Biomedical Engineering Reference
In-Depth Information
Figure 2.5
Free diffusion in the exchange of oxygen and carbon dioxide in various parts of the
body.
to apply the diffusion equation through the use of some macroscopically effective
properties, such as the local average concentration, and free solution diffusivity.
Using the fundamentals of the equation of motion (described in Chapter 4) for the
diffusivity of a rigid spherical solute of radius
r
se
in a dilute solution, Einstein (also
known as Stokes-Einstein relation) arrived at the expression
RT
D
∞
=
(2.9)
6
πμ
rN
se
A
where
D
is called free solution diffusivity [m
2
/s],
N
A
is Avogadro's number (6.023
∞
×
s, Pa.s or N.s/m
2
). Since
the gas constant is the product of Boltzmann's constant and Avogadro's number,
(2.9) is also written in terms of Boltzmann's constant. Stokes radius rse is also
referred to as the hydrodynamic radius of the diffusing molecule whose shape is
assumed to be rigid and spherical. Equation (2.9) has been shown to provide good
diffusivity predictions for large spherical molecules or particles in solvents, which
show less slippage. Equation (2.9) is also used to estimate the effective Stokes radius
of the solute if its diffusivity is known. Values for a few of the molecules are given
in Table 2.2.
To determine the diffusivity of a solute, its effective Stokes radius is estimated
assuming that it is equal to the molecular radius of the solid spherical solute, which
gives the expression
10
23
molecules/mol), and
μ
is the solvent viscosity (kg/m
⋅
1/3
⎛
⎞
3
4
MW
r
=
(2.10)
⎜
⎟
se
⎝
πρ
N
⎠
A
