TITLE:
Determination
of Diffusion Coefficient
DATE OF EXPERIMENT:
3th May 2013
OBJECTIVES:
1.
To
study the diffusion of molecules in an agar medium.
2.
To
determine the diffusion coefficient of crystal violet and bromothymol blue at
different temperature.
INTRODUCTION:
Diffusion is a process of the
movement of solute molecules in the response to applied force, occurring in a
system which is not in equilibrium. The examples of applied forces are
temperature, concentration, pressure and electric current. According to Fick’s
law, the diffusion flux is proportional to the minus gradient of
concentrations. The solute molecules move from region of higher concentration
to region of lower concentration. Fick’s law of diffusion can be used to
calculate the diffusion coefficient. Fick’s law states that the flux of
material (amount dm in time dt) across a given plane (area A) is proportional
to the concentration gradient dc/dx.
(1)
dm = -DA(dc/dx)dt
where
D is the diffusion coefficient or diffusivity for the solute, m2s-1.
If a solution containing neutral
particles with the concentration M0, is placed within cylindrical
tube next to a water column, diffusion can be stated as
(2) M = M0 exp (–x2/4Dt)
where
M is the concentration at distance x from the intersection between water and
solution that is measure at time t.
By
changing equation (2) to its
logarithmic form, we get
ln M = ln M0 –x2/4Dt
OR
(3) 2.303 x 4D (log10 M0 –
log10 M) t = x2
Thus a
plot of x2 against t can produce a straight line that passes through
the origin with the slope 2.303 x 4D (log10 Ma – log10 M).
From here D can be calculated.
If the particles in the solution are
assumed to be spherical, their size and molecular weight can be calculated by
the Stokes-Einstein equation.
(4) D = kT/6pha
where
·
k
is the Boltzmann constant 1.38 x 1023 Jk-1,
·
T
is the temperature in Kelvin,
·
p is the viscosity of the solvent
in Nm-2s, and
·
a
is the radius of particle in M.
The
volume of a spherical particle is 4/3 pa3, thus its weight M is
equivalent to 4/3 pa3Nr (r = density)
It is
known that molecular weight M=mN (N is Avogadro’s number 6.023 x 1023
mol-1).
(5) M = 4/3 pa3Nr
If charged particles are involved
in the diffusion, equation (3) needs
to be modified to include potential gradient effect that exists between the
solution and solvent. However, this can be overcome by adding a little sodium
chloride into the solvent to prevent the formation of this potential gradient.
Agar gels contain a partially strong
network of molecules that are penetrated by water. The water molecules form a
continuous phase around the gel. Thus, the solute molecules can diffuse freely
in the water if chemical interactions and adsorption effects do not exist
entirely. Therefore, the gel forms a suitable support system to be used in
diffusion studies for molecules in a medium of water.
APPARATUS:
10ml,
50ml, 100ml and 250ml measuring cylinders, 1000ml beaker, 14 test tubes with
stoppers, weighing boat, spatula, glass rod, electronic balance, hot plate and
stirrer, test tube rack, dropper, filter funnel,marker and water bath.
MATERIALS:
Jelly
Powder, Ringlet’s solution, 1 : 200 crystal violet indicator, 1 : 400 crystal
violet indicator, 1 : 600 crystal violet indicator, 1 : 500 000 crystal violet
indicator, 1 : 200 bromothymol blue solution, 1 : 400 bromothymol blue
solution, 1 : 600 bromothymol blue solution, 1 : 500 000 bromothymol blue solution,
and distilled water.
PROCEDURES:
1.
Agar
in Ringer’s solution was prepared as four tablets were crushed by using pestle
and mortar.
2.
Then,
the crushed tablets were transfered into the 1000ml beaker and 500ml of
distilled water was added into the beaker to dissolve the tablets.
3.
7g
of agar powders were weighed on electronic balance and 425ml Ringer’s solution
was measured in a suitable measuring cylinders.
4.
Agar powders and Ringer’s solution were mixed
in the beaker. The mixed solution was heated until the solution completely
dissolved and boiled.
5.
The solution was continuously stirred during
the heating process to prevent the jelly powder from embed in the mixture.
