APPLICATIONS OF TECHNOLOGY IN TEACHING CHEMISTRY
An On-Line Computer Conference
June 14 TO August 20, 1993
PAPER 6
INDIVIDUAL COMPUTER-GENERATED GRAPHICAL PROBLEM SETS
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| Frank M. Lanzafame Department of Chemistry |
| Monroe Community College 1000 East Henrietta Rd. |
| Rochester, NY 14623 (716) 292-2000 Ext. 5130 |
| Internet: flanzafame@eckert.acadcomp.monroecc.edu |
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SCHEDULE: Short questions on this paper: July 5, 1993
Discussion of this paper: July 12 and 13, 1993
ABSTRACT:
A computer program has been written to generate individual graphical
problem sets for students in general chemistry. Its purpose is to
develop graphing skills through take-home assignments requiring each
student to do his own work. Data is generated for several problems
including: the equation relating different thermometer systems, Gay-
Lussac P-T data, vapor pressure versus temperature data, and
radioactive decay/first order kinetic data. Experimental statistical
fluctuations are simulated. Answers are calculated for questions
accompanying the data. Errors in all quantities are provided using
regression errors in slope and intercept and the propagation of these
errors to calculated quantities.
The generation and use of these problems in freshman chemistry will be
discussed. The program runs in QuickBasic 4.5 on IBM PC compatibles.
It is expected that the program as well as accompanying questions will
be available via anonymous FTP to conference attendees who are
interested in obtaining a copy.
I SOME STUDENT DIFFICULTIES IN WORKING WITH GRAPHS:
One of the deficiencies many students bring to general chemistry is
their lack of facility with graphing and working with linear
relationships. Some of the student problems with graphs are:
A. The inability to plot linear data, finding slope and intercept.
B. The inability to use the equation of a line to interpolate data
C. After drawing the line, students often work with two actual data
points rather than the line.
D. Counting squares of graph paper to determine slopes. This is
sometimes learned in math class.
E. The inability to conceive of a graph with a slope of several
thousand units. In math class it usually ranged between 0.5
and 2.
F. The failure to realize that slopes and intercepts have
dimensions as well as magnitudes.
It was decided to use the computer to generate individual problem
assignments involving linear relationships after noting:
1. The difficulties that many students have with graphs and
linear relationships
2. The difficulty in thoroughly testing this area on exams due
to time constraints
3. The tendency of some students to copy assignments.
The random number generation capability enabled the construction of
unique problems with simulated experimental fluctuations in the
data points.
II ADVANTAGES OF COMPUTER GENERATED PROBLEMS:
A. Each student has his own unique numbered data set. Different
sets are generated each semester with their own numbers. Since
each student has a unique problem set, there is less concern
about copying than when a single problem set is assigned.
B. Grading is simplified since the computer generates all answers
along with the standard deviations of slope and intercept, as
well as the expected range of error in derived quantities.
Alternatively, one can take the student's slope and intercept,
and use a programmable calculator such as the HP 28S to generate
the actual answers rather than use the range of values provided
by the computer. It has been found that by working carefully,
students can generally obtain results within one standard
deviation of the regression values. It is also possible to
grade with a combination of the computer generated results for
students that are within the one standard deviation and for
which computer answers are appropriate, and use the programmable
calculator to check the work of students with larger variations
in slope and intercept.
C. These problem sets can serve to test in a more secure take-home
fashion areas which can be a bit long for an hourly exam.
Credit can be given as a lab exercise, quiz, or take-home
portion of an exam. Having seen copying of laboratory reports
over the years, one is reluctant to give take home assignments
for credit. This type of individualized problem is less of a
problem, although a student might still be able to coerce
another into doing his work.
D. Inclusion of simulated experimental errors results in realistic
data which mimics experiments.
E. While many students believe that they understand how to
graphically obtain results from linear data, the results of
grading these exercises over a number of years indicates that
many overestimate their abilities. The ability to check the
results of such data analysis by students is not easily
available unless the instructor is willing to analyze each
student's results from a lab experiment or endure the copying
that takes place when a single data set is assigned.
