by the environment is the same as the energy absorbed as heat

C) the increase in internal energy is the same as the heat absorbed

D) the increase in internal energy is the same as the work done by the gas

E) the increase in internal energy is the same as the work done by the environment

28. The pressure of an ideal gas is doubled during a process in which the energy given up as

heat by the gas equals the work done on the gas. As a result, the volume is:

A) doubled

B) halved

C) unchanged

D) need more information to answer

E) nonsense, the process is impossible

29. A real gas is changed slowly from state 1 to state 2. During this process no work is done on

or by the gas. This process must be:

A) isothermal

B) adiabatic

C) occurring at constant volume

D) occurring at constant pressure

E) a closed cycle with point 1 coinciding with point 2

30. A quantity of an ideal gas is compressed to half its initial volume. The process may be

adiabatic, isothermal or occurring at constant pressure. Rank those three processes in order of the

work required of an external agent, least to greatest.

A) adiabatic, isothermal, constant pressure

B) adiabatic, constant pressure, isothermal

C) isothermal, adiabatic, constant pressure

D) constant pressure, adiabatic, isothermal

E) constant pressure, isothermal, adiabatic

31. The speeds of 25 molecules are distributed as follows: 5 in the range from 2 to 3 m/s, 10 in

the range from 3 to 4 m/s, 5 in the range from 4 to 5 m/s, 3 in the range from 5 to 6 m/s, 1 in the

range from 6 to 7 m/s, and 1 in the range from 7 to 8 m/s. Their average speed is about:

A) 2 m/s

B) 3 m/s

C) 4 m/s

D) 5 m/s

E) 6 m/s

32. According to the kinetic theory of gases, the pressure of a gas is due to:

A) change of kinetic energy of molecules as they strike the wall

B) change of momentum of molecules as they strike the wall

C) average kinetic energy of the molecules

D) force of repulsion between the molecules

E) rms speed of the molecules

33. The force on the walls of a vessel of a contained gas is due to:

A) repulsive force between gas molecules

B) slight loss in average speed of a gas molecule after collision with wall

C) change in momentum of a gas molecule due to collision with wall

D) elastic collisions between gas molecules

E) inelastic collisions between gas molecules

34. A gas is confined to a cylindrical container of radius 1 cm and length 1 m. The pressure

exerted on an end face, compared with the pressure exerted on the long curved face, is:

A) smaller because its area is smaller

B) smaller because most molecules cannot traverse the length of the cylinder without

undergoing collisions

C) larger because the face is flat

D) larger because the molecules have a greater distance in which to accelerate before they

strike the face

E) none of these

35. Air is pumped into a bicycle tire at constant temperature. The pressure increases because:

A) more molecules strike the tire wall per second

B) the molecules are larger

C) the molecules are farther apart

D) each molecule is moving faster

E) each molecule has more kinetic energy

36. Five molecules have speeds of 2.8, 3.2, 5.8, 7.3, and 7.4 m/s. Their root-mean-square speed

is closest to:

A) 2.5 m/s

B) 5.3 m/s

C) 5.7 m/s

D) 28 m/s

E) 32 m/s

37. In a system of N gas molecules, the individual speeds are v1, v2, …, vN. The rms speed of

these molecules is:

A) 1

𝑁√𝑣1+ 𝑣2+ ⋯ + 𝑣𝑁

B) 1

𝑁√𝑣1

2+ 𝑣2

2+ ⋯ + 𝑣𝑁

2

C) √(𝑣1

2+ 𝑣2

2+ ⋯ + 𝑣𝑁

2)/𝑁

D) √[(𝑣1+ 𝑣2+ ⋯ + 𝑣𝑁)/𝑁]2

E) √(𝑣1+ 𝑣2+ ⋯ + 𝑣𝑁)2/𝑁

38. The root-mean-square speed of molecules in a gas is:

A) the most probable speed

B) that speed such that half the molecules are moving faster than vrms and the other half are

moving slower

C) the average speed of the molecules

D) the square root of the square of the average speed

E) none of the above

39. Oxygen has a molar mass of 32 g/mol. If 12 moles of oxygen are in a 0.1-m3 container with

an rms speed of 480 m/s, what is the pressure of the gas?