6.
During the heating process, 14 test tubes were
labelled according to the concentrations and types of the solutes, which are
bromotyhmol blue and crystal violet. The heating process was stopped when the
mixture become clear dissolved solution.
7.
20ml of agar solution was prepared in 14
labelled test tubes.
8.
5ml
of 1:500 000 crystal violet solution was added into a test tube containing hot
agar solution labelled with 1:500 000 crystal violet at 28°C whereas 5ml of
1:500 000 bromothymol blue solution was added into test tube containing the hot
gel solution labelled with 1:500 00 bromothymol blue at 37°C. These two test
tubes will be used as the standard to measure the colour distance resulting
from the solute diffusion.
9.
The
agar solution in the rest of test tubes were allowed to cool at room
temperature until they become solid agar.
10.
5ml
of each crystal violet solution was added into the gels that were prepared in 6
test tubes according to their concentration.
11.
This
step was repeated by using bromothymol blue as an indicator. All test tubes
were closed to prevent evaporation and stored at temperature 28°C and 37°C.
12.
The
distance of crystal violet and bromothymol blue solution diffused in the agar
were measured and recorded for seven days.
RESULTS:
a) Crystal
Violet
System
|
Time (seconds)
|
x, cm
|
x2, cm2
|
Slope of graph
|
D, cm2s-1
|
Temp., o C
|
Average Diffusion Coefficient, D cm2s-1
|
1:200
|
0
|
0
|
0
|
3.057
X 10-5
|
9.7662 x
10-7
|
28
|
2.8652 x 10-7
|
86400
|
1.3
|
1.69
|
|||||
172800
|
2.1
|
4.41
|
|||||
259200
|
2.8
|
7.84
|
|||||
345600
|
3.3
|
10.89
|
|||||
432000
|
3.7
|
13.69
|
|||||
518400
|
4.1
|
16.81
|
|||||
604800
|
4.3
|
18.49
|
|||||
1:400
|
0
|
0
|
0
|
1.911
X 10-5
|
6.6986 x
10-7
|
28
|
|
86400
|
1.2
|
1.44
|
|||||
172800
|
1.8
|
3.24
|
|||||
259200
|
2.4
|
5.76
|
|||||
345600
|
2.7
|
7.29
|
|||||
432000
|
3.0
|
9.00
|
|||||
518400
|
3.3
|
10.89
|
|||||
604800
|
3.4
|
11.56
|
|||||
1:600
|
0
|
0
|
0
|
1.488X
10-5
|
5.5281 x
10-7
|
28
|
|
86400
|
0.75
|
0.56
|
|||||
172800
|
1.6
|
2.56
|
|||||
259200
|
2
|
4.0
|
|||||
345600
|
2.3
|
5.29
|
|||||
432000
|
2.5
|
6.25
|
|||||
518400
|
2.8
|
7.84
|
|||||
604800
|
3.0
|
9.0
|
|||||
1:200
|
0
|
0
|
0
|
3.652 X 10-5
|
1.16669
x10-6
|
37
|
9.2097 x 10-7
|
86400
|
1.6
|
2.56
|
|||||
172800
|
2.4
|
5.76
|
|||||
259200
|
3.0
|
9.00
|
|||||
345600
|
3.6
|
12.96
|
|||||
432000
|
4.0
|
16.00
|
|||||
518400
|
4.4
|
19.36
|
|||||
604800
|
4.7
|
22.09
|
|||||
1:400
|
0
|
0
|
0
|
2.646
X 10-5
|
9.27497
x 10-7
|
37
|
