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Report on:
The Diffusivity of Glucose
in Water at 25°C
Submitted to:
Professor John Bonham
Department of Chemical Engineering
Bron Yr Aur University
Prepared by:
John Paul Jones
Jimmy Page
Robert Plant
ChE 3211 Chemical Engineering Laboratory
25 September 1980
Abstract
A temperature controlled Stokes diaphragm cell was used to investigate the diffusivity of
glucose in water. A sintered glass frit was used for the diaphragm. The cell was calibrated
with a single diffusion experiment using ethanol in water. In this way, the cell constant was
indirectly measured to be (2170 ± 90) m-2 at 25°C. The diffusion coefficient for dilute
glucose in water at 25°C was then determined from five diffusion experiments to be
(7 ± 1)×10-10 m2∙s-1 [95% Confidence].
- ii -
Table of Contents
Abstract ..................................................................................................................................... ii
Introduction ................................................................................................................................1
Theory ........................................................................................................................................1
Experimental Methods ...............................................................................................................3
Results ........................................................................................................................................4
Conclusions ................................................................................................................................5
References ..................................................................................................................................5
Notation......................................................................................................................................6
Appendix A: Work Plan.............................................................................................................7
Appendix B: Data Sheet.............................................................................................................8
Appendix C: Sample Calculations .............................................................................................9
Appendix D: Uncertainty Analysis .........................................................................................11
Appendix E: Safety and MSDS ..............................................................................................14
List of Figures
Figure 1. Stokes magnetically stirred diaphragm cell ...............................................................1
Figure 2. Calibration data for the diffusion of ethanol in water at 25°C ..................................4
Figure A.1. Gantt Chart Summary for Work Plan .....................................................................7
List of Tables
Table I. Experimental results for diffusivity of glucose in water at 25°C. ................................5
Table B.I. Stokes cell calibration data using ethanol in water at 25°C.....................................8
Table B.II. Experimental results for glucose diffusivity at 25°C after 48 hours ......................8
Table C.I. Stokes cell calibration data analysis from Excel .....................................................9
Table D.I. Experimental measurement uncertainties ..............................................................11
- iii -
Introduction
Diffusion coefficients are important properties in the study and design of rate controlled
separation processes. The Stokes diaphragm cell is probably the best tool to start
research on diffusion in liquids. It is inexpensive to build, rugged enough to use in an
undergraduate lab, yet capable of accuracies as high as 1%. In this work, a diaphragm
cell, similar to the one described by Cussler [1], was used to investigate the diffusivity of
glucose in water at 25°C. The Stokes cell consisted of two compartments separated by a
horizontal semi-permeable diaphragm, as shown in Figure 1. The upper and lower
compartments were initially filled with pure water and a glucose-water mixture,
respectively.
Figure 1. Stokes magnetically stirred diaphragm cell. (A) solution A; (B) solution B; (M) magnet;
(D) porous diaphragm; (R) and (S) glass stirrers enclosing iron wire; (W) level of thermostat
water; (P) vented glass stopper; (Q) glass stopper with stopcock.
Theory
It is assumed that the flux of solute across a semi-permeable diaphragm quickly reaches
its steady-state value; even though the concentrations in the upper and lower
compartments are changing with time [2]. In this pseudo-steady state, the flux across the
diaphragm is proportional to the solute concentration difference across the diaphragm:
1
(
Here,
(1)
)
is the solute diffusion coefficient, is the tortuosity of the diffusion path length
through the diaphragm,
is the nominal diaphragm thickness, and
is the concentration
of the diffusing solute 1 in the upper or lower compartment. The overall mass balances
on the adjacent compartments are:
