PROJECT FINAL REPORT COVER PAGE
GROUP NUMBER: ____R2____
PROJECT TITLE:_PERMEABILITY OF DIALYSIS TUBING TO POTASSIUM CHLORIDE
DATE SUBMITTED: ____5-7-01____
ROLE ASSIGNMENTS
ROLE
GROUP MEMBER
FACILITATOR……………………….._______Edward Hwu_______
TIME & TASK KEEPER………………______Amelia Zellander____
SCRIBE………………………………..______Angela Xavier______
PRESENTER………………………….______Anoop Kowshik_____
PLANNER…………………………….._____Kinnari Chandriani____
SUMMARY OF PROJECT
The average permeability constant of potassium chloride through Fisher Scientific #21-15218 dialysis tubing is found to be 0.00063  7.63E-05 (95% CI). The precision of the seven trials
was 13.1%. Because literature values were unavailable, accuracy had to be determined by another
method. Given this method, the accuracy was determined to be 2.54%. However, the seven trials
conducted yielded significantly different permeability constants. Statistical differences were
concluded from t-tests and non-overlapping 95% CI. Although initial concentration of KCl solution
inside the membrane was not expected to affect the permeability constant, the results show the
contrary. The unexpected results and the presence of no trends between any trials indicate that a
complex mechanism must be affecting the diffusion of KCl through the membrane.
OBJECTIVE:
The objective of the experiment was to determine the permeability constant of potassium
chloride through Fisher Scientific #21-152-18 dialysis tubing within a precision of 5%. An aim of
this experiment was to observe the affects of varying initial concentrations (0.5 M, 1.0 M, and 3.0
M) of potassium chloride solutions on the permeability constant. Because the permeability constant
describes the filtering capabilities of dialysis tubing, the value obtained would provide valuable
information for use in dialysis. The permeability constant is directly related to the time needed for
purification of a solution, which can be used in various laboratory and medical applications (such as
purification of proteins and hemodialysis).
BACKGROUND:
Before interpreting the data, several information sources were used. To relate concentration
to the equivalent conductance of the solution, the CRC Handbook of Chemistry1 was consulted. The
handbook lists several concentrations (M) of KCl and their associated conductances (S-cm2/mol).
From the provided data, a relationship between conductance and concentration of KCl was
determined.
To convert the measured conductance, Physical Chemistry for the Chemical and Biological
Sciences2 was used. It provided the necessary equations required to calculate the equivalent
conductance, which was then used to calculate the concentration of the solution. In addition, the
book presented basic background of the nature of conductance and physical chemistry of solutions.
The Cole-Parmer Model 19101-10 digital conductivity meter manual3 was also consulted to
understand how the machine worked. It provided information regarding the cell constant used in
calculations. Lastly, Robert Gagnon’s website4 provided the platform to derive equations to solve
the cubic equation in converting conductance to concentration.
The permeability constant of a membrane is a quantitative measure of the diffusion rate
across a cell boundary. Because it greatly pertains to biological systems, there has been a great deal
of research regarding membrane stretching and a consequent increase in permeability. Research on
mitochondrial membranes has concluded that swelling is a major contributor to changes in the
structuring of the membrane and its permeability.5 Other research in the past has also indicated
that swelling of the polymer due to preparation conditions accounts for changes in the permeability
constant.6
THEORY AND METHODS OF CALCULATIONS:
The permeability constant (kp) is a characteristic of a membrane that is measured with
respect to the flux and concentration gradient of the two solutions. Given only a conductivity
meter, by using Kohlrausch’s relationship and the definition of equivalent conductance a
relationship between conductance and concentration can be constructed.
Kohlrausch’s Relationship
Λ = Λo –B√c
m
(1)
C kcell
c
(2)
For a solution of KCl, the molar conductance is equal to the equivalent conductance. Using
values from the CRC Handbook of Chemistry and combining the above two equations, a cubic
equation in terms of concentration is derived. Because Microsoft® Excel cannot solve this
equation, a formula is supplied in the appendix to help with the conversion from conductance to
concentration.