|
86400
|
1.4
|
1.96
|
|||||
172800
|
2.0
|
4.00
|
|||||
259200
|
2.6
|
6.76
|
|||||
345600
|
3.0
|
9.00
|
|||||
432000
|
3.4
|
11.56
|
|||||
518400
|
3.8
|
14.44
|
|||||
604800
|
4.0
|
16.00
|
|||||
1:600
|
0
|
0
|
0
|
1.800
X 10-5
|
6.6871 x 10-7
|
37
|
|
86400
|
1.0
|
1.00
|
|||||
172800
|
1.7
|
2.89
|
|||||
259200
|
2.2
|
4.84
|
|||||
345600
|
2.5
|
6.25
|
|||||
432000
|
2.8
|
7.84
|
|||||
518400
|
3.1
|
9.61
|
|||||
604800
|
3.3
|
10.89
|
 |
b) Bromothymol
Blue
System
|
Time (seconds)
|
x, cm
|
x2, cm2
|
Slope of graph
|
D, cm2S-1
|
Temp, o C
|
Average Diffusion Coefficient, D cm2S-1
|
A (1:200)
|
0
|
0
|
0
|
2.917 X 10-5
|
9.31886 x 10-7
|
28
|
1.25302 x
10-6
|
86400
|
1.5
|
2.25
|
|||||
172800
|
2.3
|
5.29
|
|||||
259200
|
2.7
|
7.29
|
|||||
345600
|
3.2
|
10.24
|
|||||
432000
|
3.6
|
12.96
|
|||||
518400
|
3.9
|
15.21
|
|||||
604800
|
4.2
|
17.64
|
|||||
B (1:400)
|
0
|
0
|
0
|
2.388 X 10-5
|
8.3703 x 10-7
|
28
|
|
86400
|
1.3
|
1.69
|
|||||
172800
|
1.9
|
3.61
|
|||||
259200
|
2.5
|
6.25
|
|||||
345600
|
2.9
|
8.41
|
|||||
432000
|
3.2
|
10.24
|
|||||
518400
|
3.6
|
12.96
|
|||||
604800
|
3.8
|
14.44
|
|||||
C (1:600)
|
0
|
0
|
0
|
5.357 X 10-6
|
1.99017 x 10-6
|
28
|
|
86400
|
0.4
|
0.16
|
|||||
172800
|
0.8
|
0.64
|
|||||
259200
|
1.1
|
1.21
|
|||||
345600
|
1.3
|
1.69
|
|||||
432000
|
1.4
|
1.96
|
|||||
518400
|
1.6
|
2.56
|
|||||
604800
|
1.8
|
3.24
|
|||||
D (1:200)
|
0
|
0
|
0
|
3.652 X 10-5
|
1.16669 x 10-6
|
37
|
1.3313 x 10-6
|
86400
|
1.7
|
2.89
|
|||||
172800
|
2.5
|
6.25
|
|||||
259200
|
3.1
|
9.61
|
|||||
345600
|
3.6
|
12.96
|
|||||
432000
|
4
|
16
|
|||||
518400
|
4.4
|
19.36
|
|||||
604800
|
4.7
|
22.09
|
|||||
E (1:400)
|
0
|
0
|
0
|
2.388 X 10-5
|
8.3703 x
10-7
|
37
|
|
86400
|
1.3
|
1.69
|
|||||
172800
|
2.0
|
4.00
|
|||||
259200
|
2.5
|
6.25
|
|||||
345600
|
2.9
|
8.41
|
|||||
432000
|
3.2
|
10.24
|
|||||
518400
|
3.5
|
12.25
|
|||||
604800
|
3.8
|
14.44
|
|||||
F (1:600)
|
0
|
0
|
0
|
8.003 X 10-6
|
1.9901 x 10-6
|
37
|
|
86400
|
0.7
|
0.49
|
|||||
172800
|
1.1
|
1.21
|
|||||
259200
|
1.4
|
1.96
|
|||||
345600
|
1.6
|
2.56
|
|||||
432000
|
1.9
|
3.61
|
|||||
518400
|
2.1
|
4.41
|
|||||
604800
|
2.2
|
4.84
|
|||||
CALCULATIONS:
From equation:
2.303 x 4D (log 10 Mo – log 10 M) t = X²
Hence the
slope of the graph = 2.303 x 4D (log 10 Mo - log 10 M)
1.
Crystal violet system with
dilution 1:200 (28 0C)
Slope= 3.057 x 10-5cm2/sec
M = 1:500000 M0 = 1:200
= 1
/ 500000 = 1 /
200
2.303x4D
[log 10 (5x10-3)-log 10 (2x10-6)] =
3.057×10-5 cm2/sec
D = 9.7662×10-7 cm2/sec
2.
