Title
Practical 4: Determination of Diffusion Coefficient
Objective
To
determine the diffusion coefficient of crystal violet and bromothymol blue.
Introduction
Diffusion
is the process of material transport by atomic motion where the movement of
solutes from a high concentration area to a lower concentration area occur
spontaneously. The Fick’s Law states that the diffusion
flux of material ( amount dm in time dt ) across a given plane ( area A ) is proportional
to the concentration gradient dc/dx. The first Fick’s Law equation is
dm = -DA(dc/dx)dt -------------------- (Equation 1)
where
D is the diffusion coefficient or diffusivity for the solute, in unit m2s-1.
Diffusion
coefficient is generally prescribed for a given pair of species. For a
multi-component system, it is prescribed for each pair of species in the
system. The higher the diffusivity (of one substance with respect to another),
the faster they diffuse into each other. For dc/dx which is the concentration gradient is
often called the driving force in
diffusion and the minus sign in the equation means that diffusion is down
the concentration gradient. This first law is
applied to systems in which the conditions are not steady but for the second
law is applied to system in which the conditions are non-steady. In most
real situations the concentration profile and the concentration gradient are
changing with time in the second law.
If a
solution containing neutral particles with the concentration Mo, is
placed within a cylindrical tube next to a water column, the diffusion can be
stated as
M = M0 eksp (-x²/ 4 Dt) --------------------- (Equation 2)
Where
M is the concentration at distance x from the intersection between and solution
that is measured at time t. by changing the Equation 2 to its logarithmic form
which is
ln M = ln M0 – x2
/ 4Dt or
2.303 x 4D (log10M0
– log10 M) t = x2 ----------------------- (Equation 3)
Thus,
a graph of x2 against t can produce a straight line that passed
through the origin with the slope 2.303
x 4D (log10M0 – log10 M). As a result, D
can be calculated. If the particle in the solution are assumed to be spherical,
their size and molecular weight can be calculated by the Stokes – Einstein
equation which is
D = kT / 6пŋa
Heat causes
atoms to vibrate and the Vibration amplitude increases with Temperature.
Melting occurs when vibrations are sufficient to rupture bonds. the Vibrational
frequency is approximately 1013 H Average atomic / electronic energy due to
thermal excitation is of order kT
where k is the Boltzmann
constant constant 1.38 x 1023 Jk-1
T is the
temperature in Kelvin
П is the
viscosity of the solvent in Nm-2s
a is the radius
of particle in M.
The
volume of spherical particle is 4/3 пa3, thus its weight M is
equivalent to 4/3 пa3Np (p = density). It is known
that molecular weight M = mN (N is the Avogadro’s number 6.023 x 1023
mol-1). Therefore,
M = 4/3
пa3Np ------------------------ (Equation 4)
If the diffusion is for
charged particles, the Equation 3 should be modified which we need to include
potential gradient effect that exist between the solution and solvent. However,
by adding a little sodium chlorine into the solvent, the problem where
formation of this potential gradient can be solved.
Agar gel contains a
semi-solid network of molecular that can be penetrated by water molecules. The
water molecules will form a continuous phase around the agar gel. Then, the
solute molecules can be diffused freely in the water, if there are no chemical
interactions and adsorption effects occur. Thus, the agar gels will provide a
suitable supportive system that can be used in the experiment for diffusion of
certain molecules in a aqueous medium.
Procedures
1 1) 7g of agar powder is weighed and mixed with 425ml of
Ringer solution in the 500mL beaker.
2 2) The mixture in the beaker is
stirred and boiled on a hot plate until a transparent yellowish solution was
obtained.
3. About 20ml of the agar
solution is pour into 6 test tubes. The test tubes are then put into the fridge
to let them cool.
4. An agar test tube which contained 5ml of 1:500,000 crystal violet is
being prepared where it is used as a standard system to measure the distance of
the colour as a result of the diffusion of crystal violet.
5. After the agar solutions in the test tubes solidifying, 5ml of each
1:200, 1:400, 1:600 crystal violet solution is pour into each test tube.
6. The test tubes are closed immediately to prevent the evaporation.
7. Three test tubes are put in room temperature (28 ºC) while another three
are put in water bath of 37 oC.
8. The distance between the agar surface and the end of crystal violet
where that area has the same colour as in the indicator was measured
accurately.
