Physical Chemistry Third Edition

402 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

This integral is in the form of theerror function, defined in Appendix C by^1

erf (x)

2

√

π

∫x

0

e−t

2

dt (9.3-38)

Our probability is therefore

(probability)

1

2

erf (1.0055) 0. 4225  42 .25%

Where the numerical value was obtained from the table of the error function in Appendix C.

We can obtain an approximate probability of a small finite range∆vxby replacing

the infinitesimal intervaldvxin Eq. (9.3-5) by∆vx:

(

probability that

vxlies in∆vx

)

≈f(v′x)∆vx (for small∆vx) (9.3-39)

wherev′xis a value ofvxwithin the range∆vx.

EXAMPLE 9.6

Find the probability thatvxfor an argon atom in a system at 273.15 K is in the range

650.00 m s−^1 <vx<651.00 m s−^1.

Solution

From Eq. (9.3-39)

probability≈

(

m

2 πkBT

) 1 / 2

e−mv

x^2 /^2 kBT

∆vx

(

M

2 πRT

) 1 / 2

e−Mv

x^2 /^2 RT

∆vx

≈

(

0 .039948 kg mol−^1

2 π(8.3145 J K−^1 mol−^1 )(273.15 K)

) 1 / 2

×exp

(

−(0.039948 kg mol−^1 )(650 m s−^1 )^2

2(8.3145 J K−^1 mol−^1 )(273.15 K)

)

(1.00 m s−^1 )

≈ 4. 05 × 10 −^5

Exercise 9.9

Find the probability thatvxfor an argon atom in a system at 273.15 K is in the range 650 m s−^1

<vx<652ms−^1.

Exercise 9.10

a.What fraction of the molecules has x-components of the velocity between −σvx

andσvx?

(^1) M. Abramowitz and I. A. Stegun, eds.,Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, U.S. Govt. Printing Office, Washington, DC, 1964, p. 297ff.