Crystal violet system with dilution
1:400 (28ºC)
Slope = 1.911×10-5 cm2/sec
M = 1:500000 Mo = 1:400
= 1
/ 500000 = 1 /
400
= 2 x 10-6 = 2.5 x 10-3
2.303x4D
[log 10 (2.5x10-3)-log 10 (2x10-6)]
= 1.911×10-5 cm2/sec
D = 6.6986×10-7 cm2/sec
3. Crystal violet system with dilution 1:600 (28 0C)
Slope=1.488×10-5
cm2/sec
M = 1:500000 Mo = 1:600
= 1
/ 500000 = 1 /
600
= 2 x 10-6 = 1.67 x 10-3
2.303x4D
[log 10 (1.67x10-3)-log 10 (2x10-6
)] = 1.488×10-5 cm2/sec
D = 5.5281×10-7 cm2/sec
Average
of Diffusion Coefficient, m²/sec for Crystal violet system at 28ºC
= (9.7662×10-7 cm2/sec+6.6986×10-7
cm2/sec+5.5281×10-7 cm2/sec) / 3
= 2.8652×10-7 cm2/sec
4.
Crystal violet system with
dilution 1:200 (37ºC)
Slope =3.652×10-5 cm2/sec
M = 1:500000 M0 = 1:200
= 1
/ 500000 = 1 /
200
= 2 x 10-6 = 5 x 10-3
2.303x4D [log 10 (5x10-3)-log 10 (2x10-6
)] = 3.652×10-5 cm2/sec
D=1.16669×10-6 cm2/sec
5. Crystal
violet system with dilution 1:400 (37ºC)
Slope =2.646×10-5 cm2/sec
M = 1:500000 Mo = 1:400
= 1
/ 500000 = 1 /
400
= 2 x 10-6 = 2.5 x 10-3
2.303x4D
[log 10 (2.5x10-3)-log 10 (2x10-6)]
= 2.646×10-5 cm2/sec
D = 9.267497×10-7 cm2/sec
6. Crystal violet system with
dilution 1:600 (37ºC)
Slope =
1.8×10-5 cm2/sec
M = 1:500000 Mo = 1:600
= 1
/ 500000 = 1 /
600
= 2 x 10-6 = 1.67 x 10-3
2.303x4D
[log 10 (1.67x10-3)-log 10 (2x10-6
)] = 1.8×10-5 cm2/sec
D = 6.871×10-7 cm2/sec
Average of Diffusion
Coefficient, m²/sec for Crystal violet system at 37ºC
= (1.16669×10-6 cm2/sec
+ 9.267497×10-7 cm2/sec +6.871×10-7 cm2/sec)
/ 3
=
9.2097×10-7 cm2/sec
# Value of D37°C for crystal violet using the
equation
D28°C/D37°C = T28°C/T37°C
2.8652×10-7/D37°C =
(28°C + 273.15K)/(37°C + 273.15K)
D
37°C = 2.9508×10-7 cm2/sec
7. Bromotymol
blue system with dilution 1:200 (28ºC)
Slope
= 2.917×10-5
cm2/sec
M
= 1:500000 Mo =
1:200
= 1 / 500000 = 1 / 200
=
2 x 10-6 =
5 x 10-3
2.303x4D [log 10 (5x10-3)-log
10 (2x10-6)] = 2.917×10-5 cm2/sec
D = 9.1886×10-7 cm2/sec
8. Bromotymol blue system with
dilution 1:400 (28ºC)
Slope = 2.388×10-5 cm2/sec
M = 1:500000
Mo = 1:400
= 1
/ 500000 = 1 /
400
= 2 x 10-6 = 2.5 x 10-3
2.303x4D
[log 10 (2.5x10-3)-log 10 (2x10-6
)] = 2.388×10-5 cm2/sec
D= 8.3703×10-7
cm2/sec
9. Bromotymol blue system with dilution
1:600 (28ºC)
Slope =5.357×10-6 cm2/sec
M = 1:500000 Mo = 1:600
= 1
/ 500000 = 1 /
600
= 2 x 10-6 = 1.67 x 10-3
2.303x4D [log 10 (1.67x10-3)-log
10 (2x10-6 )] = 5.357×10-6 cm2/sec
D = 1.9907×10-7 cm2/sec
Average of Diffusion Coefficient, m²/sec for
Bromothymol blue system at 28ºC
= (9.1886×10-7 cm2/sec
+ 8.3703×10-7 cm2/sec
+11.9907×10-7 cm2/sec) / 3
= 1.253×10-6 cm2/sec
10. Bromotymol blue system with
dilution 1:200 (37ºC)
Slope = 3.625×10-5
cm2/sec
M = 1:500000 Mo = 1:200
= 1
/ 500000 = 1 /
200
= 2 x 10-6 = 5 x 10-3
2.303x4D
[log 10 (5x10-3)-log 10 (2x10-6 )]
= 3.625×10-5 cm2/sec
D = 1.1669×10-6 cm2/sec