A) 2.9 x 105 Pa

B) 2.1 x 106 Pa

C) 3.4 x 107 Pa

D) 2.9 x 108 Pa

E) 2.1 x 109 Pa

40. The pressure of an ideal gas is doubled in an isothermal process. The root-mean-square

speed of the molecules:

A) does not change

B) increases by a factor of √2

C) decreases by a factor of 1/√2

D) increases by a factor of 2

E) decreases by a factor of 1/2

41. The temperature of low pressure hydrogen is reduced from 100C to 20C. The rms speed

of its molecules decreases by approximately:

A) 89%

B) 79%

C) 46%

D) 21%

E) 11%

42. The mass of an oxygen molecule is 16 times that of a hydrogen molecule. At room

temperature, the ratio of the rms speed of an oxygen molecule to that of a hydrogen molecule is:

A) 16

B) 4

C) 1

D) 1/4

E) 1/16

43. The rms speed of an oxygen molecule at 0C is 460 m/s. If the molar mass of oxygen is 32

g and of helium is 4 g, then the rms speed of a helium molecule at 0C is:

A) 160 m/s

B) 330 m/s

C) 650 m/s

D) 1300 m/s

E) 3700 m/s

44. A sample of argon gas (molar mass 40 g) is at four times the absolute temperature of a

sample of hydrogen gas (molar mass 2 g). The ratio of the rms speed of the argon molecules to

that of the hydrogen is:

A) 1

B) 5

C) 1/5

D) √5

E) 1/√5

45. If the molecules in a tank of hydrogen have the same rms speed as the molecules in a tank

of oxygen, we may be sure that:

A) the pressures are the same

B) the hydrogen is at the higher temperature

C) the hydrogen is at the greater pressure

D) the temperatures are the same

E) the oxygen is at the higher temperature

46. A system consists of N gas molecules, each with mass m. Their rms speed is vrms. Their

total translational kinetic energy is:

A) (1/2)m(Nvrms)2

B) (1/2)N(mvrms)2

C) (1/2)mv2rms

D) (1/2)Nmv2rms

E) N[(1/2)mvrms]2

47. An ideal gas is at a temperature of 320 K. What is the average translational kinetic energy of

one of its molecules?

A) 9.2 x 10–24 J

B) 1.4 x 10–23 J

C) 2.1 x 10–23 J

D) cannot tell without knowing the molar mass

E) cannot tell without knowing whether the gas is monatomic or diatomic

48. The temperature of a gas is most closely related to:

A) the kinetic energy of translation of its molecules

B) its total molecular kinetic energy

C) the sizes of its molecules

D) the potential energy of its molecules

E) the total energy of its molecules

49. In a certain gas the molecules are 5.0  10–9 m apart on average, have a mean free path of

5.0  10–6 m, and have an average speed of 500 m/s. The rate at which a molecule has collision

with other molecules is about:

A) 10−11 s−1

B) 10−8 s−1

C) 1 s−1

D) 108 s−1

E) 1011 s−1

50. Evidence that molecules of a gas are in constant motion is:

A) winds exert pressure

B) two gases interdiffuse quickly

C) warm air rises

D) energy as heat is needed to vaporize a liquid

E) gases are easily compressed

51. The mean free path of a gas molecule is:

A) the shortest dimension of the containing vessel

B) the cube root of the volume of the containing vessel

C) approximately the diameter of a molecule

D) average distance between adjacent molecules

E) average distance a molecule travels between intermolecular collisions

52. The mean free path of molecules in a gas is:

A) the average distance they travel before escaping

B) the average distance they travel between collisions

C) the greatest distance they travel between collisions

D) the shortest distance they travel between collisions

E) the average distance they travel before splitting apart

53. The average speeds v and molecular diameters d of five ideal gases are given below. The

number of molecules per unit volume is the same for all of them. For which is the collision rate

the greatest?

A) v = v0 and d = d0

B) v = 2v0 and d = d0/2

C) v = 3v0 and d = d0

D) v = v0 and d = 2d0

E) v = 4v0 and d = d0/2

54. The mean free path of air molecules at room temperature and atmospheric pressure is

about:

A) 10–3 m

B) 10–5 m

C) 10–7 m

D) 10–9 m

E) 10–11 m

55. The mean free path of molecules in a gas is proportional to:

A) the molecular cross-sectional area

B) the reciprocal of the molecular cross-sectional area

C) the root-mean-square molecular speed

D) the square of the average molecular speed

E) the molar mass

56. The mean free path of molecules in a gas is proportional to:

A) the molecular diameter

B) the reciprocal of the molecular diameter

C) the molecular concentration

D) the reciprocal of the molecular concentration

E) the average molecular speed