Physical Chemistry Third Edition

9.6 Effusion and Wall Collisions 417

Therefore

⎛

⎝

total number

striking area

Ain timeτ

⎞

⎠NAτ

(

kBT

2 πm

) 1 / 2



1

4

NAτ

(

8 kBT

πm

) 1 / 2



1

4

NAτ〈v〉 (9.6-5)

The number of particles that strike the areaAin timeτis proportional to the areaA,

proportional to the length of timeτ, proportional toN, the number of particles per unit

volume, and proportional to the mean speed of the particles. The number of molecules

striking unit area per unit time is denoted byν:

ν

1

4

N〈v〉

N

4

(

8 kBT

πm

) 1 / 2

(9.6-6)

This equation for the rate of wall collisions also gives the rate of effusion per unit area

in the case of a small hole in the wall. The effusion rate predicted by Eq. (9.6-6) is

inversely proportional to the square root of the mass of the particles, and thus to the

square root of the density, in agreement with Graham’s law of effusion.

EXAMPLE9.13

Estimate the number of molecules of air striking a person’s eardrum in 1.00 s at 298 K and

1.00 atm. Assume air to be 79 mol% nitrogen and 21 mol% oxygen by moles, and assume

that the area of the eardrum is 0.50 cm^2.

Solution

〈v(O 2 )〉444ms−^1 from Example 9.4 and〈v(N 2 )〉475ms−^1 by a similar calculation.

N(N 2 )N(N 2 )/VP(N 2 )/kBT



(

101325 N m−^2 atm−^1

)

( 0 .79 atm)

(

1. 3807 × 10 −^23 JK−^1

)

(298 K)

 1. 9 × 1025 m−^3

whereP(N 2 ) is the partial pressure of N 2 ,N(N 2 ) is the number of N 2 molecules and

N(N 2 ) is the number density of N 2 molecules.

N(O 2 )N(O 2 )/V 5. 2 × 1024 m−^3 by a similar calculation

ν(N 2 )

1

4

(1. 9 × 1025 m−^3 )(475 m s−^1 ) 2. 3 × 1027 m−^2 s−^1

ν(O 2 ) 5. 7 × 1026 m−^2 s−^1 by a similar calculation

(number per second)ν(N 2 )+ν(O 2 )(5. 0 × 10 −^5 m^2 ) 1. 4 × 1023 s−^1

Exercise 9.19

a.Estimate the number of air molecules striking the palm of your hand in 24 hours.

b.A certain solid catalyst has a surface area of 55 m^2 per gram of catalyst. A mixture of gases

with carbon monoxide mole fraction equal to 0.0050 passes over the catalyst at 350 K and

1.00 atm. Find the amount of CO in moles striking 1 gram of the catalyst per second.