Physical Chemistry , 1st ed.

This integral has a known form and can be simplified.* We get

average velocity  

2

k

T

m



1/2

(19.49)

Thus we have

rate of effusion 

d

d

N

t

 

2

k

T

m



1/2

The only other factors involved are the number (density) of gas particles in the

system,N/Vor , and the area Athat the gas particles are actually passing

through. Both factors are directly related to the effusion rate: as either in-

creases, the effusion rate increases. We can therefore write



d

d

N

t

A 

2

k

T

m



1/2

(19.50)

as the rate of effusion of gas particles through an area A. Using the ideal gas

law pVNkT, we can substitute for N/Vand get



d

d

N

t

A

k

p

T

 

2

k

T

m



1/2

which simplifies to



d

d

N

t

Ap 

2 m

1

kT



1/2

(19.51)

Curiously, a part of this expression is very similar to part of the translational

partition function of a monatomic gas, and is proportional to the thermal de

Broglie wavelength (see Chapter 17).

Example 19.8

Iron metal is confined to a small cell that has a small hole of 0.5 mm diameter.

At a temperature of 2050 K, the vapor pressure of the Fe inside the cell is

1.00 mmHg. Estimate the number of atoms of Fe that effuse through the hole

per second.

Solution

As with many of these examples, having consistent units is important. The

area of the hole is [(0.5 mm)/2]^2 , or 0.196 mm^2 , which is 1.96  10 ^7 m^2.

A pressure of 1.00 mmHg should be converted to pascals (1 Pa 1 N/m^2 

0.00750 mmHg), making it 133 Pa. Using the mass of a single Fe atom:

19.5 Effusion and Diffusion 673

*Specifically, one would perform a substitution and recognize the resulting integral as

the form





0

xneaxpdx^

p

(

a

k

k

)

k

n

p

^1

(k) is called a gamma functionand is a well-known mathematical function. By definition,

(1) 1. Apply this formula to equation 19.48 and see if you can get equation 19.49. Do

not confuse this use of the gamma variable with the three-dimensional probability function!



d

d

N

t

(1.96  10 ^7 m^2 )  133 

m

N

 (^2)

1 1/2



2 (9.27  10 ^26 kg)(1.381  10 ^23 J/K)(2050 K)