B c
3
2
  0 c  C kcell
0
(3)
Where
Λ
= Equivalent conductance
Λo = Equivalent conductance at an infinite dilution
B
= a constant dependent on the ion
c
= concentration
Λm = Molar conductance
C
= Conductance
kcell = Cell Constant
By using the definition of flux and the conservation of mass formula, an equation in terms of
concentration can be achieved.
Flux
d
mass
dt
ci( 0)  Vi
kp A   ci( t)  co( t) 
(4)
ci( t)  Vi( t)  co( t)  Vo
(5)
Where
Kp = Permeability Constant
A
= Surface Area
ci(t) = Concentration of the inside compartment
c0(t) = Concentration of the outside compartment
Vo = volume in outside compartment
Vi = volume in inside compartment
Combining these equations together, and then reducing the equation, kp can be determined.
ln Vi ci ( 0)  co ( t)  Vo  Vi
ln Vi ci ( 0)  co ( t)  Vo  Vi
 A Vo  Vi  t  ln V  c  ( 0)

 i i 
 Vo Vi

K p 
K p 
A
 Vi
 t   lnVi ci ( 0)

(7)
(8)
Using equation (8), the concentration of KCl can be calculated at any time during the experiment.
MATERIALS/APPARATUS:
1. Potassium Chloride (KCl) provided by Fisher Scientific
2. DI Water
3. Cole-Parmer Model 19101-10 Digital Conductivity Meter
4. Fisher Scientific #21-152-18 Dialysis Tubing
5. Spectrum dialysis tubing clamps
6. Fishing Line and swivel apparatus
7. Fisher Scientific Mini-Pump Variable Flow
8. Magnetic stirrer, rod, and mouse pad
9. Bucket large enough to hold the dialysis tubing apparatus. A 2.0 L container was used.
10. Mettler H72 electronic balance, Mettler BD6000 electronic balance
11. 10 mL plastic pipette, P-1000 air displacement pipette
12. Suspension Apparatus:
a. Ring stand
b. Fishing Line
PROCEDURE:
Preparation of Solutions
Three potassium chloride solutions (0.5 M, 1.0 M, 3.0 M) were prepared using solid Fisher
Scientific KCl. The largest concern was contamination, especially from skin contact, so gloves
were worn and all apparatus and materials were always rinsed with deionized water before use.
Large volumes of the potassium chloride solutions were made to reduce the effects of mass transfer
loss. In addition, DI water was used to rinse all the KCl from the weighing boat into the flask.
18.6375 g KCl were weighed out for the 0.5 M solution, 37.275 g KCl were weighed out for the 1.0
M solution, and 111.8445 g KCl were weighed out for the 3.0 M solution all on the Mettler H72
balance. The salt was added to a 500 mL volumetric flask and deionized water was added to the
mark. The flasks were covered with Parafilm to prevent evaporation of the solution and the solution
was allowed to equilibrate to room temperature.
Preparation of Dialysis Tubing and Bucket
The dialysis tubing was cut into approximately 12 cm pieces. The tubing lengths were
soaked in DI water and stored in the refrigerator until the morning when they were to be used. For
each trial, the inside of the tubing was rinsed with KCl solution of the same concentration to be used
for that trial. The tubing was then clamped on one end and 10.0 mL of KCl solution was transferred
in using a 10 mL plastic pipette. The tubing was then clamped on the top as close to the solution as
possible with minimal loss of solution, where usually one air bubble remained. Fishing line was
tied to the top clamp so that its free end could hang from the hook on the ring stand. This allows for
the free rotation of the dialysis tubing suspension. The entire tubing setup, which includes the
tubing filled with solution, the clamps, and the fishing line were then weighed on the Mettler H72
electronic balance. Finally, the entire setup was rinsed with DI water. The tubing setup was
reweighed after the completion of the trial to check for any volume change. In addition, the volume
of the KCl solution inside of the tubing at the end of the trial was measured by removing it with a
10mL plastic pipette.