11. Bromotymol blue system with
dilution 1:400 (37ºC)
Slope =2.388×10-5
cm2/sec
M = 1:500000 Mo = 1:400
= 1
/ 500000 = 1 /
400
= 2 x 10-6 = 2.5 x 10-3
2.303x4D [log 10 (2.5x10-3)-log
10 (2x10-6)] = 2.3888×10-5 cm2/sec
D = 8.3703×10-7 cm2/sec
12. Bromotymol blue system with
dilution 1:600 (37ºC)
Slope =8.003×10-6
cm2/sec
M = 1:500000 Ma = 1:600
= 1
/ 500000 = 1 /
600
= 2 x 10-6 = 1.67 x 10-3
2.303x4D
[log 10 (1.67x10-3)-log 10 (2x10-6
)] = 8.003×10-6 cm2/sec
D = 1.9901×10-6
cm2/sec
Average of Diffusion Coefficient, m²/sec for Bromotymol blue system at 37ºC
= (1.1669×10-6 cm2/sec
+8.3703×10-7cm2/sec +1.9901×10-6 cm2/sec)
/ 3
= 1.3313×10-6
cm2/sec
# Value of D37°C for bromothymol blue using
the equation
D28°C/D37°C = T28°C/T37°C
1.253×10-6 /D37°C =
(28°C + 273.15K)/(37°C + 273.15K)
D 37°C = 1.2904×10-6 cm2/sec
DISCUSSION:
Diffusion is the movement of
solute particles in a solid from a high concentration area to a low
concentration area, resulting in the uniform distribution of the substance.
Diffusion is a process which is not due to the force action but due to the
random movements of atoms. In each diffusion reaction the flux of matter is
equal to the conductivity multiplied by a driving force. Conductivity is the
mobility of the diffusing species or known as diffusivity. The presence of
concentration gradient acts as the driving force for the solute particles to move.
Fick’s First Law states that the
flux, dm, across a membrane of unit area, A, is proportional to the
concentration gradient, dc/dx, and is expressed by
dm = -DA(dc/dx)dt
D is the
diffusion coefficient or diffusivity for the solute. The value of diffusivity
describes the rate of diffusion and therefore its unit is m2s-1.
Diffusion coefficient depends on several factors such as temperature, pressure,
composition, physical state, structure of the phase, and oxygen fugacity. The
higher the value of diffusion coefficient, the easier for that solute to
penetrate through the continuous phase.
The
negative sign indicates that the direction of the diffusive flux is down the concentration
gradient. Diffusive flux goes from high concentration region to low
concentration region with the gradient going the opposite direction, from low
to high concentration. Fick’s First Law only applies to steady state flux where
the concentration gradient is uniform.
In this practical, a solution
containing neutral molecules with the concentration M0 is placed
within a cylindrical tube. The diffusion can be expressed as
M = Mo exp (x²/4Dt)
The
equation is derived into logarithmic form and produce new equation,
ln M = ln Mo – (x²/4Dt)
or
2.303 x 4D (log 10 Mo- log 10 M) t = x²
Therefore,
a graph of x2 against time produces a straight line that passes
through origin with the slope of 2.303 x 4D (log 10 Mo- log 10
M). D can be calculated from the gradient obtained.