9. Average of the readings are obtained, this value is x in meter.
10. The x values are recorded after 2 hours and at appropriate intervals for
2 weeks.
11. Steps 3 to 10 are repeated for bromothymol blue solutions.
12. Graph of x² values (in m²) versus time (in hours) are plotted.
13. The diffusion coefficient , D was determined from the graph gradient for
both 28 ºC and 37 ºC and the molecular mass of crystal violet and bromothymol
blue are also determined by using N and V equation.
Result :
Crystal Violet system
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:200
|
0
|
0
|
0
|
3.5x10-5
|
1.12 x
10-6
|
28
|
1.0617
x 10-6
|
86400
|
1.5
|
2.25
|
|||||
172800
|
2.0
|
4.00
|
|||||
259200
|
2.6
|
6.76
|
|||||
345600
|
3.2
|
10.24
|
|||||
432000
|
3.8
|
14.44
|
|||||
518400
|
4.2
|
17.64
|
|||||
604800
|
4.5
|
20.25
|
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:400
|
0
|
0
|
0
|
3.24x10-5
|
1.136 x 10-6
|
28
|
1.0617
x 10-6
|
86400
|
1.0
|
1.00
|
|||||
172800
|
1.8
|
3.24
|
|||||
259200
|
2.5
|
6.25
|
|||||
345600
|
3.0
|
9.00
|
|||||
432000
|
3.7
|
13.69
|
|||||
518400
|
4.0
|
16.00
|
|||||
604800
|
4.2
|
17.64
|
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:600
|
0
|
0
|
0
|
2.5x10-5
|
9.291 x 10-7
|
28
|
1.0617
x 10-6
|
86400
|
0.8
|
0.64
|
|||||
172800
|
1.2
|
1.44
|
|||||
259200
|
2.0
|
4.00
|
|||||
345600
|
2.7
|
7.29
|
|||||
432000
|
3.0
|
9.00
|
|||||
518400
|
3.5
|
12.25
|
|||||
604800
|
3.8
|
14.44
|
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:200
|
0
|
0
|
0
|
3.41x10-5
|
1.089 x 10-6
|
37
|
1.073
x 10-6
|
86400
|
1.8
|
3.24
|
|||||
172800
|
2.2
|
4.84
|
|||||
259200
|
2.5
|
6.25
|
|||||
345600
|
3.0
|
9.00
|
|||||
432000
|
3.4
|
11.56
|
|||||
518400
|
4.1
|
16.81
|
|||||
604800
|
4.7
|
22.09
|
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:400
|
0
|
0
|
0
|
3.043x10-5
|
1.067 x 10-6
|
37
|
1.073
x 10-6
|
86400
|
1.5
|
2.25
|
|||||
172800
|
2.1
|
4.41
|
|||||
259200
|
2.6
|
6.76
|
|||||
345600
|
3.2
|
10.24
|
|||||
432000
|
3.5
|
12.25
|
|||||
518400
|
3.7
|
13.69
|
|||||
604800
|
4.4
|
19.36
|
System
|
Time (seconds)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:600
|
0
|
0
|
0
|
2.86x10-5
|
1.063 x 10-6
|
37
|
1.073
x 10-6
|
86400
|
0.8
|
0.64
|
|||||
172800
|
1.5
|
2.25
|
|||||
259200
|
2.4
|
5.76
|
|||||
345600
|
2.8
|
7.84
|
|||||
432000
|
3.3
|
10.89
|
|||||
518400
|
3.7
|
13.69
|
|||||
604800
|
4.0
|
16.00
|
Bromothymol Blue system
System
|
Time (s)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:200
|
0
|
0
|
0
|
3.704x10-5
|
1.1833x10-6
|
28
|
1.10873
x 10-6
|
86400
|
1.4
|
1.96
|
|||||
172800
|
2.2
|
4.84
|
|||||
259200
|
2.5
|
6.25
|
|||||
345600
|
3.1
|
9.61
|
|||||
432000
|
3.8
|
14.44
|
|||||
518400
|
4.2
|
17.64
|
|||||
604800
|
4.9
|
24.01
|
System
|
Time (s)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:400
|
0
|
0
|
0
|
3.472x10-5
|
1.2170x10-6
|
28
|
1.10873
x 10-6
|
86400
|
1.2
|
1.44
|
|||||
172800
|
1.6
|
2.56
|
|||||
259200
|
2.4
|
5.76
|
|||||
345600
|
3.0
|
9.00
|
|||||
432000
|
3.4
|
11.56
|
|||||
518400
|
3.7
|
13.69
|
|||||
604800
|
4.3
|
18.49
|
System
|
