Measuring the Diffusivity of Acetone and Ethanol in Air at 40 and 50 Degrees
Celsius with Winkelmann’s Method
Introduction
Background
Diffusion refers to the process by which molecules move from a higher concentration to a lower
concentration through a gradient, without the use of energy, until equilibrium is attained (“Diffusion and
Osmosis”). Diffusion is the outcome of random molecular motion. When the temperatures are high,
molecules move quicker this makes diffusion occur more rapidly. The behavior of diffusion does not rely
on chemical reactions. As long as there is a presence of a concentration gradient, diffusion will happen.
An experiment will be conducted to show the vapor diffusivity of acetone and ethanol into air. During the
experiment unimolecular diffusion will be sustained without the occurrence of a chemical reaction. By
monitoring the rate of loss of the liquid phase, the Stephen-Winkelmann method is applied to determine
the rate of mass flux into the gas phase.
Temperatures of specific systems directly affect transport properties such as velocity, diffusion,
and thermal conductivity. To perceive the separation of procedures such as distillation and chemical
reactor design, the concept of molecular diffusion has to be used. Molecular diffusion has many
educational applications as well. Molecular diffusion is commonly used in various areas such as;
chemistry, physics, biology, and engineering. One main example of molecular diffusion is osmosis; this is
a process seen throughout nature in many biological processes. Diffusion is very important to understand
because it is encountered in every human’s lifetime (Whitmer).
Objective
The purpose of this lab was to determine the diffusivities of acetone and ethanol by measuring the
rate at which the liquid level of both compounds dropped in a capillary tube at 40 ℃ and 50 ℃. The
experimental value of acetone’s diffusivity at 50 ℃ was calculated using equations derived from Fick’s
Law and was statistically compared to acetone’s experimental diffusivity at 40 ℃, ethanol’s experimental
diffusivity at 50 ℃, a value found in literature, and a value calculated through mass transfer equations.
1
Theory
Diffusivity of acetone was calculated with Winkelmann’s method by measuring the time it takes
for a liquid in a tube to lower a certain distance. The liquid acetone evaporated out of a narrow diameter
glass tube at constant temperature and pressure while air flowed over the top of the tube. The diffusivity
of acetone was calculated with:
𝐷𝐴𝐵 =
𝜌𝐴 ∗(𝑧𝐹2 −𝑧02 )∗𝑅∗𝑇∗𝑝𝐵𝑀
2∗𝑀𝐴 ∗𝑡𝐹 ∗𝑃∗(𝑝𝐴1 −𝑝𝐴2 )
(eq. 1)
where 𝜌𝐴 is the density of acetone, 𝑧𝐹 is the final height of the liquid, 𝑧0 is the initial height of the liquid,
R is the ideal gas constant, T is absolute temperature, 𝑃𝐵𝑀 is given by equation 2, 𝑀𝐴 is the molar mass of
acetone, 𝑡𝐹 is the time it took the liquid to move from 𝑧0 to 𝑧𝐹 , P is the atmospheric pressure, 𝑝𝐴1 is the
partial pressure of acetone above the liquid acetone, and 𝑝𝐴2 is the partial pressure of acetone at the top of
the tube. The value of 𝑝𝐴2 can be assumed to be zero since the airflow over the top will keep the vapor
pressure of acetone very low at the top of the tube. The quantity 𝑝𝐵𝑀 was calculated with:
𝑝𝐵𝑀 =
𝑝𝐴1 −𝑝𝐴2
(𝑃−𝑝𝐴2 )
]
(𝑃−𝑝𝐴1 )
𝑙𝑛[
(eq. 2)
where 𝑝𝐴1 is the partial pressure of acetone above the liquid acetone, and 𝑝𝐴2 is the partial pressure of
acetone at the top of the tube. The partial pressure of acetone was calculated directly above the liquid in
units of bar with Antoine’s equation:
𝐵
𝑝𝐴1 = 10(𝐴−𝐶+𝑇)
(eq. 3)
where A, B, and C are constants equal to 4.42448, 1312.253, and -32.445 respectively and T is
temperature in Kelvin (Ambrose, 1974). Antoine’s equation was also used for ethanol with the constants
A, B, and C equaling 4.92531, 1432.526, and -61.819 respectively (Ambrose, 1975).