(2)
(3)
where
is the nominal surface area of the diaphragm and is the diffusion time.
Subtract Equation 3 from Equation 2, rearrange, and combine with Equation 1 to give
(
)
(
)
(4)
The cell calibration constant, ,
(
)
(5)
is characteristic of the particular diaphragm cell. The cell constant is determined
experimentally by calibrating the cell with a compound of known diffusivity. Equation 4
is integrated subject to the initial condition
(6)
to give
(7)
Finally, Equation 7 is rearranged for the diffusivity:
(
)
2
(8)
Given the cell constant, Equation 8 may be used to determine diffusion coefficients by
measuring the concentration difference between the compartments with time.
Experimental Methods
Details of the laboratory procedure are provided in the Work Plan (Appendix A). The
diaphragm cell consists of two glass compartments separated by a horizontal sintered
glass frit. As shown in Figure 1, the frit lies in the horizontal plane to minimize the
effects of free convection [3]. The temperature was maintained by immersing the cell in
a constant temperature water bath, controlled to within ± 0.1°C. The two compartments
were stirred at about 60 RPM with a magnet rotating around the bath and cell. Initially
the two compartments were filled with solutions of different concentrations. When the
experiment was complete, the two compartments were emptied and the difference in the
two solution concentrations was measured with a differential refractometer. The
diffusion coefficient was then calculated according to Equation 8.
The cell constant was determined by calibrating the device with the well-known
value for the diffusivity of ethanol in water, (1.28 ± 0.05)×10-9 m2·s-1, reported in the
International Critical Tables [4]. The cell constant was determined from a linear leastsquares regression of the data for concentration difference versus time:
(
)
Here, the independent variable is time, , the slope of the line is the product
(9)
, and the
dependent variable is the natural logarithm of the ratio of initial to final concentration
differences between the two compartments. The cell constant, , was then found by
dividing the slope of the line by the known standard diffusivity of ethanol in water.
The glucose-in-water diffusion experiments were performed with an initial
concentration of 1 M glucose in water in the lower compartment, and pure water in the
upper compartment. All of the experimental runs were stopped after 48 hours. Due to
the relatively long run times, only the initial and final concentration differences were
measured.
3
Results
The cell calibration data using ethanol in water are listed in Appendix B and plotted in
Figure 2. A linear least squares regression of the data gave a value for the slope of
(0.0099 ± 0.0002) h-1(Appendix C). The uncertainty in the slope was taken as the
standard error of the linear fit. The cell constant was calculated by dividing the slope by
= (2170 ± 90) m-2. The
the diffusivity of ethanol at the experimental conditions for
experiment to determine the diffusivity of glucose in water at 25°C was repeated six
times. One datum point was eliminated on the basis of Chauvenet’s criterion, as outlined
in Appendix C. The raw experimental diffusion data are listed on the laboratory Data
Sheet in Appendix B. Results for glucose diffusivity in water are given in Table 1. The
mean value for the diffusivity was (7 ± 1)×10-10 m2∙s-1 [95% Confidence]. Details of the
uncertainty analysis for the cell constant and diffusivity calculations are provided in
Appendix D.
0.06
0.05
0.04
ln(D Co/D C) 0.03
0.02
0.01
0
0
2
t / hr
4
6
Figure 2. Calibration data for the diffusion of ethanol in water at 25°C.
4
Table I. Experimental results for diffusivity of glucose in water at 25°C.
/ m2∙s-1
7.06
6.90
7.25
6.79
6.82
Average 7.1  0.2
Conclusions
A Stokes diaphragm cell was used to determine the diffusion coefficient of glucose in
water at 25°C. The cell was calibrated using ethanol in water. The cell calibration
constant was determined to be (2170 ± 90) m-2. Six glucose-in-water diffusion
experiments were completed. The result from one experiment was rejected based on
Chauvenet’s criterion. An analysis of the accepted data resulted in a value of the
diffusion coefficient for glucose in water: (7 ± 1)×10-10 m2∙s-1 [95% Confidence].
References
[1]
Cussler, E. L., Diffusion Mass Transfer in Fluid Systems, 2nd ed., Cambridge
University Press, Cambridge, p. 130, 1997.
[2]
Robinson, R.A. and Stokes, R. H., Electrolyte Solutions. London: Butterworth,
1960.
[3]
Toor, H.L., “Convection and Transport in an Inclined Diaphragm Cell,” Industrial
and Engineering Chemistry Fundamentals, vol. 6, pp. 454-457, 1967.
[4]
International Critical Tables, vol. 5, p. 63, 1926.
5
Notation
diaphragm surface area, m2
concentration of species 1, mol·dm-3 or M
diffusion coefficient, m2·s-1
flux of species 1, mol·m-2·s-1
contact time, h
volume of sample in upper or lower chamber, m3
Greek Letters
diaphragm characteristic constant, m-2
diaphragm thickness, m
diaphragm porous tortuosity
Subscripts and Superscripts
condition in the lower compartment
condition in the upper compartment
