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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5948_Библиотеки_им_академика_М_И_Перельмана
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CHAPTER 6
Consolidation Parameters
Diffusion parameters, Dissolution parameters and Pharmacokinetic parameters, Heckel plots,
Similarity factors – f2 and f1, Higuchi and Peppas plot, Linearity Concept of significance,
Standard deviation, Chi square test, students’ t-test, ANOVA test.
Diffusion Process
Diffusion is a process. It can be defined as a process of mass transfer of molecules of a
substance brought about by random molecular motion and is controlled by a driving force of
concentration gradient . Release of drug from a variety of drug delivery system, absorption
and elimination of drug, dialysis, osmosis, and ultra-filtration are some of the examples of
diffusion process. Many pharmaceutical phenomena are related to such mass transfer of a
solvent such as water or a solute such as drug. For example, diffusion of a drug across the
biological membrane is necessary for its absorption into and elimination from the body, even
after reaching the site of action within a particular cell. If the container or closer fails to
preserve the drug, or if the container fails to prevent the transmission of water vapor from
the container, the shelf-life of the drug product can be reduced. Thus, diffusion is involved in
many important processes with necessary thermodynamic background 1 .
Fig. 6.1 Diffusion of molecules of dye in water
If a drop of dye is put in a beaker containing water at a particular temperature, the dye tends
to diffuse throughout the water; ultimately the water becomes a uniform-colored solution. At
the molecular level, the dye molecules are in a continuous random motion. Thus, each
molecule of dye can move in any direction with equal probability. Initially large number of
dye molecules move away from the source. At equilibrium, when the color is uniformly
distributed throughout the water, no net movement of dye molecule can be detected. The
process can be illustrated in Fig 6.1.
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Thus, diffusion is a process of spontaneous transference of solute from the region of higher
concentration to the region of lower concentration until the solute is uniformly distributed
throughout the medium .
Fick’s laws of diffusion
The rate of diffusion of a solute is expressed by Fick’s law as:
Where, dm = Amount of solute diffusing in time, dt.
A = Area under the influence of concentration gradient, dc/d x ,
D = Diffusion coefficient (area per unit time)
Diffusion is not constant; it varies with concentration and on temperature. The value of D
determined from any measurement should be considered as a mean value for the
concentration range. The diffusion coefficient for spherical particles of colloidal dimensions
can be given by
Where,
r is the radius of the spherical particle,
η is the viscosity of the liquid medium,
R is the gas constant,
T is the thermodynamic temperature, and
N A is the Avogadro’s constant.
The equation 6.2 is applicable only for spherical particle. The rates of diffusion of small
molecules and ions are generally greater than those of colloidal particles. Since diffusion of
the former particles is hindered by the viscosity of the medium, the eqn.6.2 is not applicable
to such system.
Diffusion is an effective transport mechanism over small distances. To evaluate the validity
of this parameter, it is necessary to have some means of calculating the distance a molecule
or a particle moves in a particular duration of time. Firstly, it may be thought that
determination of the mean distance travelled by the particle would be useful. But in a
diffusion process, the molecules move randomly. So the mean distance moved may be zero.
This problem can be overcome by calculating the mean squared distance travelled in a
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particular time interval and then taking the positive square root of this value. A relation
between the mean squared distance and the time can be found by using the general equation
of Fick’s second law and solving it specifically for an instantaneous plane source placed in
liquid perpendicular to the direction of mass transport as shown in Fig 6.2.
Fig. 6.2 Typical plot of normalized concentration, C/no against position, x
The Fig 6.2 shows that the diffusion is taking place from a plane perpendicular source. The
curves t 1 and t 2 represent the distribution of diffusing molecules at times t 1 and t 2 . At t =0,
all the diffusing molecules (no) are present at x = 0. This represents a volume equal to zero
so that the concentration at t = 0 is infinity.
For this case, the general solution to Fick’s second law can be made as follows:
Where, k o and D are constants. By substituting the eqn. 6.3 into the Fick’s second law, the
validity of this equation can be verified. If the total number of solute molecules involved, ko
can be determined as:
Let P(x, t) dx is the probability that a solute molecule will be found between (x) and (x +
dx). By using this definition of probability, P(x, t) dx is equal to the number of solute
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molecules present in the space between (x) and (x + dx) divided by the total number of
solute molecules (n o ).