The diffusivity was also calculated computationally with:
𝐷𝐴𝐵 =
1/2
1
1
+ )
𝑀𝐴 𝑀𝐵
2
1/3
𝑃[(∑ 𝑣𝐴 ) +(∑ 𝑣𝐵 )1/3]
1.00𝑥10−1 𝑇 1.75 (
(eq. 4)
where T is absolute temperature, Ma is molar mass of acetone, MB is molar mass of air, P is absolute
pressure in bar, ∑ 𝑣𝐴 is the sum of the diffusion volume for acetone, and ∑ 𝑣𝐵 is the sum of the diffusion
volume for air (Fogler).
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Methods
Apparatus
In this study, the Armfield Gaseous Diffusion Coefficient Apparatus was used to perform the
vapor diffusion experiment according to Winkelmann’s method. The apparatus pictured in Figure 1
utilized a glass capillary tube to contain the liquid species being tested. A heating block held the capillary
tube and maintained the liquid at a constant temperature, which was set using the digital display. A fan
provided a steady flow of air into the apparatus and a flexible tube to the top of the capillary tube carried
this air. The airflow was used to maintain the partial pressure of the liquid species at zero at the top of the
capillary tube. A camera was used to take images of the level of the liquid contained in the capillary tube.
The readings of the liquid levels at the preferred temperature appeared on the digital display and a
personal computer was attached to the apparatus by a USB port to capture and record these images.
Figure 1: This figure represents the Armfield Gaseous Diffusion Coefficient Apparatus and the various
important components that will be utilized.
Experimental Design
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In each experiment, the independent variable in the design was the time while the height from the
top of the tube was the response variable. The density of the the liquid being tested, ideal gas constant,
temperature, partial pressures, molar mass, and pressure were assumed constant throughout the
experiments. The temperature was varied, and the type of compound was changed between acetone and
ethanol. The calculated variable was the diffusivity in all the trials. For the first experiment, the
diffusivity of acetone was measured at a temperature of 50 °C. This experiment with acetone at 50 °C was
repeated for a total of three times. For the second experiment, the diffusivity of acetone was measured at
40 °C once. In the third experiment, acetone was substituted for ethanol, the temperature was kept at 50
°C, and the diffusivity of ethanol at 50 °C was measured once.
Method of Analysis
For each time interval, the height and time values for acetone and ethanol were input into
equation 1 to determine the diffusivity at the successive time points. The diffusivity values of acetone
were compared, using two separate unequal variance t-tests, to a diffusivity value found in literature and a
diffusivity value calculated using equation 4. It was determined, using the t-tests, if there was a
statistically significant difference between the measured diffusivities and those found in literature.
Likewise, it was determined if there was a statistically significant difference between the measured
diffusivities and those that were calculated using equation 4. The diffusivity values of acetone for the tests
at 40 ℃ and 50 ℃ were compared to one another using equal variance t-tests to determine if there was a
statistically significant difference between the data. Another equal variance t-test was used to determine if
there was a statistically significant difference between the diffusivity of acetone and the diffusivity of
ethanol at 50 ℃.
A plot was made of the height of liquid acetone vs time for the 40 ℃ and 50 ℃ tests. This plot
displayed time on the x-axis and (𝑧𝐹2 − 𝑧02 ) ∗ 𝐶 on the y-axis where 𝐶 =
𝜌𝐴 ∗𝑅∗𝑇∗𝑝𝐵𝑀
.
2∗𝑀𝐴 ∗𝑃∗(𝑝𝐴1 −𝑝𝐴2 )
A least squares
regression was fit to the data. From rearranging equation 1, the slope of the regression corresponds to the
experimental 𝐷𝐴𝐵 value.
Safety
Safety glasses are very important objects to be used in the lab. They should be worn as soon as
one enters the lab. Acetone is a very flammable substance, the use of gloves is advised when handling
acetone and capillarity tubes. Also, it is advised not to touch the apparatus with bare hands in order to
avoid burns; this is because surfaces might be hot on the apparatus. Extreme care should be taken when
4
handling glass capillary tubes because the tubes are very delicate and difficult to replace. It is important to
lubricate capillary tube with silicone before inserting the tube into rubber gland; this will help the tube to
slide in smoothly. It is advised to use temperatures below 60 °C as a set point, so the equipment being
used is not damaged. To avoid the acetone from boiling, it is important to set the temperature controller
no higher than 50 °C (Wettstein).