initial condition
6
Appendix A: Work Plan
1.
2.
Check the operation of the apparatus, including the stop watch, constant temperature bath,
magnetic sitter, and differential refractometer. (2 h)
Clean the diaphragm cell. (1 h)
3.
Collect the ethanol and glucose. Review MSDS for potential hazards, lab safety
precautions and waste treatment. (Appendix E). (0.5 h)
4.
Calibrate the cell with an ethanol-water mixture. (6 h)
Time/h
0
1
2
3
4
5
∆C/M
1
5.
Analyze the calibration data for the cell constant. (1 h)
6.
7.
Run the experiments for glucose diffusivity at 25°C for 48 hours. (2 d)
Check the data. Is it reasonable? (1 h)
8.
Repeat step 6 at least five times. (2 w)
Experiment
1
2
3
4
5
6
9.
10.
∆C˚/M
∆C/M
Analyze the data. (2 h)
Write and bind the report. (2 d)
Plan Experiment
Prelab Design Review
Collect Data
Analyze Data
Write Report
Bind Report
1/1
1/8
1/15
Figure A.1. Gantt Chart Summary for Work Plan.
7
1/22
Appendix B: Data Sheet
Table B.I. Stokes cell calibration data using ethanol in water at 25°C.
/h
(
)/ M
0
1.000
1
0.989
2
0.978
3
0.970
4
0.963
5
0.951
Table B.II. Experimental results for glucose diffusivity at 25°C after 48 hours.
Experiment
(
)/ M (
)/ M
1
0.991
0.761
2
1.011
0.782
3
1.013
0.771
4
1.002
0.233
5
0.981
0.762
6
1.021
0.791
Mean
1.002
0.682
St. Deviation
0.015
0.220
Standard Error
0.006
0.090
8
Appendix C: Sample Calculations
1. Stokes Cell Calibration Analysis
The spreadsheet software Excel 5.0 was used to regress the calibration data by the linear
least squares method. The spreadsheet results are given in the following table.
Table C.I. Stokes cell calibration data analysis from Excel.
t/h
ln(∆C°/∆C)
0
0.000
1
0.011
Multiple R
0.99731682
2
0.022
R Square
0.99464084
3
0.030
Adjusted R Square
0.79464084
4
0.038
Standard Error
0.00132939
5
0.050
Observations
Regression Statistics
Coefficients
Standard Error
Intercept
0
X Variable 1
6
N/A
0.00994545
0.00017925
The slope of the line is the coefficient of the X Variable 1.
2. Diffusion Data Analysis
The diffusion coefficient for glucose in water was calculated according to Equation 8:
(
)
Substituting the data for Experiment 1 from Table B.II gives
(
)(
)(
(
)
)
The other values for diffusivity were calculated in a similar manner.
3. Chauvenet’s Criterion for Data Rejection
Outlying, or suspicious data, may be systematically rejected through the application of
Chauvenet’s criterion; that is, a suspect datum point may be rejected if the expected
9
number of outlying data points, which are at least as bad as the suspect data, is less than
1/2. Follow these steps to apply Chauvenet’s criterion.
1.
Assuming the data is normally distributed, first calculate the mean, ̅ , and
standard deviation, , of
2.
data points (including any suspicious data,
Next, calculate the number of standard deviations that the suspect datum
point,
, lies from the mean:
|
3.
).
̅|
(C.1)
Then find the probability that a legitimate datum point will deviate from x by
t or more standard deviations:
(
̅
)
̅
∫
√
(C.2)
(See Normal Error Integral Tables for approximate values of the integral).
4.
Finally, calculate the number of data points expected to be at least as bad as
, and apply Chauvenet’s rejection criterion:
(
5.
̅
̅
)
(C.3)
If Chauvenet’s criterion holds, reject the suspect datum point and recalculate
the mean and standard deviation. Although not generally recommended, you
may repeat these steps to eliminate other suspect data.
Chauvenet’s criterion was applied to the diffusion data in Table B.II to eliminate
the single outlying datum point resulting from Experiment 4.
|
(
(
̅
|
̅
)
̅
̅
)
The suspect datum point was rejected.
10
Appendix D: Uncertainty Analysis
This appendix on uncertainty analysis is divided into two sections: (1) diaphragm cell
calibration, and (2) glucose diffusion measurements. The uncertainties in the
experimental measurements are listed in Table D.I.
Table D.I. Experimental measurement uncertainties.
Physical Quantity
Uncertainty and Units
± 0.1 h
± 0.005  0.006 M
2
2
= ± 0.008 M
±0.5 m2·s-1
1. Diaphragm Cell Calibration Uncertainty
The diaphragm cell was calibrated with ethanol in water. The known value for the
diffusivity of ethanol in water is 1.28 × 10-9 m2·s-1. A linear least squares regression of
the calibration data gave
(D.1)
√(
√(
)
)
(
(
(
(D.2)
)
)
)
(
2. Diffusion Coefficient Uncertainty
The Root Sum Squared formula was applied to Equation 8 to calculate the systematic
uncertainty propagation:
11
)
{*
(
)+
* (
)
* (
)
*
(
)+
(
)
(
)
+
(D.3)
+ }
Substitute the numerical values:
{*
(
) (
)
[(
)
[(
)
(
)+
[
(
)(
(
)(
)(
)
(
)(
)(
)
)
(
)]
]
] }
)
√(
Note that the largest contribution to the uncertainty is the calibration constant.
However, the concentration difference uncertainties are of the same order of magnitude.
Hence, improvements in the cell calibration are not expected to improve the uncertainty
in the diffusion calculations.
The total systematic and random uncertainties from the standard uncertainty (Table
I) give
√(
)
(
)
Degrees of Freedom from Welch-Satterthwaite formula:
12
v
DD 4
 D  C1,olow  C1,oup
 2 ln 
  t  C1,low  C1,up
nc  1
 D  C1,olow  C1,oup  
 D  C1,low  C1,up  