Given that the definition of the mean squared distance x 2 is just the sum over all x 2 weighted
by its probability of occurrence, we have
Substituting for P(x, t) dx from eqn. 6.10, the eqn.6.11 becomes
The eqn. 6.12 can be written in reduced form as
The equation 6.13 relates the mean squared distance moved in a time t to the diffusion
coefficient, D, was first derived by Einstein 2 . Typical diffusion coefficients for molecules in
liquids or membranes range from 10
–5
to 10
–12
cm 2 sec
–1
.
Example: What would be the time required for a protein molecule of molecular weight
20000 to cross a RBC whose diameter is 7 × 10
–4
cm or to move a length of a 10 cm long
nerve cell? Assume that the diffusion coefficient of the protein is 8 × 10
–7
cm 2 sec
–1
.
Therefore, diffusion is an efficient means of transport, provided the distance to be traversed
is small.
Size of molecule and its ability to diffuse
It has been observed that if the molecular weight of solutes increases, the ability of the
molecule to diffuse would decrease. This makes a sensitive observation because with
increasing molecular weight, the frictional resistance to its movement will be increased. This
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concept was first placed by Stokes. The calculations made by Stokes are based on the
following assumptions:
The diffusing solute molecule can be considered as a slowly moving solid sphere.
The solute molecule is larger than the solvent molecule. For this reason, it can be considered
that the solvent is the continuous phase.
Steady-state conditions are also attained.
The solvent is isotropic and incompressible.
With these assumptions the following result (Stokes’ law) is obtained
Where,
F f is the frictional forces,
r is the radius of the solute molecule,
V is the velocity of the solute molecule, and
η is the viscosity of the diffusing medium
The frictional force can also be expressed as
Where,
f is the frictional coefficient.
Since f = 6πrη, and
Then,
The equation 6.15 is known as Stokes-Einstein equation.
Example: A spherical colloidal particle has a diameter of 1× 10
–7
cm. Calculate how long it
will take this particle to diffuse through 1 cm in water at 20 o C. The viscosity of water is
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1.002×10
–3
kg/m/sec and Boltzmann’s constant is 1.38×10
–23
JK
–1
. (J = kg.m 2 .sec
–2
)
Solution:
Diffusion is practically a slow process. If a molecule is spherical, then its molecular weight
(MW) would be
Where, r is the van der Waal’s radius of the molecule, and
ρ is the molecular density.
It can be considered that D does not depend fairly on the changes in molecular weight, MW.
This is true, particularly when drug molecules have the molecular weight in the narrow
range of 100 – 500. In some cases, calculated diffusion coefficients vary clearly from the
experimental values. Because in such cases, the assumptions based on which Stokes’ law is
derived, are not valid. For example, the shape of DNA molecule is rod, not spherical and the
size of CH 3 OH molecule is similar to that of the solvent water (H 2 O).
Solute binding and its effect on diffusion
When a molecule diffuses through a membrane lattice, the possibility of binding may be
found. Fick’s first law is applicable and valid only when free diffusion of solute molecules
takes place. The effect of binding can be considered as:
Where,
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C f = concentration of freely diffusible solute,
C b = concentration of bound solute.
If it is assumed that the concentration of freely diffusible solute (C f ) is directly proportional
to the concentration of bound solute (C b ), then
Where K’ is the proportionality constant. Now, in one dimension
Since, C = C f + C b (eqn 6.18) and C b = K' C f , (eqn. 6.19) we can write
Or, the eqn. 6.20 can be rewritten as
Therefore, reversible adsorption that takes place within the membrane can decrease the flux
by a factor of
The binding of solute may be the reason for smaller diffusion coefficients of water and other
polar molecules for the skin.
Variable diffusion coefficient and the concept of mean diffusion coefficient
There are some situations when the diffusion coefficient cannot be considered a constant.
For example, during hydration, when a polymer swells or the skin becomes hydrated after
application of an occlusive bandage. In both the cases, the diffusion coefficient can become a
function of time and/ or position and it becomes easy to replace the diffusion coefficient
which may be the function of the position, time, or concentration of the solute with its mean
value. Let the diffusion coefficient be a function of an arbitrary variable x with respect to
position, time, or concentration of solute. Then the mean value of diffusion coefficient can
be written as:
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The Fig 6.3 illustrates the physical meaning of Where, is the mean diffusion
coefficient over the time interval from x 2 to x 1 . From the Fig 6.3 it is understood that
represents the area under the rectangle of height and width ( x 2 – x
1
). As such represents the average value of D( x ). This technique 3 can be utilized
when transportation of drug across the skin is being investigated.