A 2.0 L bucket was placed on the Mettler BD6000 electronic balance and the balance was
tared. 1.5 L of DI water was added by mass to the bucket. A stirrer bar was placed at the bottom of
the bucket and the entire bucket was placed on the magnetic stirrer with a mouse pad separating the
two, serving as insulation from heat. Before the onset of the trials, the DI water in the bucket was
allowed to equilibrate to room temperature. Again, the bucket was reweighed after the trial was
completed to check for volume change.
Preparation of the Flow Pump, Conductivity Meter and Calibration Curve
The Model 19101-10 Digital Conductivity Meter Operations Manual was consulted for
determining the cell constant and calibrating the meter. The meter was always kept in “ATC ON”
mode during the trials. A solution of known concentration and conductivity for the same cell
constant should be used to calibrate the meter. The supplied wires were used to connect the
recorder port on the back of the conductivity meter to the analog-to-digital converter on the board.
The flow pump and conductivity cell were cleaned before each trial by running DI water
through the tubing while the pump was on. The electrode was placed in the solution with the flow
pump tubing attached to it above. The speed of the flow pump was set to 8.0 on “fast” mode.
Since the CRC handbook data was used in calculations, a calibration curve was produced
using our conductivity meter and solutions to make sure it correlated with that of the CRC
handbook. Dilutions from the 1.0 M KCl solution as prepared above were made ranging from 0.001
M to 0.01 M in increments of 0.001 M. Dilutions of such low concentrations were prepared
because the concentrations in the trials would only reach small values in this range. A P-1000 air
displacement pipette and 100 mL volumetric flask were used for the 0.006 M - 0.01 M dilutions,
while the same pipette and a 200 mL volumetric flask were used for the 0.001-0.005 M dilutions.
RESULTS:
In order to relate conductance to concentration, seven sample solutions were made to
correlate concentration with conductance. The results deviated from the CRC values by 5%.
Because of the small error of 5%, the relationship from the CRC values was applied. Figure 1,
below, relates concentration with conductance.
FIGURE 1: CALIBRATIONS
CRC Handbook
equiv conductance
[m S*cm ^2/m ol]
155000
y = -89068x + 149749
150000
R2 = 0.9997
145000
140000
135000
130000
125000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
root conc [M^0.5]
REGRESSION STATISTICS FOR FIGURE 1
Standard
Coefficients
Error
Intercept
149749.1 41.63883
X Variable 1
-89067.9 1032.932
t Stat
P-value Lower 95% Upper 95%
3596.38 7.73E-08 149569.9 149928.2
-86.2282 0.000134 -93512.2 -84623.5
Because the values of concentrations expected during the experiment were at most 0.003  0.001M,
only the CRC values below 0.005M were used. The linear regression for those points is shown
above. The slope was found to deviate by 4.9% of the mean, while the y-intercept deviated by 0.1%
of the mean.
To calculate the kp values and observe trends, many measurements were taken before and after each
trial was conducted. Table 1, below, displays the measurements that were taken for each trial.
Table 1: Initial and Final Measurements.
Results
Trial 1
Trial 2
Trial 3
Trial 4
Trial 6
Trial 7
Trial 9
Molarity
1.0
0.5
1.0
3.0
3.0
0.5
0.5
Vout (L)
Initial
1.588
1.500
1.500
1.500
1.500
1.500
1.500
Vout (L)
Final
1.55
1.483
1.449
1.494
1.502
1.493
1.495
% change
of volume
of tank
-2.39
-1.13
-3.40
-0.40
0.13
-0.47
-0.33
Vin (L)
Initial
0.006
0.010
0.010
0.010
0.010
0.010
0.010
Vin (L)
Final
0.0060
0.0110
0.0097
0.0107
0.0118
0.0096
0.0120
% change Tubing
of volume Length
in tubing Final (cm) SA (cm2)
0.00
5.9
29.5
10.00
7.4
37.0
-3.00
7.4
37.0
7.00
7.0
35.0
18.00
6.9
34.5
-4.00
7.3
36.5
20.00
8.0
40.0
Table 1 shows that there was a decrease in the outside volume over time, with an accompanied
increase of internal volume over time. The average percent change for the outside volume is 1.14%
and 6.86% for the inside volume. The percent change in volume also differs for each trial, showing
that there were no trends, even between trials of similar concentration. Further note should be made
that during the course of each trial, less than 3.2% of the total water mass was lost over time.