Time (ss)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:600
|
0
|
0
|
0
|
2.49x10-5
|
9.2589x10-7
|
28
|
1.10873
x 10-6
|
86400
|
1.0
|
1.00
|
|||||
172800
|
1.4
|
1.96
|
|||||
259200
|
2.0
|
4.00
|
|||||
345600
|
2.7
|
7.29
|
|||||
432000
|
3.1
|
9.61
|
|||||
518400
|
3.5
|
12.25
|
|||||
604800
|
3.8
|
14.44
|
System
|
Time
(s)
|
x,
M
|
x2,
M2
|
Slope
of graph
|
D,
M2S-1
|
Temperature,
oC
|
Average
Diffusion Coefficient, D (M2S-1)
|
1:200
|
0
|
0
|
0
|
2.716x10-5
|
8.6768x10-7
|
37
|
8.8445
x 10-7
|
86400
|
1.2
|
1.44
|
|||||
172800
|
1.7
|
2.89
|
|||||
259200
|
2.0
|
4.00
|
|||||
345600
|
2.6
|
6.76
|
|||||
432000
|
3.4
|
11.56
|
|||||
518400
|
4.0
|
16.00
|
|||||
604800
|
4.5
|
20.25
|
System
|
Time (s)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:400
|
0
|
0
|
0
|
2.862x10-5
|
1.0032x10-6
|
37
|
8.8445
x 10-7
|
86400
|
1.0
|
1.00
|
|||||
172800
|
1.5
|
2.25
|
|||||
259200
|
2.1
|
4.41
|
|||||
345600
|
2.5
|
6.25
|
|||||
432000
|
3.3
|
10.89
|
|||||
518400
|
3.7
|
13.69
|
|||||
604800
|
4.0
|
16.00
|
System
|
Time (s)
|
x, M
|
x2, M2
|
Slope of graph
|
D, M2S-1
|
Temperature, oC
|
Average Diffusion Coefficient, D (M2S-1)
|
1:600
|
0
|
0
|
0
|
2.106x10-5
|
7.8248x10-7
|
37
|
8.8445
x 10-7
|
86400
|
0.8
|
0.64
|
|||||
172800
|
1.3
|
1.69
|
|||||
259200
|
2.0
|
4.00
|
|||||
345600
|
2.5
|
6.25
|
|||||
432000
|
2.8
|
7.84
|
|||||
518400
|
3.2
|
10.24
|
|||||
604800
|
3.4
|
11.56
|
Graph
of x2 against time (Crystal violet at 280C)
 |
| Click to enlarged |
Graph
of x2 against time (Crystal violet at 370C)
 |
| Click to enlarged |
Graph
of x2 against time (Bromothymol Blue at 280C)
 |
| Click to enlarged |
Graph
of x2 against time (Bromothymol Blue at 370C)
 |
| Click to enlarged |
Calculation:
The
slope = 2.303 x 4D (log10 M0 – log10 M)
Crystal
violet system with dilution 1:200 (280C)
M
= 1:500000 M0 =
1:200
=1/500000 = 1/200
= 2.0 x 10-6 = 5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.5 x 10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 3.5 x 10-5
D
= 1.12 x 10-6 cm2/ s
Crystal
violet system with dilution 1:400 (280C)
M
= 1:500000 M0 =
1:400
=1/500000 = 1/400
= 2.0 x 10-6 = 2.5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.24x 10-5
2.303
x 4D (log10 2.5 x 10-3 – log102 x 10-6)
= 3.24 x 10-5
D
= 1.136 x 10-6 cm2/ s
Crystal
violet system with dilution 1:600 (280C)
M
= 1:500000 M0 =
1:600
=1/500000 = 1/600
= 2.0 x 10-6 = 1.67 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.5 x 10-5
2.303
x 4D (log10 1.67 x 10-3 – log102 x 10-6)
= 2.5 x 10-5
D
= 9.29 x 10-7 cm2/ s
Average
of Diffusion Coefficient, m²/hour for Crystal Violet system at 28ºC
=
(1.12 x 10-6 cm2/ s
+ 1.136 x 10-6 cm2/ s +9.29 x 10-7 cm2/
s) / 3
=
1.0617 x 10-6 cm2/ s
Crystal
violet with dilution 1:200 (370C)
M
= 1:500000 M0 =
1:200
=1/500000 = 1/200
= 2.0 x 10-6 = 5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.41 x 10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 3.41 x 10-5
D
= 1.089 x 10-6 cm2/ s
Crystal
violet with dilution 1:400 (370C)
M
= 1:500000 M0 =
1:400
=1/500000 = 1/400
= 2.0 x 10-6 = 2.5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.043 x 10-5
2.303
