18.8. Solve: (a) The average speed is

25

avg

15

1 220 m/s m/s 20.0 m/s

11 11

n

n

v n

(b) The root-mean-square speed is

2

rms avgv v

1225

2

15

1 m/s

11

n

n

n

124510

20.2 m/s11

18.9. Solve: (a) In tabular form we have

Particle vx (m/s) vy (m/s) 2xv (m/s)

2 2

yv (m/s)2 v

2 (m/s)

2 v (m/s)

1 20 30 400 900 1300 36.06

2 40 70 1600 4900 6500 80.62

3 80 10 6400 100 6500 80.62

4 60 20 3600 400 4000 63.25

5 0 50 0 2500 2500 50.00

6 40 20 1600 400 2000 44.72

Average 0 0 3800 59.20

The average velocity is avg 0 0v i j .

(b) The average speed is avg 59 m/sv .

(c) The root-mean-square speed is 2 2 2rms avg3800 m /s 62 m/sv v .

18.10. Solve: (a) The most probable speed is 4.0 m/s. (b) The average speed is

avg

2 2 m/s 4 4 m/s 3 6 m/s 1 8 m/s4.6 m/s

2 4 3 1v

(c) The root-mean-square speed is

2 2 2 2

rms

2 2 m/s 4 4 m/s 3 6 m/s 1 8 m/s4.9 m/s

2 4 3 1v

18.11. Solve: (a) The atomic mass number of argon is 40. This means the mass of an argon atom is

27 2640 u 40(1.661 10 kg) 6.64 10 kgm

The pressure of the gas is

21rms3

Np mv

V

225 3 261

32.00 10 m 6.64 10 kg 455 m/s 9.16 10

4 Pa

(b) The temperature of the gas in the container can be obtained from the ideal-gas equation in the form pV NkBT:

4

25 3 23B

9.16 10 Pa332 K

2.00 10 m 1.38 10 J/K

pVT

Nk

18.20. Solve: The formula for the root-means-square speed as a function of temperature is

Brms T

3k Tv

m

The ratio at 20 C and at 100 C is

rms 100

rms 20

373 K1.13

293 K

v

v

18.21. Solve: Assuming ideal-gas behavior and ignoring relativistic effects, the root-mean-square speed of a molecule is

Brms

3k Tv

m

The temperature where vrms is the speed of light (c) for a hydrogen molecule

227 822

12rms

23B B

2 1.66 10 kg 3.0 10 m/s2 u7.22 10 K

3 3 3 1.38 10 J/K

cmvT

k k