Physical Chemistry Third Edition

9.4 The Distribution of Molecular Speeds 409

Exercise 9.13

Find the most probable speed of helium atoms at 298 K.

Themean speedis given by

〈v〉

∫∞

0

vfv(v)dv 4 π

(

m

2 πkBT

) 3 / 2 ∫∞

0

v^3 e−mv

(^2) / 2 kBT
dv
〈v〉

√

8 kBT

πm



√

8 RT

πM

(9.4-6)

Exercise 9.14

Look up the integral and verify Eq. (9.4-6).

EXAMPLE9.10

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.

Themean-square speedis defined to be

〈

v^2

〉

 4 π

(

m

2 πkBT

) 3 / 2 ∫∞

0

v^4 e−mv

(^2) / 2 kBT
dv
3 kBT
m



3 RT

M

(9.4-7)

where we have looked up the integral in the table of Appendix C. Theroot-mean-square

speedvrmsis the square root of the mean-square speed:

vrms

〈

v^2

〉 1 / 2



√

3 kBT

m



√

3 RT

M

(9.4-8)

EXAMPLE9.11

Find the value of the root-mean-square speed of oxygen molecules at 298 K.

vrms

⎛

⎝

( 3 )

(

8 .3145 J K−^1 mol−^1

)

(298 K)

(

0 .0320 kg mol−^1

)

⎞

⎠

1 / 2

482 m s−^1