Anticipated Results
Table 1: A table of the various T-tests performed against the experimental acetone at 50 °C and the
corresponding P-values.
P-value at 95% Confidence Interval
Unequal Variance T-Test: Experimental Diffusivity Acetone at 50
2.6 ∗ 10−12
°C versus Calculated Diffusivity
Unequal Variance T-Test: Experimental Diffusivity Acetone at 50
0.052
°C versus Diffusivity of Acetone in Literature at 50 °C
Equal Variance T-Test: Experimental Diffusivity Acetone at 50 °C
0.31
versus Experimental Diffusivity Acetone at 40 °C
Equal Variance T-Test: Experimental Diffusivity Acetone at 50 °C
0.017
versus Experimental Diffusivity Ethanol at 50 °C
Two unequal variance T-tests were performed comparing the experimental diffusivity data of
acetone at 50 °C against the calculated diffusivity and a diffusivity value found in literature. The Pvalues, which are all listed in Table 1, were 2.6 ∗ 10−12 and 0.052 respectively, so the experimental
diffusivity of acetone was found to be statistically different at the 95% confidence level than the
calculated diffusivity. In contrast, the experimental diffusivity and value found in literature were not
found to be statistically different at the 95% confidence level.
An equal variance T-test was performed on the experimental diffusivity of acetone at 50 °C
against the experimental diffusivity of acetone at 40 °C, and a P-value of 0.31 was obtained. The P-value
was greater than 0.05, so the data was not statistically different at the 95% confidence level. Another
equal variance T-test was performed on the experimental diffusivity of acetone at 50 °C and the
5
experimental diffusivity data of ethanol at 50 °C. The P-value was 0.017 between acetone and ethanol at
50 °C, so the data was statistically different at the 95% confidence level.
A regression analysis was performed in Figure 2 that returned a value of 10.06 for the slope,
which corresponds with the diffusivity in millimeters squared per second. The slope is within one
standard deviation of the mean, 10.28, for the experimental data. The R2 value was also 0.999, so Figure 2
shows a strong linear relationship between time and the distance down the tube squared.
Distance from Top of Tube Squared
times C (mm2 *(min/s))
2000
y = 10.056x
R² = 0.99888
1600
1200
800
400
0
0.00
50.00
100.00
150.00
200.00
Time (min)
Figure 2: The liquid level in the tube is from the top of the tube squared against the time it took for the
level to drop.
6
References
Ambrose, D.; Sprake, C.H.S.; Townsend, R., Thermodynamic properties of organic oxygen compounds
XXXIII. The vapour pressure of acetone, The Journal of Chemical Thermodynamics, 1974, 6, 7,
693-700.
Ambrose, D.; Sprake, C.H.S.; Townsend, R., Thermodynamic Properties of Organic Oxygen
Compounds. XXXVII. Vapour Pressures of Methanol, Ethanol, Pentan-1-ol, and Octan-1-ol
from the Normal Boiling Temperature to the Critical Temperature, J. Chem. Thermodyn., 1975,
7, 2, 185-190.
The College Board. “Diffusion and Osmosis.” Cellular Processes: Energy and Communication,
media.collegeboard.com /digitalServices/pdf/ap/bio-manual/Bio_Lab4-DiffusionandOsmosis.pd.
Fogler, and Gurmen. “Prediction of Binary Gas Diffusivities.” Chapter 11,
www.umich.edu/~elements/fogler&gurmen/html/course/lectures/eleven/exam4.htm. Accessed
25 Sept. 2017
Wettstein, Stephanie, Dr. Gas Diffusion Apparatus. N.d. Description of Procedure For ECHM 442
Vapor Diffusivity Experiment.
