9.4 The Distribution of Molecular Speeds
409
Exercise 9.13
Find the most probable speed of helium atoms at 298 K.
The mean speed is given by
v
∞
0
vf v (v) dv 4π
m
2πk B T
3/2 ∞
0
v
3 e
−mv 2 /2k B T dv
v
8k B T
πm
8RT
πM
(9.4-6)
Exercise 9.14
Look up the integral and verify Eq. (9.4-6).
E X A M P L E 9.10
Find the mean speed of oxygen molecules at 298 K.
v
(8)
8.3145 J K −1 mol −1
(298 K)
π
0.0320 kg mol −1
444 m s −1
This speed corresponds to 993 miles per hour or 1598 kilometers per hour.
The mean-square speed is defined to be
v
2
4π
m
2πk B T
3/2 ∞
0
v
4 e
−mv 2 /2k B T dv
3k B T
m
3RT
M
(9.4-7)
where we have looked up the integral in the table of Appendix C. The root-mean-square
speed v rms is the square root of the mean-square speed:
v rms
v 2 1/2
3k B T
m
3RT
M
(9.4-8)
E X A M P L E 9.11
Find the value of the root-mean-square speed of oxygen molecules at 298 K.
v rms
⎛
⎝
(3)
8.3145 J K −1 mol −1
(298 K)
0.0320 kg mol −1
⎞
⎠
1/2
482 m s −1
409
Exercise 9.13
Find the most probable speed of helium atoms at 298 K.
The mean speed is given by
v
∞
0
vf v (v) dv 4π
m
2πk B T
3/2 ∞
0
v
3 e
−mv 2 /2k B T dv
v
8k B T
πm
8RT
πM
(9.4-6)
Exercise 9.14
Look up the integral and verify Eq. (9.4-6).
E X A M P L E 9.10
Find the mean speed of oxygen molecules at 298 K.
v
(8)
8.3145 J K −1 mol −1
(298 K)
π
0.0320 kg mol −1
444 m s −1
This speed corresponds to 993 miles per hour or 1598 kilometers per hour.
The mean-square speed is defined to be
v
2
4π
m
2πk B T
3/2 ∞
0
v
4 e
−mv 2 /2k B T dv
3k B T
m
3RT
M
(9.4-7)
where we have looked up the integral in the table of Appendix C. The root-mean-square
speed v rms is the square root of the mean-square speed:
v rms
v 2 1/2
3k B T
m
3RT
M
(9.4-8)
E X A M P L E 9.11
Find the value of the root-mean-square speed of oxygen molecules at 298 K.
v rms
⎛
⎝
(3)
8.3145 J K −1 mol −1
(298 K)
0.0320 kg mol −1
⎞
⎠
1/2
482 m s −1