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Plotting Functions: Trigonometry and the RPF
Chemistry 200 – Fall 2004 – Assignment #3
Due: September 17, 2004, noon
There are big differences when it comes to plotting functions instead of experimental data. For
experimental data, each data point is what is important. Error bars should be included when
possible. A trendline can be included, but the trendline is derived from the data points. When you
are plotting a smooth function, the line shape is more important than the individual data points. In
these activities, you will practice your equation entering skills and produce plots of smooth
functions.
All of these exercises should be completed in a single Excel document. In this document, each
exercise should be completed on a separate worksheet and you should change the name of the
worksheet to reflect the worksheet content. Your Excel document should be submitted
electronically via the dropbox on the Chem200 Blackboard website. Your file should be named
userid_excel2.xls, where userid is your JMU email ID. For all plots, you should use good plotting
practices, as described in the tutorial on tips for making aesthetically pleasing graphs. All plots
should be printed (hardcopy) and turned in to Dr. Reisner. Question 1e should be answered on a
separate sheet.
1. Plotting Trigonometric Functions, Part I. Trigonometric functions are encountered
throughout chemistry. To refresh your memory of the behavior of these functions, you will plot
a variety of functions over two periods.
a. Set up a spreadsheet in worksheet 1 that looks like the template given in Table 1.
b. Using the fill function, enter your data in degrees. Starting at 0°, enter numbers that
correspond to 2 full rotations around a circle in increments of 1°.
c. In the following columns, convert your data to radians and evaluate the trigonometric
functions. (When Excel evaluates trig functions, it assumes that your data is in radians.
However, since we’re used to dealing with things in degrees, we’ll start there.)
d. Plot the following combinations of functions. Express your x-axis in degrees, not radians.
Plot 1: sin(x) and cos(x)
Plot 2: sin(x), 2sin(x), and sin(2x)
Plot 3: sin(x), sin2(x), cos2(x), and sin2(x) + cos2(x)
In your legend, be sure to differentiate between the functions. Use good plotting
techniques for all graphs.
e. In a separate document, answer the following question about plot 2. Explain why/how the
three functions plotted differ in terms of amplitude, wavelength and frequency.
2. Plotting Trigonometric Functions, Part II. Two interesting periodic waves are square waves
and triangular waves. Each comes from the sum of an infinite series of sine waves of varying
amplitude and period. The first term of each is a simple sine wave, but as additional terms are
added, they differ more and more from each other.
1
1
1
1
sin( 3x )  sin 5x  sin7 x  sin9x  ...
3
5
7
9
square functions are used to represent potentials in quantum mechanics.
square wave
sin( x ) 
sin( x ) 
1
2
sin(3 x ) 
1
2
sin 5 x 
1
2
sin 7 x 
1
sin 9 x  ...
3
5
7
92
I can’t think of a physical quantity represented by a triangular wavefunction, but it’s
really cool!
triangular wave

By increasing the number of terms used in the infinite series, they begin to resemble square
and triangular waves. In this problem, you will see the effects that adding terms has on the
shape of the wavefunction.
a. Set up a spreadsheet in worksheet 2 that looks like the template shown in Table 2.
b. In column A, enter your data in degrees. Go between 0° and 1440° in increments of 1°.
Convert to radians in column B.
c. Choosing EITHER the triangular OR the square wavefunction, calculate the function with 1
term in column C, 2 terms in column D, etc...
d. Plot you wavefunction with 1, 2, 3, 4, 5, and 6 terms on a single graph. Express your xaxis in degrees, not radians.
3. Plotting Radial Probability Functions. The properties of an element are determined
primarily by valence electrons. The position of these electrons is described by their atomic
orbiatls, which is a solution of the Schrodinger equation. It is not possible to solve the
Schrodinger equation exactly for a many electron system. Fortunately, we can make
approximations and use the hydrogenic wavefunctions of orbitals to approximate the orbitals
of a many electron system. (Remember, hydrogenic implies that you have an atom or ion with
a single electron.) In this exercise, you will be plotting the RPF (radial probability functions)
for the five lowest energy hydrogenic orbitals. These are derived from radial portion of the
wavefunction.
a. Set up a spreadsheet in worksheet 3 that looks like the template given in Table 2.
b. Define names for the variables Z and a0 so you can use variables (rather than values) in
your functions.
c. In column 1, enter the distance from the nucleus (radius) in pm. Start at 0 pm and go from
0 pm to 1200 pm in increments of 2 pm.
d. In columns B to G, evaluate the hydrogenic wavefunctions given below. Normalized
wavefunctions have been used so that your expression for r2R2 is the RPF.


2
  3 / 2
Zr 
Z
r R (1s )  r 2  exp

 a0 
a0 



2
2
2 


3 /2

2
Z   Zr  Zr 
r R ( 2s )  r 
 2  exp

2a0   a0  2a0 



3 /2
 
Z  
18Zr 2Z2 r 2
2 2
2  2
r R ( 3s )  r
27



 
27 3a0  
a0
a02



3 /2
2
2  Z   Zr Zr  Zr 
r R ( 3p )  r

 6   exp

 27 3a0   a0 a0  3a0 



3 /2
2  Z  Z2 r 2
r R ( 3d )  r

  2
27 5 3a0  
 a0

2
2
2 
2
 Zr 


exp3a 
  0 


2
2
2 2



2
2
2  2
2
 Zr 


exp3a 
  0 
a0 = Bohr Radius = 52.9 pm
Z = atomic number of the hydrogenic atom
e. Document your formulas in cells B6 to G6.
f.
Create two line plots. The first should contain all wavefunctions where the angular
momentum quantum number = 0. The second should contain all wavefunctions where the
principal quantum number = 3.
Table 1. Template for trigonometric functions problem.
1
A
B
angle
(deg)
angle
(rad)
C
D
sin(x)
E
cos(x)
2sin(x)
F
G
H
I
sin(2x)
sin2(x)
cos2(x)
sin2(x) +
cos2(x)
2
3
4
5
6
7
8
9
10
Table 1. Template for square/triangular wavefunctions problem.
1
A
B
C
D
E
F
G
H
angle
(deg)
angle
(rad)
1 term
2 terms
3 terms
4 terms
5 terms
6 terms
2
3
4
5
6
7
8
9
10
Table 3. Template for RPF problem.
A
1
2
3
4
5
6
7
8
9
10
11
12
B
Z
a0
1s
Function =
radius (pm)
0
2
4
6
8
C
D
E
F
G
1
52.9
2s
2p
3s
3p
3d