Because the mass loss and change in volume are insignificant, calculations were done assuming that
the volume did not change over time.
Using equation 8, a graph was constructed where the slope was defined to be kp. Figure 2 displays
the graphs of the permeability constants for all the trials.
Figure 2: Graph for Permeability Constant for the 7 Trials
Figure 2, above, displays the change in the permeability constant over time. The permeability
constant (the slope) is linear only within the first hour. After the first hour the permeability constant
gradually decreases. After 10 hours, the permeability constant for trial 3 has been reduced by a
factor of 80. The chart also shows that the starting points for the trials are clustered by
concentration.
Table 2: Permeability Constants
Permeability Constants
Trial 1
Trial 2
Trial 3
Trial 4
Trial 6
Trial 7
Trial 9
Molarity
1.0
0.5
1.0
3.0
3.0
0.5
0.5
Kp
5.56E-04
7.19E-04
5.27E-04
6.86E-04
7.19E-04
6.46E-04
5.56E-04
Descriptive Statistics for Kp
95% CI
1.60E-05
3.30E-06
3.30E-06
4.40E-06
1.90E-06
1.60E-06
2.30E-06
Mean
Standard Error
Median
Standard Deviation
Sample Variance
Range
Minimum
Maximum
Confidence Level(95.0%)
0.00063
3.12E-05
0.000646
8.25E-05
6.8E-09
0.000192
0.000527
0.000719
7.63E-05
Table 2, shows the experimental permeability constants for each trial with their 95% CI. The table
clearly shows that the 95% CI values do not consistently overlap one another. A precision of 13.1%
was calculated for the 7 trials.
Table 3: Y-intercepts
Trial 1
Trial 2
Trial 3
Trial 4
Trial 6
Trial 7
Trial 9
Molarity
1.0
0.5
1.0
3.0
3.0
0.5
0.5
Y-intercept*
Y-intercept
(theoretical)
-5.11600
-5.29832
-4.60517
-3.50656
-3.50656
-5.29832
-5.29832
(experimental)
-5.2515
-5.4222
-4.7463
-3.5709
-3.5992
-5.4122
-5.4637
95%CI
0.02273
0.004798
0.004814
0.003022
0.002732
0.003049
0.004883
% error
2.65
2.34
3.06
1.83
2.64
2.15
3.12
Average
2.54
*Y-intercept = ln(Vi*Ci(0))
Table 3, above, shows the theoretical and experimental Y-intercept values of Figure 2. As can be
seen, the experimental Y-intercepts do not match the theoretical Y-intercepts within the 95% CI,
and are consistently smaller than the theoretical values. Table 3 also indicates an average percent
error of 2.54%, indicating a high degree of accuracy.
Figure 3: Normalized Data Series
Figure 3 normalizes the seven trials to trial 9. Trial 9 was chosen because it was run for the shortest
amount of time. The data shows that for the first 16 minutes, the seven trials are all very linear.
After that time, the seven trials begin to diverge. The first to diverge are the two 1.0 M trials.
To compare the data by concentrations, three t-tests were done on the different concentrations
(1.0M vs. 3.0M, 0.5M vs. 1.0M, and 0.5 vs. 3.0M).
Table 4: T-Tests for 1.0M and 3.0M solutions
T-Test: Two-Sample Assuming Unequal Variances
1.0 M
3.0 M
Mean
0.000541 0.000702
Variance
4.14E-10 5.43E-10
Observations
2
2
Hypothesized Mean Difference
0
df
2
t Stat
-7.35848
P(T<=t) one-tail
0.008986
t Critical one-tail
2.919987
P(T<=t) two-tail
0.017972
t Critical two-tail
4.302656
Table 5: T-Tests Between Concentrations
Comparison
T-stat
T-critical
0.5 M vs 1.0 M
2.004455 4.302656
0.5 M vs 3.0 M
-1.24291 4.302656
1.0 M vs 3.0 M
-7.35848 4.302656
Table 4, displays the results for the 1.0M and 3.0 M trials. The results indicate that the kp values for
the two concentrations are significantly different. Table 5 displays the t-test results for the other
concentrations. The 0.5M vs 1.0M and 0.5M vs 3.0M solutions were found not to be significantly
different.
