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Vapour Diffusion Coefficient of Acetone

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Vapour Diffusion Coefficient of Acetone in Air
KEK 150085 Sin Jia Huey
Abstract
The objective of this gas diffusion experiment is to find out the unimolecular diffusion
coefficient of acetone vapor in air at atmospheric pressure and 50°c(323K). In this experiment
, the method that is going to use is Winkelmann method , which is commonly used in diffusion
coefficient determination. The experiment is set up by inserting a capillary tube carrying
acetone liquid in a water bath of 50°c. Air flow is supplied at the mouth of the capillary tube
to ensure zero partial pressure of vapor acetone. It is assumed that the evaporation rate of
acetone in the air is the indicator of the diffusion rate of the acetone in the air. By observing
the decrease in the level of acetone liquid in every fifteen minutes intervals , the new level of
meniscus are collected and from this the diffusion coefficient can be calculated. Fick’s first law
which saying that the molar diffusion flux of a component A into B in z direction is proportional
 t 

to the negative concentration gradient of A in z-direction is being applied. A graph of 

z

z
0 

versus ( z  z 0 ) is plotted and the slope gradient is obtained for diffusion coefficient
calculation. The experimental value for the diffusivity of acetone in air at 50°c and 1atm is
1.419 × 10−5
𝑚2
𝑠
. The experimental value is 14.02% deviated from the actual one.
1.0 Introduction
Diffusion is a process where molecules move from a area of high concentration to a area of
low concentration until equilibrium is achieved. Diffusion can happen in gas phase, liquid
phase and solid phase. The differences in the concentration is the main driving force for this
phenomena to happen. Gas diffusion is the mixing of molecules of one gas component to
another by kinetic properties. Gas diffusion is relatively rapid as compare to liquid and solid
diffusion as the gas molecules have higher kinetic energies and further apart from each other.
The rate at which the molecules diffuse from one point to another point depends on the
concentration gradient of the molecules on that point and to the direction of diffusion. The
higher the concentration gradient, the higher the rate of diffusion.
Figure 1: Molecular Diffusion Process
1
The concentration of molecules A is higher on the left side of the plane than on the right side.
As the partition is removed, the molecules A diffuse from left to right which is higher
concentration area to a lower concentration area until the equilibrium is reached.
Diffusion phenomena can be explain by Fick’s First Law of diffusion , saying that when the
temperature and pressure is constant , the molar diffusion flux of molecules A into B in zdirection is proportional to the negative concentration of A in z-direction.
𝑑𝐶𝐴
JAB,z = −𝐷𝐴𝐵
𝑑𝑧
Where DAB is proportional coefficient, known as diffusion coefficient of A into B .
However , there is bulk flow contributing to the molar diffusion flux when there is externally
force promoting the mixture flow such as gravity force or pumping energy. In this case, the
diffusion of molecules A into a stationary molecule B , the total molar diffusion flux of A into