Whitmer, Phil. “Why Is Diffusion Important to the Life of a Cell?” Sciencing, 24 Apr. 2017,
sciencing.com/diffusion-important-life-cell-8759126.html. (intro 2)
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Appendix
Table A1: Statistical Analysis for t-test assuming Unequal Variances
for acetone diffusivity at 50 °C versus calculated diffusivity
Variable 1
Variable 2
Mean
10.28200296
12.1883
Variance
1.789583583
0
45
2
Observations
Hypothesized Mean Difference
df
t Stat
0
44
-9.559184329
P(T<=t) one-tail
1.31684E-12
t Critical one-tail
1.680229977
P(T<=t) two-tail
2.63367E-12
t Critical two-tail
2.015367574
Table A2: Statistical Analysis for t-test assuming Unequal Variances
for acetone diffusivity at 50 °C versus diffusivity in Literature
Variable 1
Variable 2
Mean
10.28200296
10.68
Variance
1.789583583
0
45
2
Observations
Hypothesized Mean Difference
df
t Stat
0
44
-1.995768217
P(T<=t) one-tail
0.02608623
t Critical one-tail
1.680229977
P(T<=t) two-tail
0.05217246
t Critical two-tail
2.015367574
8
Table A3: Statistical Analysis for t-test assuming Equal Variances
for acetone diffusivity at 50 °C versus acetone diffusivity at 40 °C
Variable 1
Variable 2
Mean
10.28200296
9.955009074
Variance
1.789583583
2.856109236
45
45
Observations
Pooled Variance
2.32284641
Hypothesized Mean Difference
df
0
88
t Stat
1.017702047
P(T<=t) one-tail
0.155804711
t Critical one-tail
1.662354029
P(T<=t) two-tail
0.311609421
t Critical two-tail
1.987289865
Table A4: Statistical Analysis for t-test assuming Equal Variances
for acetone diffusivity at 50 °C versus ethanol diffusivity at 50 °C
Variable 1
Variable 2
Mean
10.28200296
11.36352162
Variance
1.789583583
7.135198177
45
45
Observations
Pooled Variance
Hypothesized Mean Difference
df
t Stat
4.46239088
0
88
-2.428518792
P(T<=t) one-tail
0.008597721
t Critical one-tail
1.662354029
P(T<=t) two-tail
0.017195441
t Critical two-tail
1.987289865
9
Table A5: Blank Data Sheet for acetone at 40 °C
40 °C Acetone
Time
Height Z0
Height Zf
Time
Height Z0
Height Zf Acetone
(min)
Acetone (mm)
Acetone (mm)
(min)
Acetone (mm)
(mm)
4
96
8
100
12
104
16
108
20
112
24
116
28
120
32
124
36
128
40
132
44
136
48
140
52
144
56
148
60
152
64
156
68
160
72
164
76
168
80
172
84
176
88
180
92
10
Table A6: Blank Data Sheet for acetone at 50 °C
50 °C Acetone
Time
Height Z0
Height Zf
Time
Height Z0
Height Zf Acetone
(min)
Acetone (mm)
Acetone (mm)
(min)
Acetone (mm)
(mm)
4
96
8
100
12
104
16
108
20
112
24
116
28
120
32
124
36
128
40
132
44
136
48
140
52
144
56
148
60
152
64
156
68
160
72
164
76
168
80
172
84
176
88
180
92
11
Table A7: Blank Data Sheet for ethanol at 50 °C
50 °C Ethanol
Time
Height Z0
Height Zf
Time
Height Z0
Height Zf Acetone
(min)
Acetone (mm)
Acetone (mm)
(min)
Acetone (mm)
(mm)
4
96
8
100
12
104
16
108
20
112
24
116
28
120
32
124
36
128
40
132
44
136
48
140
52
144
56
148
60
152
64
156
68
160
72
164
76
168
80
172
84
176
88
180
92
12
Sample Calculations:
𝜌 ∗(𝑧𝐹2 −𝑧02 )∗𝑅∗𝑇∗𝑝𝐵𝑀 (784 𝑘𝑔/𝑚3 )∗(322 𝑚𝑚2 −02 )∗(8.314 𝐽/𝑚𝑜𝑙/𝐾)∗(323 𝐾)∗(50112 𝑃𝑎)
=
=
2∗(0.05808 𝑘𝑔/𝑚𝑜𝑙)∗(10800 𝑠)∗(101000 𝑃𝑎)∗(80900 𝑃𝑎)
𝐴 ∗𝑡𝐹 ∗𝑃∗(𝑝𝐴1 −𝑝𝐴2 )
𝐴
Equation 1: 𝐷𝐴𝐵 = 2∗𝑀
𝑚𝑚2
𝑠
10.54
Equation 2: 𝑝𝐵𝑀 =
𝑝𝐴1 −𝑝𝐴2
(𝑃−𝑝𝐴2 )
]
(𝑃−𝑝𝐴1 )
𝑙𝑛[
Equation 3: 𝑝𝐴1 = 10
(𝐴−
𝐵
)
𝐶+𝑇
=
80900
𝑙𝑛[
= 50112 Pa
(101000))
]
(101000−80900)
=10
1312.253
)
−32.445+323
(4.42448−
1
1
+
)1/2
