Biomedical Engineering Reference
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where M i and M j are the molecular weights of the gases, T is the absolute temperature, and p is the
pressure. The collision diameter s ij is the arithmetic average of the characteristic diameter of the gas
molecules s i and s j :
s i þ
s j
s ij ¼
:
(2.35)
2
can be taken from the diagram depicted in Fig. 2.1 . In this diagram, the
dimensionless temperature is calculated based on the interaction energy, which is the geometric
average of the individual characteristic energies:
The collision integral
U
3 i 3 j p (2.36)
Example 2.5 ( Estimation of diffusion coefficient of gases ). Estimate the diffusion coefficient of
hydrogen in air at 282 K. Lennard-Jones potential parameters of air and hydrogen are ( s 1 ¼
0.3711 nm,
3 ij ¼
, respectively. The molecular
weights of air and hydrogen are 29 and 2 respectively. The experimental value is 0.710
ð˛=
k B Þ 1 ¼
78
:
6
Þ
and ( s 2 ¼
0.2827 nm,
ð˛=
k B Þ 2 ¼
59
:
7
Þ
10 4 m 2 /sec.
Solution. Although air consists mainly of oxygen and nitrogen, we assume air as a gas molecule.
According to (2.35) , the collision diameter between air and hydrogen is:
10 9 m
s 12 ¼ð
s 1 þ
s 2 Þ=
¼ð
:
þ
:
Þ=
¼
:
:
2
0
3711
0
2827
2
0
3269
According to (2.36) , the dimensionless temperature is:
k B T
3 12 ¼
k B T
3 i 3 2
T
ð
282
78
p
¼
p
¼
p
¼
4
:
12
:
3 1 =
k B Þð
3 1 =
k B Þ
:
6
59
:
7
According to Fig. 2.1 , the collision potential is approximately 0.88. Thus, the diffusion coefficient
of hydrogen in air at 282 K is:
p
1
p
1
10 27 T 3 = 2
10 27 282 3 = 2
1
:
86
=
M i þ
1
=
M j
1
:
86
=
29
þ
1
=
2
D
¼
¼
ps ij U
2
1
ð
0
:
3269
Þ
ð
0
:
88
Þ
10 5 m 2
:
The estimated diffusion coefficient is 33% lower than the measured data.
¼
4
:
76
=
sec
2.2.3.2 Diffusion coefficient in liquids
While diffusion coefficients of gases are on the order of 10 5 m 2 /sec, diffusion coefficients of liquids
are on the order of 10 9 m 2 /sec. The diffusion coefficients of large molecules can be on the order of
10 11 m 2 /sec. Diffusion coefficient of a molecule i in a solute j with a viscosity m j can be estimated by
the Stokes-Einstein equation:
k B T
3 pm j s i ;
D ij ¼
(2.37)
where s i is the diameter of the molecule i . In the above equation, the term in the numerator represents
the kinetic energy of the molecule, while the denominator represents the friction force acting on the
molecule. Equation (2.37) breaks down if the size of the solute i is 5 times less than that of the solvent.
For a small solute, the factor 3 p in (2.37) can be replaced by the factor 2 p because there is less friction
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