ANALYSIS:
The slope at a particular point in time of the graph displayed in Figure 2 was defined to be
the permeability constant at that time (by equation 8). The slope has a linear region for
approximately the first forty minutes, and then decreases towards zero gradually over time.
Although the assumption was that the permeability constant would remain constant over time, the
graph shows the permeability constant gradually decreasing to zero. The slope would be expected
to be linear while there was a concentration gradient. However, this is not the case with the
experimental data. After 2 hours, none of the solutions had reached equilibrium, yet the kp values
had drastically changed.
Table 2 lists the observed kp values with their 95% confidence limits. The mean
permeability constant was calculated to be 0.00063 (cm/hr)  7.63E-05 (95% CI). However, this
number should not be associated with the KCl permeability constant of the dialysis tubing because
of the statistical differences in trials.
The kp values obtained for each trial are displayed in Figure 2, as well as tabulated with their
95% confidence intervals, in Table 2. Comparing the values obtained for the trials with the same
initial concentrations, the kp values do not overlap each other within 95% CI, indicating that they
are significantly different. Also three different t-tests for unequal variances were performed on the
data as indicated in Tables 4 and 5. By comparing their respective t-stat and t-critical values, the
indication is that there is no significant difference between the kp values when using the initial
concentrations of 0.5M and 3M, or between 0.5M and 1M. However, the kp values obtained for 3M
and 1M initial concentrations are significantly different. Looking at the 0.5M solutions, it is also
can be observed that the 0.5M solutions had the greatest amount of spread and therefore less
internal consistency, and therefore no conclusions can be made about the 0.5M solutions. The
prediction that the permeability constant would not depend on initial concentration inside the
membrane and would be constant for all trials has been disproved. This information proves that
there is no validity to the calculated average kp value of 0.00063 (cm/hr)  7.63E-05 (95% CI). The
poor precision of 13.1% further indicates that the values deviate from each other significantly.
Table 3 shows the theoretical and experimental y-intercept values of Figure 2. Since
literature values were not available, an alternate method for checking the accuracy of the
permeability constant was devised. Because equilibrium is reached after more than twelve hours, it
was not practical to compare the final concentration of the solution to the theoretical equilibrium
concentration. Therefore, the initial concentration was used to check for accuracy. The y-intercept
represents the natural log of the initial number of moles placed into the membrane. Since the
number of initial moles for each trial is known, this value can also be calculated and compared to
the experimental y-intercept of Figure 2. The average accuracy of the y-intercepts for the trials is
2.54%. The experimental results were consistently smaller than the theoretical values, indicating
that not all of the mass is accounted for in equation 8.
The normalized data plot shown in Figure 3 further proves that there is a significant
difference between each trial. The graphs do not follow the same pattern or linearity throughout the
plot. Observance of the plot for each initial concentration indicates that the two trials for 1.0M
solutions are similar in pattern and linearity. However, there is no pattern between the trials for
3.0M or 0.5M. Using the graph and Table 1, no trends between different concentrations or within
the same concentration are observed, leading to no conclusion about the effects of concentration.
Deviation of kp over time can be attributed to two explanations. First, the dialysis tubing
could have stretched, thus changing its physical properties. As listed in Table 1, there was an
average increase of 6.86% in volume of the solution in the dialysis tubing at the end of the trials.
The change in volume due to swelling supports the claims made in past research. An increase in
volume, straining the membrane, could result in the stretching of the shape and size of the pores.
Stretching by plastic deformation occurs in polymers. (The dialysis tubing is a polymer.) The
mechanism for this deformation is lining up of the polymer chains. Because the gaps in the
polymer chains are the pores of the membrane through which molecules move, the lining up of
these chains would alter the pore shape and size. The change in pore shape will limit the KCl
passing out of the tubing. Additionally, the volume increase in the tubing increases pressure which
could increase the force applied to the inside solution by the membrane. From Figure 3, the 3.0M
solutions approaches equilibrium slower than the 1.0M, supporting this hypothesis. The comparison
between the 0.5M and 1.0M solutions disproves this hypothesis. Furthermore, the change in
volumes of the solution does not correlate with the concentrations of the solutions.