III USES OF COMPUTER GENERATED GRAPHICAL DATA IN GENERAL CHEMISTRY:
A. TEMPERATURE CONVERSION--FIRST REVIEW OF LINEAR RELATIONSHIPS:
Early in the course, basic linear relationships are reviewed by
deriving the equation for conversion between Celsius and
Fahrenheit. (This has been done experimentally in our
preparatory course using a Fahrenheit and Celsius thermometer.)
To emphasize and drill the concepts, equations are derived for
various thermometer systems with arbitrary reference points.
Finally, the plotting of experimental data is emphasized by
using computer simulated data for calibrating arbitrarily
referenced thermometer systems. One of the two problems
intentionally generates data with positive and negative data to
test students' handling of positive and negative numbers.
B. COMPREHENSIVE GRAPHICAL EXERCISE:
Later in the semester, a more comprehensive graphical exercise
is done. It is placed after gas laws are completed and before
phase relationships are covered. The exercise has three
problems.
The following is an example of the student data which is
generated for the graphical exercise. It is printed in
compressed mode on the left side of a sheet of paper. It is
followed by the answers which are printed on the right side of
the same sheet of paper. The instructor need only cut the
answers away from the questions, and use them for grading
student results. They can be returned to the student with his
graded exercise.
SAMPLE STUDENT PROBLEM DATA
No. 9303 A. Gay-Lussac P (Torr) Versus T (Trebors)
Pressure 869.2 892.1 941.9 982.8 1036.6 1081.4
Temperature 96.4 103.3 120.3 137.5 154.1 168.3
No. 9303 B. Radioactive Decay Activity versus Time (Years)
Counts 5456 4488 2856 2321 2054 1463 1039
Time 25.1 51.0 77.4 103.3 125.7 152.7 174.1
No. 9303 C. Pvap (Torr) versus Temp. (deg. C)
Pressure (Torr) 24.4 41.8 51.2 93.3 156.6 200.4
Temperature (C) 7.3 18.8 21.8 35.5 46.2 53.6
SAMPLE PROBLEM ANSWERS
No. 9303 A. Gay-Lussac P (Torr) Versus T (Trebors)
value error (rel.err)
Slope = 2.898 0.062 (0.021)
Intercept = 590.6 8.2 (0.014)
t (P=380) = -72.7 4.5 (0.063)
t (P=0) = -203.8 7.2 (0.036)
No. 9303 B. Radioactive Decay T(1/2) = 64.17 Years
value error (rel.err)
Slope = -k = -.00469 0.00026 (0.055)
Intercept = 3.8617 0.0289 (0.007)
A (t=0) = 7273 484 (0.067)
t (A=500) = 247.9 11.1 (0.045)
No. 9303 C. Pvap (Torr) versus Temp. (deg. C)
value error (rel.err)
Slope = -4237 76 (0.018)
Intercept 11.65 0.25 (0.021)
t (P=760) = 90.6 1.7 (0.019)
t (P=10) = -8.0 0.8 (0.103)
dHvap = 35.2 kJoules dSvap = 96.9 Joules
1. The A Problem is a Gay-Lussac problem which reviews a
laboratory experiment done earlier. The students measured
pressure versus temperature in Celsius and then extrapolated
to an absolute temperature scale. Here, the computer
generates pressure in Torr versus a fictitious temperature
called a Trebor. This is named in honor of one of the
'older' faculty members at Monroe Community College, Robert
Flanigan. ;-)
FIGURE 1 Graph of sample "A." problem: Pressure in Torr.
versus Temperature in Trebor The figure shows the
scatter in the points along with the best line
(linear regression) through the points.
2. The B Problem utilizes the time integrated radioactive decay
or first order kinetics equation. This relationship will not
be seen until the second semester, but as with the Clausius-
Clapeyron equation, is presented as an equation which can
reduce the non-linear data to the equation of a straight line
by plotting the log of the activity versus time. Log base 10
is used here for pedagogic reasons even though the equation
naturally involves the natural logarithm. Once the equation
is determined, it is used to predict the activity at a given
time, and the time required to achieve a given activity. The
time units are randomly chosen by the program.
FIGURE 2 Graph of sample "B." problem: log (Activity) versus
Time (here, in years) . The figure shows the scatter
in the points along with the best line.