58 𝑔/𝑚𝑜𝑙 28.97 𝑔/𝑚𝑜𝑙
(1 𝑏𝑎𝑟)[(66.86)1/3 +(20.1)1/3 ]2
1
1
+ )1/2
𝑀𝐴 𝑀𝐵
𝑣𝐴 )1/3 +(∑
𝑣𝐵 )1/3 ]2
1.00𝑥10−1 𝑇 1.75 (
Equation 4: Calculated diffusivity value, 𝐷𝐴𝐵 =
1.00𝑥10−1 (323𝐾)1.75 (
= 0.809 bar
𝑃[(∑
=
𝑚𝑚2
𝑠
=12.19
Experimental Protocol
1. Put on closed toed shoes, long pants, and safety glasses.
2. Plug in the Armfield Gaseous Diffusion Coefficient Apparatus to turn on the unit.
3. Connect the computer to the apparatus with the USB port and start the software program, eScope.
4. Turn on the temperature moderator on the PID controller (digital display) and set the desired
temperature to 50 °C by pressing the F button, using the up and down arrows to set the
temperature, hitting the F button again, and waiting eight seconds.
5. Wash the capillary tube with acetone.
6. Use a hypodermic syringe to fill the capillary tube with acetone to a height of 35 mm.
7. Carefully insert the capillary tube into the apparatus through the ring at the top left of the
apparatus. Add lubricant if necessary.
8. Check if the liquid level can be seen on the digital display screen and if the resolution (on the top
menu bar) is set to 640x480. Adjust if necessary.
9. Carefully, connect the air tubing from the fan located on the top of the apparatus to the tube piece
located above the capillary tube.
10. Check that the air is blowing across the tube.
11. On the computer, left click on the measure drop down icon and click ‘Calibration’.
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12. Left click on the scale on the left of the image, and count down a distance of 5 mm. Left click
again on the scale, and type 5 mm in the actual dimension box. Click the ok box.
13. Left click the clock image for the time lapse icon. Select the time interval between each image
capture as 240 seconds. Select the total number of images as 45.
14. Left mouse click the ‘Start’ button to begin the time lapse feature.
15. Record all height changes from the image captures by double left clicking on the images on the
left and then clicking the ‘measure’ icon and select ‘straight angle line’ to measure the level at the
meniscus. Left click the mouse at the lowest point of the meniscus and left click at the top of the
screen. Then move the text where it can be read and record the data.
16. When the liquid runs out, carefully remove the air tubing from the tube piece located above the
capillary tube.
17. Carefully remove the capillary tube.
18. Save the data by clicking the floppy disc icon at the top of the screen. Name and save the file.
19. Repeat steps 4-18 two more times.
20. Set the desired temperature to 40 °C for the second experiment for acetone at 40 °C.
21. Repeat steps 5-18 for the second experiment.
22. Set the desired temperature to 50 °C for the third experiment for ethanol at 50 °C.
23. Wash the capillary tube with acetone.
24. Use a hypodermic syringe to fill ethanol in the capillary tube to a height of 35 mm.
25. Repeat steps 7-18 using ethanol as the liquid at a temperature of 50 °C.
26. Wash the capillary tube with acetone.
27. Disconnect the apparatus from the computer, close the eScope program, and unplug the Armfield
Gaseous Diffusion Coefficient Apparatus.
28. Leave the lab and take off your closed toed shoes, pants, and safety glasses.
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