In addition, clogging of the pores by the potassium chloride ions could also explain the
changing kp. As the K+ and Cl- permeate through the membrane, they gradually build up a residue
around the pores. As the build up increases, additional charge repulsion could hinder the passage of
ions. Figure 3 indicates that the 1.0M KCl approaches equilibrium earlier than the 3.0M and 0.5M
solutions. However, assuming that KCl would hinder the passage of molecules, the 3.0M solution
would reach equilibrium first, which is not observed. The 0.5M KCl would approach equilibrium
last which is supported by Figure 3.
Since neither of the above two explanations are supported by the data, the change in kp can
not be attributed to just one factor. Rather, a more complex mechanism than the ones examined are
probably responsible for the change in kp over time. While unable to quantitatively determine the
mechanism involved, in agreement with past research, swelling and the preparation conditions are
found to contribute to the change in kp.
The average uncertainty of the permeability constants was 21%. Such a high uncertainty
suggests that the data from this experiment is inconsistent. This is further supported by the
statistical differences in all trials. Overall, the nature of the available apparatus and methods used is
insufficient for correct calculation of the permeability constant. 5% of this uncertainty is attributed
to the linear regression values derived from the values in the CRC Handbook. 8% of the uncertainty
was found to come from the preparation techniques involving the 10mL pipette. Furthermore, the
initial setup was complicated by a constant mixing of solutions. For future reference, it is suggested
to find a better method of setting up the filled dialysis tubing. In addition, measuring the volume
inside the tubing after each trial was found to be incredibly difficult. To reduce error, all
measurements should also have been done by weight for increased accuracy.
CONCLUSIONS:
1. The data was inconsistent with expected results because the permeability constant did not
remain constant over time and varies inconsistently with differing concentrations.
2. The initial permeability constant of KCl at 26oC is calculated to be between 5.3E-04 and
7.2E-04.
3. Changes in volume and mass occurred during each trial, which may be the cause of the
unexpected variance in the permeability constant.
REFERENCES:
1. Lide, David R. CRC Handbook of Chemistry and Physics. CRC press, Boca Raton,
80th edition, 1999.
2. Chang, Raymond. Physical Chemsitry for the Chemical and Biological Sciences.
University Science Books, Sausalito, CA, 3rd edition, 2000.
3. Cole-Parmer Model 19101-10 digital conductivity meter manual.
4. Gagnon, Robert. http://www.1728.com/cubic.htm.
5. Garlid, Keith D., Beavis, Andrew D. Swelling and Contraction of the Mitochondrial Matrix.
The Journal of Biological Chemistry. 260(25):13434-13441, 1985 November 5.
6. Okada J. Masuyama Y. Kondo T. An analysis of first-order release kinetics from albumin
microspheres. Journal of Microencapsulation. 9(1):9-18, 1992 Jan-Mar.
Appendix:
Solving the cubic equation: [taken from http://www.1728.com/cubic.htm ]
AUTHOR - Robert Gagnon
By using the depressed cubic equation and given a cubic equation in the form ax3 + bx2 + cx + d =
0, the following formula will solve for x (which is equal to sqrt(c))
f = [3c/a –(b/a)2]/3
g = [2*(b/a)3 – 9bc/a2 + 27d/a]/27
h = [(g/2)2 + (f/3)3]
if h > 0
m = [-(g/2) + sqrt(h)]1/3
n = [-(g/2) - sqrt(h)]1/3
X1 = (m + n) - (b/3a)
X2 = X3 = 0
if h < 0
r = [sqrt((g/2)2-h)]1/3
θ = arcos(-g/2r)
s = -r
t = cos(θ/3)
v = sqrt(3)*sin(θ/3)
w = -b/3a
X1 = 2r*cos(θ/3) + s
X2 = s(t+u) + w
X3 = s(t-u) + w