x 4D (log10 2.5 x 10-3 – log102 x 10-6)
= 3.043 x 10-5
D
= 1.067 x 10-6 cm2/ s
Crystal
violet with dilution 1:600 (370C)
M
= 1:500000 M0 =
1:600
=1/500000 = 1/600
= 2.0 x 10-6 = 1.67 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.86 x 10-5
2.303
x 4D (log10 1.67 x 10-3 – log102 x 10-6)
= 2.86 x 10-5
D
= 1.063 x 10-6 cm2/ s
Average
of Diffusion Coefficient, m²/hour for Crystal Violet system at 37ºC
=
(1.089 x 10-6 cm2/
s + 1.067 x 10-6 cm2/ s + 1.063 x 10-6 cm2/
s) / 3
=
1.073 x 10-6 cm2/ s
Bromothymol
blue system with dilution 1:200 (280C)
M
= 1:500000 M0 =
1:200
=1/500000 = 1/200
= 2.0 x 10-6 = 5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.704x10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 3.704x10-5
D
= 1.1833 x 10-6 cm2/sec
Bromothymol
blue system with dilution 1:400 (280C)
M
= 1:500000 M0 =
1:400
=1/500000 = 1/400
= 2.0 x 10-6 = 2.5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 3.472x10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 3.472x10-5
D
= 1.2170 x 10-6 cm2/sec
Bromothymol
blue system with dilution 1:600 (280C)
M
= 1:500000 M0 =
1:600
=1/500000
= 1/600
= 2.0 x 10-6 = 1.67 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.492x10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 2.492x10-5
D
= 9.2589 x 10-7 cm2/sec
Average
of Diffusion Coefficient, m²/hour for Bromothymol Blue system at 28ºC
=
(1.1833 x 10-6 cm2/sec + 1.2170 x 10-6 cm2/sec
+ 9.2589 x 10-7 cm2/sec) / 3
=
1.10873 x 10-6 cm2/sec
Bromothymol
blue system with dilution 1:200 (370C)
M
= 1:500000 M0 =
1:200
=1/500000
= 1/200
= 2.0 x 10-6 = 5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.716 x 10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 2.716 x 10-5
D
= 8.6768 x 10-7 cm2/sec
Bromothymol
blue system with dilution 1:400 (370C)
M
= 1:500000 M0 =
1:400
=1/500000 = 1/400
= 2.0 x 10-6 = 2.5 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.862 x 10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 2.862 x 10-5
D
= 1.0032 x10-6 cm2/sec
Bromothymol
blue system with dilution 1:600 (370C)
M
= 1:500000 M0 =
1:600
=1/500000
= 1/600
= 2.0 x 10-6 = 1.67 x 10-3
2.303
x 4D (log10 M0 – log10 M) = 2.106 x 10-5
2.303
x 4D (log10 5 x 10-3 – log102 x 10-6)
= 2.106 x 10-5
D
= 7.8248 x 10-7 cm2/sec
Average
of Diffusion Coefficient, m²/hour for Bromothymol Blue system at 37ºC
=
(8.6768 x 10-7 cm2/sec + 1.0032 x10-6 cm2/sec
+ 8.8445 x 10-7 cm2/sec) / 3
=
8.8445 x 10-7 cm2/sec
Discussion:
Fick's
first law relates the diffusive flux to
the concentration under the assumption of steady
state. It postulates that the flux goes
from regions of high concentration to regions of low concentration, with a
magnitude that is proportional to the concentration gradient (spatial
derivative). In one (spatial) dimension, the law is
During this experiment, we are
preparing a standard system with the dilution 1:500 000 that is known as M. the
(log 10 Mo- log 10
M) will increased when Mo is increased. This will cause the rate of diffusion
increase because the concentration gradient become larger and therefore the
driving forces for the occurance of diffusion would be larger also.