o
o

t
C

C

t
C

C







1,
low
1,
up
1,
low
1,
up


 


ne  1
ne  1
4
4
 5.2 10 
4
11 4

 1.7 2 0.0042 0.62 1.12  10142





5
5
5   3600 4
 4
 11.85  (4  5  5  5  19)  11
t-statistic coverage factor k = 2.2 for 11 degrees of freedom.
Expanded Uncertainty
-
13
-
(D.4)
Appendix E: Safety and MSDS
Laboratory safety measures:
The apparatus will be operated at a safe-to-touch temperature.
Glucose is not considered a hazardous substance under conditions of normal
industrial use. Use eye protection at all times and respiratory protection when in solid
form. The waste solutions may be disposed of down the drain.
The following summarizes important safety information from the MSDS.
Potential Health Effects

Eye: Dust may cause mechanical irritation.

Skin: Dust may cause mechanical irritation. Low hazard for usual industrial
handling.

Ingestion: May cause irritation of the digestive tract.

Inhalation: No hazard expected in normal industrial use. May cause Respiratory
tract irritation.
Handling

Use with adequate ventilation. Minimize dust generation and accumulation.
Dusts at sufficient concentrations can form explosive mixtures with air.

Extinguishing Media: Use water spray, dry chemical, or carbon dioxide.

Spills/Leaks: Vacuum or sweep up material and place into a suitable disposal
container.

Avoid contact with skin and eyes. Avoid ingestion and inhalation.

Storage: Store in a cool, dry, well-ventilated area away from incompatible
substances.

Chemical Stability: Stable under normal temperatures and pressures.

Incompatibilities with Other Materials: Sodium peroxide + potassium nitrate,
strong oxidizers.

Hazardous Decomposition Products: Carbon monoxide, carbon dioxide.
14