B can be expressed as (Weltly&Wicks,1984),
N𝐴,𝑧 = 𝑦𝐴 𝑁𝑧 + 𝐽𝐴,𝑧
For unimolecular diffusion , N𝑧 = 𝑁𝐴,𝑧
N𝐴,𝑧 = −
𝑐𝐷𝐴𝐵 𝑑𝑦𝐴
(1 − 𝑦𝐴 ) 𝑑𝑧
There are several methods of experimental determination of gas diffusion coefficient such as
Stefan-Winkelmann method (Coca et al.,1980 )
and gas chromatography method
(Castells,2004). The method that is going to use in determining the gas diffusion coefficient of
acetone in air is Winkelmann method. The model consists of a T tube , the volatile liquid where
its diffusion coefficient need to be determined is placed inside the capillary tube. An air flow
is kept above the T tube to keep the partial pressure of vapor to be almost zero. The temperature
of the liquid acetone is kept constant by immersing it into a water bath and the pressure is
assumed to be constant because it is always open to the air. By observing the fall in the liquid
level in the capillary tube , the evaporation rate of the volatile liquid can be determined whereas
the diffusion coefficient of the volatile liquid can be calculated from here.
The objective of this experiment is to find out the vapor diffusion coefficient of acetone in air
at pressure of 1 atm and 50°c.
2.0 Methodology
The experiment set up to study the unimolecular vapor diffusion coefficient of the acetone in
air is Winkelmann method. First of all, the capillary tube is sprayed with some detergent and
thn some distilled water. It is then rinsed with a little bit of acetone solution and is inserted with
liquid acetone until the liquid level reach approximately 40mm height. Next, it is slowly
inserted into the metal nut until it is exactly straight on the nut. After that , the capillary tube is
submerged into a water bath with temperature 50°c. The stirrer is switch on to even sure that
the water bath is evenly heated. The microscope is adjusted to let the microscope see through
2
the capillary tube. After the meniscus of liquid acetone is being spotted , the vernier scale is
adjusted to ease the measurement later. A tube allowing air flow on top of the capillary tube in
connected from an air pump on to it. The initial level of acetone is read and recorded as Z0 .
After 15 minutes, the air pump is switch off when taking the new reading of meniscus level of
liquid acetone and it is recorded as z. This action is repeated for 10 times and the respectively
readings are recorded. A plot of
𝑡
𝑧+𝑧0
versus z − z0 is made. The gradient of the graphs is
obtained to calculate the diffusion coefficient of acetone in air .
3
3.0 Results and Discussion
3.1 Results and graph
Graph of t/ (z + z0 ) versus z - z0
0,7
0,6
t/ (z + z0 )
0,5
0,4
0,3
0,2
0,1
0
0
1
2
3
4
5
6
z - z0
Figure 2 : Graph of t/ (z + z0 ) versus z - z0
Table 1: Level of acetone at every 15 minutes interval.
3.2 Derivation of equation for unimolecular diffusion coefficient
For unimolecular diffusion , the molar diffusion flux which concern bulk flow,
𝑁𝐴,𝑧=𝑏𝑢𝑙𝑘 𝑓𝑙𝑜𝑤+𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛
N𝐴,𝑧 = 𝑦𝐴 𝑁𝑧 + 𝐽𝐴,𝑧
N𝑧 = 𝑁𝐴,𝑧
4
𝑑𝑦𝐴
𝑑𝑧
𝑐𝐷𝐴𝐵 𝑑𝑦𝐴
=−
(1 − 𝑦𝐴 ) 𝑑𝑧
(1 − 𝑦𝐴 )𝑁𝐴,𝑧 = −𝑐𝐷𝐴𝐵
N𝐴,𝑧
where,
𝑦𝐴
𝑁𝑧
= mole fraction of component A
= total molar flux of solution due to diffusion in the z- direction..
Integrating from y=0 and z=0 to y=𝑦𝐴 and z=z
𝑧=𝑧
∫
𝑦=𝑦𝐴
𝑁𝐴,𝑧 𝑑𝑧 = − ∫
𝑧=0
𝑦=𝑜
𝑐𝐷𝐴𝐵
𝑑𝑦
1 − 𝑦𝐴 𝐴
Integrate and the equation became,