The results show that the diffusion
coefficient, D, for both crystal violet and bromothymol blue is increasing from
1:600<1:400<1:200 which mean that 1:200 concentration is the easiest to
diffuse as 1:200 concentration is the most concentrated solution. Therefore,
the solute with high concentration is easier to diffuse through the solid
medium compared to the low concentration. This is because the concentration
gradient is high for concentrated solute and forces the solute particles to diffuse
further through the medium.
If we compare the diffusion
coefficient in the term of temperature between 28°C and 37°C, the D for 37°C
has the larger value than that of 28°C.
The same phenomena occurred for both crystal violet and bromothymol blue
solution. Temperature is one of the factors that affect the rate of diffusion. As
the temperature increases, the amount of energy available for diffusion is
increased. There would be increase in
molecules' mobility (kinetic energy). The molecules move faster and there will
be more spontaneous spreading of the material which means that diffusion occurs
quicker. Thus, the diffusivity increases as the temperature increases.
The value of D37°C can be determined if we
already know the value for D28°C by using the equation below
D28°C/D37°C = T28°C/T37°C
where T
is the temperature in Kelvin.
However,
there are the differences between the value of D37°C that we get from the
experiment and from the theoretical value. For crystal violet, 9.2099×10-7
cm2s-1 is the experimental value of D37°C whereas 7.5502×10-7
cm2s-1 is the theoretical value that obtained from the
equation above. Meanwhile, for bromothymol blue the D37°C from theoretical and
experimental is 6.7559×10-7 cm2s-1 and
7.6703×10-7 cm2s-1 respectively. Generally, the
experimental value for D37°C is higher than theoretical value. This is maybe
due to the some errors that occurred during the experiment is carried out. The
temperature may be set higher than 37°C that cause the rate of diffusion to increase.
Besides, the errors also due to the room temperature at 28°C that is not
constant throughout the experiment and the viscosity of agar in the test tube
that is not uniform. Misreading or parallax error might occur.
According to the equation:
M
= 4/3πa3 N Þ
a3 = 3M / 4πN Þ
a = 3√ 3M / 4πN Þ
--------------------------- (1)
Where M= molecular weight, a =
radius of particle, N=Avogadro's number (6.02x1023),
Þ =density.
Another equation:
D = kT / 6 πηa
--------------------------- (2)
Substituting (1) into the equation
(2)
D = kT / 6 πηa
= kT / 6 πη 3√ 3M/4πNÞ
= kT 3√4πNÞ / 6 πη 3√ 3M -------------------------
(3)
Where D=diffusion coefficient, η=
viscosity, T=temperature, k=Boltzmann constant (1.38 x1023 Jk-1)
According
to equation (3), D is inversely proportional to M. So, D increases when M
decreases. Crystal violet with molecular formula C25N3H30Cl
has molecular weight of 407.979 g mol-1 while bromothymol
blue solution with molecular formula C27H28Br2O5S
has molecular weight of 624.38 g mol−1. Molecular
weight is how much mass each particle has
or how heavy it is. The heavier the particle, the slower it diffuses into
solidified agar solution, assuming energy of the system remains constant.
Crystal violet solution diffuses faster than bromothymol blue since the value
of diffusion coefficient for crystal violet is higher than bromothymol blue at
both temperatures. Crystal violet solution diffuses easily because its
molecular weight is much smaller compared to that of bromothymol blue and thus
easy for them to penetrate through gel medium.
CONCLUSION:
From the experiment, the diffusion
coefficient of crystal violet at 28°C is 7.3311×10-7
cm2s-1 and at 37°C is 9.2099×10-7 cm2s-1. Meanwhile, for
bromothymol blue solution the diffusion coefficient at 28°C is 6.5599×10-7
cm2s-1 and at 37°C is 7.6703×10-7 cm2s-1. The diffusion
coefficient of crystal violet is generally higher than bromothymol blue because
of the difference inmolecular weight. In term of temperature, diffusivity at
37°C is much higher than diffusivity at 28°C. This is due to the rapid movement
of the solute particles as the temperature increases.
REFERENCES:
3. physics.nyu.edu/grierlab/methods/node11.htm
hello..i want to ask you something..where did you get the value of log m ?
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