3. The C Problem is a vapor pressure versus temperature problem
which requires the soon to be covered Clausius-Clapeyron
equation. For purposes of this exercise, the equation is a
given and the reduction to a linear relationship involving
the natural log of vapor pressure in atmospheres as Y and 1/T
(Kelvin) as X is shown. The slope is presented as (-dHvap /
R) and the intercept as (dSvap / R). At this point enthalpy
has been introduced, but entropy has not. Entropy is merely
described as a thermodynamic property which will be covered
later. Its magnitude and dimensions are determined from the
dimensional analysis of the Clausius-Clapeyron equation.
Once the equation is determined, it is used to predict the
Normal boiling point and the temperature required to achieve
a given pressure.
FIGURE 3 Graph of sample "C." problem: ln (Pressure in atm)
versus 1 / T (Kelvin) . The figure again shows the
scatter in the points along with the best line for
the Clausius Clapeyron problem.
This graphical exercise is sometimes repeated at the beginning
of the second semester as a review of graphing skills which will
be needed as the student encounters the temperature dependence
of equilibrium constants, kinetic rate constants, and the
integrated rate expressions. Since not all instructors use this
exercise in the first semester, it is a review for some students
and new for others.
C. VAPOR PRESSURES FROM MANOMETER MEASUREMENTS:
Another simulation generates data for the measurement of the
vapor pressure of a volatile liquid versus temperature. This is
an actual experiment which is done in a flask containing air and
using a mercury manometer. A volatile liquid is injected into a
flask through a septum seal, and the pressure is measured using
the manometer. The temperature is controlled with a water bath.
Similar to the Clausius-Clapeyron study earlier, this version
generates the manometer readings which lead to the flask
pressure and, after correcting for air in the system, results in
the vapor pressure of the compound. Ideal gas law corrections
must be applied to determine the actual vapor pressures of the
compound. This experiment is often the last one of the
semester, sometimes coming a few days before the final exam. To
save time, the pre lab is given normally, initial data taken
from the apparatus as a demonstration, and computer generated
data is then given to the students. In this way, students can
use the time required to collect the data to begin to calculate
the results (which in the worst case may be due the next day).
The actual experiment works well, providing heats of
vaporization which differ by only a few percent from the
literature values for simple alcohols and ketones. When time is
short, the computer simulated data saves two hours of data
collection which is largely devoted to merely changing the
temperature of the system.
IV USES OF COMPUTER GENERATED GRAPHICAL DATA IN ANALYTICAL CHEMISTRY
One simulation used in the analytical chemistry course consists in
the potentiometric titration of an unknown polyprotic acid. The
students actually do a potentiometric titration of an unknown
mixture of phosphoric acid and sodium dihydrogen phosphate. At the
same time, they analyze a computer generated potentiometric
titration of an unknown mixture. The mixture can be a diprotic
acid, its monoprotic salt, or HCl in a compatible mixture. The
students must identify the composition of the mixture as well as
the concentrations of the components. pKa's are determined from
the pH of the buffer regions. The computer selects the mixture
randomly from a table of polyprotic acids which have pKa's that
favor potentiometric titration. The concentrations are also
selected, as well as the sizes of pipets whose sample is titrated.
Students not only get a very unique problem on which to work, but
one for which it would be difficult to provide an unknown solution
because of difficulty in obtaining the variety of compounds of
suitable reliability. This simulation complements the experimental
work which is done in the laboratory.
NOTE: this simulation has not been ported to QuickBasic on the IBM
PC from its original version on the Atari 800. It is therefore NOT
part of the package which is available by anonymous FTP.
V APPROACH TO PLOTTING THE GRAPHS:
The following points are emphasized with students in graphing:
A. Use the 2-5-10 Rule to layout the axes. Calculate the number of
units per division, then round to the nearest 2, 5, or 10 units
per division to facilitate the plotting of the points.
B. Label the axes, plot the points, check any outliers, and then
fix the points in ink.
C. With a sharp pencil, draw the best straight line through the
points. If the placement of the line is not satisfactory, erase
it and redraw it. Extrapolate the line over the axes at the
edges of the graph.
D. Read the two points at which the line crosses the axes. Use
these points to determine the slope and intercept of the line.
The use of pen and ink encourages the student to redraw the line
until it is the best possible representation of the points.