By analogy to eqn. 6.23 we can write
Fig. 6.3 A plot or diffusion coefficient vs. x, position, time, or concentration of solute
But the range of x covers all possible values of x . Thus, x 1 = – ∞ and x 2 = ∞
Now, the integral of a probability density [P( x )] over all possible values must be equal to
unity. Therefore, and
Dissolution Parameters
When a solid molecule (particle) dissolves in a solvent and transforms into a solution, the
process brings about a change in the environments of both solute and solvent. Generally, the
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solid molecules (particles) remain compacted among themselves. On the other hand, solvent
molecules remain separated from one another. That is, the intermolecular distance of
separation among liquid molecules is sufficiently large. Within that intermolecular space of
liquid, solute molecule can easily be accommodated. In other words, the solute molecule
surrounded by other similar molecules gets separated and becomes surrounded by solvent
molecules. Thus, the force of attraction between solute-solvent molecules is much more than
the solute-solute molecules or solvent-solvent molecules. Such process is called as
dissolution . Thus, dissolution will take place only when the force of attraction between the
solute and solvent molecules would be to a level so that the intermolecular force of attraction
between solutesolute and solvent-solvent can be overcome.
The attractive forces exist between the polar molecules are much stronger than those exist
between the polar and nonpolar molecules.
In polar solutes the intermolecular interactions is substantial; hence, transfer of solute
molecules into solution would be possible only when there would be stronger solute-solvent
intermolecular interaction. This is normally feasible when the solvent is also a polar
substance such as water. A nonpolar solvent such as benzene is unable to exert sufficient
force of attraction on a solute molecule to separate it from other solute molecules.
On the other hand, in case of a nonpolar solute such as paraffin wax, the intermolecular
attractive forces are relatively weaker. The dissolution of such substance will occur when the
solute-solvent attractive force would be stronger than that existing between the solventsolvent molecules. Thus, a marked intermolecular association between the solvent molecules
(the relatively stronger intermolecular attractive force between polar solvent molecules) will
not be able to dissolve a nonpolar solute.
The above observations can be expressed in a general manner that ‘like dissolves like’. That
is, a polar solvent will dissolve a polar solute and a nonpolar solvent will dissolve a nonpolar
solute freely. The intermolecular forces involved in the process of dissolution are influenced
by hydrogen bonding which is more influential than polarity.
Factors that influence the dissolution of a solid (solute) in liquid (solvent)
Temperature
Usually, the dissolution of solute in a liquid occurs with absorption of heat. That is, the
temperature of the solution decreases. Such process of dissolution is called endothermic
process and the heat of solution is positive. The heat required to dissolve such solutes is
collected from the surroundings; that is, from the environment. If this type of system is
heated, the temperature of the system increases, and dissolution of solute becomes faster.
On the other hand, when heat is released by the system during dissolution process, the
process is called exothermic one. Thus, in an exothermic dissolution process the heat
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involved is called negative heat of solution. If the system is cooled (heat is absorbed by the
surroundings), the temperature of the system decreases, and the process of dissolution
becomes faster.
The Fig 6.4 shows the effect of temperature on the dissolution of few solutes such KNO 3 ,
Na 2 SO 4 , Na 2 SO 4 . 10H 2 O, NaCl, and (CH 3 COO) 2 . Ca, 2H 2 O. The figure shows that
the dissolution of potassium nitrate, sodium chloride and sodium sulphate decahydrate
increases with increase in temperature. That is, their dissolution requires a positive heat of
solution. While the dissolution of sodium sulphate anhydrous and calcium acetate dihydrate
decreases with increase in temperature. It indicates that the heat of solution involved in these
processes is negative.
Fig. 6.4 Effect of temperature on the process of dissolution
Most of the solubility curves are continuous curves; sometimes, abrupt changes in slopes
may be observed, if the nature of the solid phases in contact with solution changes. For
example, sodium sulphate can exist in decahydrate form up to a temperature of 32.55 o C
(305.55K) and its dissolution in water is an endothermic process with positive heat of
solution as shown in curve AB. When the temperature crosses this limit, sodium sulphate
decahydrate changes to anhydrous form and the dissolution also changes to exothermic
process with a negative heat of solution. Therefore, its solubility curve breaks at B and takes
the form of BC.
Particle size of the solute
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