One of the other factor that will
affect the rate of diffusion is temperature. As we can see from the result we
get from the experiment, it shows that test tube that is located in the water
bath 37°C has a high rate of diffusion than test tube located at room
temperature 28°C. This result is due to the kinetic energy theory. The kinetic
energy of the molecule will increase when the temperature is increased. It will
provide the molecule the energy to free from the intermolecular forces and make
the easily escape and enter the agar.
Other than that the agar gel also
can influence the rate of diffusion. When the concentration of gel substance is
increase, the size of the hole will decrease and the diffusion rate will
decrease too as the hole size same with the size of the diffuse molecule.
Moreover, the viscosity of the solution in the hole also can influence the
diffusion rate. When the crystallinity of the gel medium is increased, the
diffusion rate will decrease. The larger the volume fraction of crystalline
material, the slower the movement of diffusion molecules. This can be happened
because crystalline regions of the gel medium represent an impenetrable barrier
to the movement of solute particles where it have to circumnavigate through it.
From the result that we obtained, at
room temperature 28°C, the diffusion coefficient for crystal violet is 1.0617x10-6 cm2/sec and
bromothymol blue is 1.10873 x 10-6 cm2/sec while for the experiment that carried out in the
water bath with temperature 37°C, the D for crystal violet and bromothymol blue
are 1.073x10-6 cm2/sec and 8.8445 x10-7cm2/sec
respectively. The D for crystal violet supposes to be higher than that of
bromothymol blue. This may due to some errors that occurs during the
experiment. The measurement taken be different people may be a little bit
different and this will lead to the inconsistency of the readings. Besides, the
colour of the dyes are not very obvious and this cause the measuring process
become difficult and in accurate.
Question:
1. From
the experiment value for D28, estimate the value of D37
using the following equation
D28
T28
------ = --------
D37
T37
Where
ƞ1 and ƞ2 is the viscosity of water at temperature 280C
and 370C.
D37
= D28 x (T37/ T28)
=
1.0617x10-6 x ((37+273)/(28+273)
= 1.093x10-6 cm2/
sec.
2. Is
the calculated value of D37 the same as the value from the
experiment? Give some explanation if it is different. Is there any difference
between calculated molecular weight with the real molecular weight?
No. D37 from this experiment
for crystal violet is 1.073x10-6 cm2/sec and for bromothymol blue is 8.8445 x10-7cm2/sec.
This may due to some errors that occur during
the experiment. There might be an inaccuracy while measure the length of dye diffuse because our eye is not parallel to the
meniscus and the colour is hard to be seen.
Other than that viscosity of the gel may be not uniform which may affect the length of dye diffuse.
3. Between
crystal violet and bromothymol blue, which diffuse quicker? Explain if ther are
any differences in the diffusion coefficient values?
The
crystal violet will diffuse faster as its molecular size is smaller than
bromothymol blue. From the diffusion coefficient calculated, the D for crystal
violet is bigger than D of bromothymol blue. The higher the diffusion
coefficient, the faster the diffusion rate. Thus, the statement that crystal
violet diffuse faster is proven correct.
.
Conclusion:
From
this experiment, we can determine the factor which effect the diffusion rate
which are temperature and concentration of diffusing molecules. Therefore, the
rate of diffusion is increase in high temperature and faster in the
concentration of diffusing molecules 1:200> 1:400> 1:600. The rate of
diffusion of crystal violet also higher than bromotymol blue. Diffusion
coefficient, D for Crystal Violet system at 28ºC is 9.85x10-7 cm2/sec
while at 37ºC is 1.088x10-6 cm2/sec. The
diffusion coefficient, D for Bromothymol Blue system at 28ºC is 1.10873 x
10-6 cm2/sec while at 37ºC is 8.8445 x10-7cm2/sec.
References:
1) http://www.eng.utah.edu/~lzang/images/lecture-5.pdf
2) http://web.utk.edu/~prack/mse201/Chapter%205%20Diffusion%20.pdf
References:
1) http://www.eng.utah.edu/~lzang/images/lecture-5.pdf
2) http://web.utk.edu/~prack/mse201/Chapter%205%20Diffusion%20.pdf
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