Extrapolation off the edges of the graph encourages the student to
take two convenient, widely spaced points for determining the slope
and intercept. This discourages the student from taking two data
points which are close to the line, thereby ignoring the rest of
the data points. Further, the widely spaced points gives the
largest difference between two points, maximizing the significant
figures in the calculation of slope and intercept.
VI DESCRIPTION OF THE PROGRAM
A. Random Number Generator:
The random number routines were taken from a collection of
random number routines in "Microsoft Quickbasic Programmer's
Toolbox" by John Clark Craig, Microsoft Press, ISBN 1-1-55615-
127-6, p. 353-364. It was desired to have the randomly
generated data be keyed on the problem number as a seed value.
This permits one to be able to generate the same sets of data
using the same problem numbers. This was deemed useful to
reproduce the data in case the keys were lost or some other such
disaster struck. This seeding on the problem number was not
possible using the random generators provided with QuickBasic.
While the RANDOMIZE statement appears to do this, repeated
calling of this function does not restart the generator from the
new seed value. Thus it was not possible to reliably obtain the
desired effect with the built-in function.
B. Generation of Statistical Fluctuations about a Value:
The program uses a function called ErrFactor (relative standard
deviation). This function returns a statistically generated
multiplier with a mean value of 1.00 and a standard deviation
given by the relative standard deviation specified. For
example, if it is desired to apply a 5 percent fluctuation to a
given value, the function called is ErrFactor (0.05). The
function returns a randomly generated value of 1.00 +/- 0.05
which is applied as a multiplier to the value one wishes to
randomize. Thus a multiplier between 0.95 and 1.05 is generated
approximately 2 of 3 times. Since this follows a normal
distribution, occasionally one finds the 2 or 3 or 4 sigma
variation. This produces fluctuations with points which are
outside the limit (here 5 %) about 1 of 3 times. Students are
told that the data is realistic and occasionally points are
measured which don't agree with the rest of the data set, and
must be discounted in drawing the best line through the data.
C. Modular Program:
The program has been carefully written using a good structured
BASIC. Extensive use of Functions and Subprograms provide a
modular package of routines dedicated to problem generation.
Variable and Constant names have been carefully chosen to be
descriptive of the quantities represented. The use of "magic
numbers" within the program has been avoided. For example, one
finds variables such as SetsPerPage, NumberOfSets, SLOPE, dHvap,
dSvap, Temp, Press, etc. A testing capability with on screen
graphing has been included to look at the scatter in the points
while tweaking the error factors.
It is hoped that the program will be useful to others in its
present format, and that those with some BASIC programming
experience might be tempted to modify and add their own routines
to the basic problem generating environment which has been
created.
D. The Clausius Clapeyron Problem:
As an example of the generation of a graphical problem, consider
the Clausius Clapeyron Problem:
ln Pvap = -dHvap / RT + dSvap / R
The program first selects the compound's boiling point in the
range of 80 to 120 degrees Celsius which might be typical of
compounds to be studied. Next, Trouton's rule is applied to
pick a slope for this compound and a 10 percent factor is
applied to randomize the slope. An intercept is chosen using
the standard molar entropy of vaporization of 88 Joules per mole
K also using a 10 percent factor to provide fluctuations. A set
of 6 temperatures is randomly chosen in the range of 0 to 60
degrees Celsius which represent the temperatures at which the
vapor pressures are measured. The corresponding pressure is
calculated using the selected slope and intercept and then
applying a 5 percent factor to each point to provide the
experimental variation sought. This then is the simulated
experimental data which the student must analyze.
Upon generating the simulated data, the program then analyzes
the data using linear regression, reporting for the instructor
the least squares slope and intercept as well as the standard
deviation in both quantities. From experience it has been seen
that a student working carefully to draw the "best" straight
line through the data can generally come within one standard
deviation of the slope and intercept. The intercept is a bit
more sensitive, and some allowance should be made for values
close to but outside of one standard deviation. It is
reasonable to give full credit for a student coming within this
range.