NA,z z = c𝐷𝐴𝐵 ln(1 − 𝑦𝐴 )…………………………..(1)
Expressing the molar diffusion flux in term of density and molar mass
𝑁𝐴,𝑧 =
𝑛𝐴
𝐴𝑡
𝜌
= 𝑀𝐴
𝐴
𝑑𝑧
𝑑𝑡
………………………………………………(2)
Putting (1) and (2) together
𝑧
𝜌𝐴 𝑑𝑧
= − − 𝑐𝐷𝐴𝐵 ln(1 − 𝑦𝐴 )
𝑀𝐴 𝑑𝑡
Integrating both side and rearranging
𝑧=𝑧
∫
𝑧
𝑧=𝑧0
𝑡=𝑡
𝑑𝑧
𝑀𝐴 𝑐𝐷𝐴𝐵 𝐼𝑛(1 − 𝑦𝐴 )
=∫
𝑑𝑡
𝑑𝑡
𝜌𝐴
𝑡=0
𝑧 2 − 𝑧0 2
𝑀𝐴 𝑐𝐷𝐴𝐵 𝐼𝑛(1 − 𝑦𝐴 )
=−
𝑡
2
𝜌𝐴
Simplifying,
(𝑧 + 𝑧0 )(𝑧 − 𝑧0 ) = −
2𝑀𝐴 𝑐𝐷𝐴𝐵 𝐼𝑛(1 − 𝑦𝐴 )
𝑡
𝜌𝐴
𝑡
𝜌𝐴
=−
(𝑧 − 𝑧0 )
𝑧 + 𝑧0
2𝑀𝐴 𝑐𝐷𝐴𝐵 𝐼𝑛(1 − 𝑦𝐴 )
5
Comparing to the graph
t
versus z  z 0 plotted ,the gradient of the slope,
z  z0
𝑠=−
𝜌𝐴
2𝑀𝐴 𝑐𝐷𝐴𝐵 𝐼𝑛(1 − 𝑦𝐴 )
3.3 Calculation of Diffusion Coefficient
Determine the gradient of the graph
0.66566−0.20292
s= 5.34−1.76
= 0.12926 min/mm2
60𝑠
1000𝑚𝑚
= 0.12926 min/mm2 x 1𝑚𝑖𝑛 x ( 1𝑚 )2
= 7.76 x 106 s/m2
From Perry’s Chemical Engineering handbook,
The constant for the vapor pressure of saturated acetone:
A
B
C
7.02447
1161.0
224
Using Antoine equation to calculate
𝑃∗ [𝑚𝑚𝐻𝑔] = 10
𝐴−
𝐵
𝑇(°𝑐)+𝐶
P*=612.69 mmHg
Using Raults law to calculate the molar fraction of A,
𝑦𝐴 =
𝑃∗
𝑃
612.69
= 760
= 0.806
For ideal gases,
PV = nRT
n
P
Thus, c  
V RT
101325
c = 8.314 ×323.15
6
=37.714 𝑚𝑜𝑙/𝑚3
MA= 58.08 kg/ kmol
= 0.05808 kg/ mol
ρA = 791 kg/ m3
𝜌𝐴
𝐷𝐴𝐵 = − 2.𝑀
𝐴 .𝑐.𝑠.𝐼𝑛(1−𝑦𝐴 )
791
= − 2×0.05808×37.714×7.76×106×𝐼𝑛(1−0.806)
= 1.419 x 10-5 m2 /s
3.4 Comparison of the results to theoretical value
The fuller correlation extracted from Perry’s Chemical Engineers’ handbook , stated that the
diffusion coefficient of acetone in air is (1.22± 0.11) × 10−5 m²/s.
1.419×10−5 −1.22×10−5
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 = |
1.22×10−5
| × 100%
=14.02%
The percentage of error as compare to the theoretical is 14.02% which is considered not high.
The deviation might due to the certain errors made during conducting the experiment.
Parallax error might be occurred during taking the reading of liquid acetone level. This is
because the meniscus of the liquid acetone in the capillary tube is not parallel to the reference
line. Assuming the partial pressure of the acetone as zero at the mouth of the capillary tube is
the second factors that affecting the inaccuracies in the results because the partial pressure may
not exactly be zero since some acetone vapor might not being displaced by the air flow.
The fluctuating temperature of the water is also one of the reasons that causing inaccuracies of
the results.
4.0 Conclusion
The diffusivity coefficient for the unimolecular acetone diffusion in air at 50°c(323K) and 1
atm is 1.419 ×
10−5 𝑚2
𝑠
. The experimental value is 14.02% deviated from the actual one.
5.0 References
Castells,R.C .( 2004) . Determination of gas-liquid partition coefficients by gas
7
Perry,R.H.,&Green,D.W.(2008).Perry’s chemical engineers’ handbook(8th ed.).New York:
McGraw-Hill.
Welty, J. R. & Wicks, C. E. (1984), Fundamentals of Momentum, Heat and Mass Transfer,
Wiley, pp 483, 780 — 782.
José Coca, Julio L. Bueno, Ricardo Alvarez. Gaseous Diffusion Coefficients by the StefanWinkelmann Method Using a Polymer- Solvent Mixture as Evaporation Source. Ind. Eng. Chem.
Fundamen., 1980, 19 (2), pp 219–221.
Gilliland,E.R.(1934). Diffusion Coefficients in Gaseous Systems. Ind. Eng. Chem., 26 (6), pp
681–685.DOI: 10.1021/ie50294a020
chromatography. Journal of Chromatography A ,1037(1-2),223-31.
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