The student is then asked to calculate the Normal Boiling Point,
the temperature at which the vapor pressure is 10 Torr, dHvap,
and dSvap. The generated answer key provides these values along
with the propagated error expected from the error in the slope
and intercept. These values can be used to evaluate student
results or, as was indicated earlier, a programmable calculator
can be used to generate the exact values the student should get
from his slope and intercept.
E. General Comments Regarding the Program:
Anyone who uses the program, since the source code is available
in BASIC, is free to tweak the Errfactors to provide lesser or
greater fluctuations. One should be careful in choosing larger
factors. Since the specified fractional error is ONLY the
standard deviation, one gets larger fluctuations in the
distribution from the random number generator routine.
Occasionally, more than one anomalous point will occur in the
same graph. Students are told that this can happen in any
experiment, and that they should do the best they can with the
data they obtained. Since the program calculates the errors in
slope and intercept, those values along with the student's graph
serves to flag those few cases where statistics has been unkind.
This should be taken into account in grading the student's work.
F. Hardware Requirements:
The program source code runs under QuickBasic 4.5 on an IBM PC
compatible. A compiled version, PGEN11.EXE, is also being made
available by anonymous FTP. The EXE versions runs under DOS
without requirement of having QuickBasic 4.5.
The program has been written to drive an Epson compatible
printer. However, only two special printer codes are used
during printing:
ASCII 12 sends a FormFeed command to advance to the next
sheet.
ASCII 15 sends an initialization command to turn on compressed
mode on the printer. The print format assumes 132
character compressed font available per line of
output. Student question data is printed on the left
side of the page, while the answers are printed on
the right side of the same page.
The capability of changing the printer initialization code has
been added. The printer must have a compressed font available
which provides at least 132 characters per line.
VII CONCLUSIONS:
The computer generated graphical problems has provided the author
and other faculty at Monroe Community College with a useful tool
with which to help teach chemistry and emphasize the importance of
graphing and linear relationships to students of chemistry:
1. Unique problem sets provide plenty of practice without the
temptation to copy someone else's work. Thus the instructor can
provide some credit for the work as Laboratory, Quiz, or a
portion of an Examination.
2. Grading with the combination of answer sheet and programmable
calculator provide an easy way for the instructor to check and
evaluate student performance.
3. Statistical fluctuations in the data provide practice in working
with data which is realistic.
4. It shows the students whether they do (as they often think) or
whether they do not (as is too often the case) have the ability
to work with graphical data and linear relationships.
VIII THE PROGRAM PACKAGE AVAILABLE BY ANONYMOUS FTP:
The following files will be available via anonymous FTP from
Host:info.umd.edu in the Path:info/Teaching/ChemConference
during the conference:
PGENBAS.TXT The QuickBasic 4.5 source code for problem
generation of graphical problems
PGENDOC.TXT A documentation file describing the
requirements of the package.
PGENQTMP.TXT A text version of the temperature problem
questions.
PGENQGRF.TXT A text version of the graph problem set
questions.
PGENQVAP.TXT A text version of the Vapor Pressure Lab.
for which data can be generated.
PGENQTTN.TXT A text version of the potentiometric titration
problem questions.
PGEN11.EXE An executable, stand-alone version of the
problem generator for those without access to
QuickBasic 4.5 NOTE, this file is BINARY.
The complete package will be available in a single ZIPped file from
the Simon Fraser University anonymous FTP site from
Host: truth.chem.sfu.ca in the Path: pub/chemcai
PGEN11ZP.EXE A self-extracting ZIPped collection of the 7 files
described above. NOTE, this file is BINARY
REMINDER: To download a binary file from an FTP site, the BINARY
command must be issued before GETting the file.
IX GIF FIGURES:
Three figures accompany this paper. They consist of graphs of the
data for the type A, B, and C problems discussed in Section III B.
The principal contribution they make to the paper is to give some
idea of the statistical fluctuations which is built into the data
provided to the student with these generated problems.
The graphs were generated with Quattro Pro, written to a PCX file,
and then converted to a GIF file with PICEM. They are 480 X 640
pixels, and were done in black on white to give better quality
paper copies than the colored versions provide.
X QUESTIONS:
1. Is this student math deficiency principally a Community College
problem, or do Universities see similar problems?
2. Do other participants, particularly from other countries, find a
similar math deficiency in their students or is this
predominantly a U.S. problem?