Chem 3111                                                                                                               Eric R. Bittner
Univ. of Houston                                                                                                  

Thanks to Hannes Jónsson and Gisli Johannesson , U. Washington

Introduction to using Mathematica

• Introductory Remarks

Mathematica (registered trademark of Wolfram Research Inc.) is a very   powerful tool  for solving mathematical  problems and for using graphics to   gain more  insight into mathematical results. Mathematica can, for example, provide exact or approximate solutions to   equations  involving polynomial, exponential, trigonometric, and logarithmic   functions;  solve differential equations and evaluate integrals either   numerically or  sometimes analytically. It can symbolically factor   polynomials, graph  functions of one, two or three variables, and perform   operations on matrices.  The evaluation of  special functions such as Bessel   functions becomes almost   as easy as the evaluation of sines and cosines on   an ordinary calculator. Mathematica can, furthermore, be used as a programming language, a level   above C,  Pascal, Fortran, and other such programming languages.

In this and other exercises based on Mathematica you load in a notebook, the working document in Mathematica. Read the text carefully, carry out all the calculations that are set up (i.e. make Mathematica execute the commands) one by one. You should think carefuly   about what  the commands mean and how they are constructed so as to learn to   use Mathematica from these examples.  Several questions are raised in the   notebooks.  Those are labeled with a "Q:" at the left margin.  You should   type in your  answer after each question.  Identify your answer in some way,   for example  with an "A:"  in the left margin, or by using a different   color.

Remember to save your work frequently.  This can be done by   pressing the  "CTRL" and "s" keys simultaneously.  Make a habit of saving   your notebook  every couple of minutes at least. Mathematica can crash and you will then loose the work you have done since   the last  time you saved the notbook.  

Below are section headings.  To open each sections,   double click on the blue  bar farthest to the right. You can close it again   by double clicking on the  blue bar.

• The Mathematica notebook

The graphical interface to Mathematica is called a notebook.  All notebook names get the suffix .nb    automatically when you save them.  The building blocks of notebooks are    `cells'.  The lines that are grouped together in a cell are identified by a    blue square bracket on the right hand side of the cell.  This paragraph is in    one cell.

There are many kinds of cell styles. You need to become familiar with two    of them, `text' and `input' style.  This cell is a text cell and is used for    comments (introduction, questions and answers, for example). You should    create new text cells for your answers to questions raised in the notebooks    later on.  You can create a new cell by placing the cursor at the end of a    cell and then hitting the `enter' key.  The new cell is by default an `input'    cell, not a text cell, and the contents of such cells get sent to the Mathematica's processing engine (the `kernel').  An input cell can be changed to a    text cell, as explained below.

When you have mathematical expressions you wish to evaluate, you   type them  into an input cell and then press enter on the keypad (the enter   button on  the keypad on the far right of the keyboard is the only key you   should use on  the number keypad when working with Mathematica, type in numbers from the upper row on the main keyboard).  You   can also  execute a Mathematica command by pressing Shift and Return simultaneously on the main   keyboard,  after placing the cursor in the relevant input cell.  This causes   Mathematica  to evaluate the expressions and print the results in an ouput   cell.  

Q:  Open a new cell by placing the cursor in between this cell   and the  following one (the cursor should become a horizontal line) and hit   enter.  Type in 2+2 and then hit enter on the keypad.  You should get the   result of  the calculation from Mathematica (the first command often takes a while to execute because Mathematica has to initialize various things).  This notebook is written   with the  `classroom stylesheet', so the background of input and output cells   gets  highlighted with different colors.

To change the style of a cell, you must first   highlight the cell. by  clicking on the blue square bracket on the right hand   side of the cell (that  is, place the cursor over the square bracket and   press the left mouse  button).  Next you click on "Format" on the tool bar.    A pop-up window  appears.  Place the cursor over "Style" in the pop-up   menu, and a second  pop-up window shows the available styles.  Cilck on the   desired style.

Q:  Open another cell and change the style to "text" style.  Again   type in  2+2 and hit enter. How does Mathematica respond now?

You can change the color of the text by selecting the   text or the cell and  choosing "Text Color" from the "Format" menu.

It is a good idea to divide your notebook into many cells.  This way you can keep track of intermediate results and limit the source of possible errors, if any problems arise.

You can divide a cell in two by placing the cursor where you want the second cell to begin and then select "Cell" from the menu bar, and then divide cell.  This can be done by keystrokes only, as written to the right in the pop-up window ( Shift-Ctrl-D).  If you wish to combine two or more adjacent cells into one, you must select both cells (hold down the shift key when selecting the second one) and then select "Merge Cells" from the "Cell" menu.  You can also select more than one cell by clicking the square bracket on the right hand side of the top or bottom cells, and then draggin the cursor over the rest of the cells while holding down the left mouse butto

You can delete a cell by selecting it and then selecting "Clear" or "Delete" from the "Edit" menu (or pressing Ctrl-X on the keyboard)

Q:  Create some new cells and practice dividing, merging and deleting   cells.

•The Mathematica Help Browser

Mathematica has a very useful help directory built into the program.  Click on "Help   " on the tool bar and select "Help..." on the pop-up menu.  Then the Help    Browser window appears.  As shown in the pop-up menu at the right margin,  you   can alternatively press Shift and F1 simultaneously on the keyboard to  get   the help browser.   You can, for example, type the word you need  information   about in the narrow horizontal window.  As you type, the help  browser will   try to complete the rest of the word.  When you have finished  typing (or the   browser has found the word for you) select "Go To".  Then  you get some   information about the word and links (highlighted in blue,  which you can   access by clicking on them) to additional information.  The  help browser has   six categories: Built-in Functions, Add-ons, The Mathematica Book, Getting Started/Demos, Other Information, and the Master Index.     You should experiment with each of these, to get an idea of how to get the    desired information as fast as possible.  An essential part of learning to    use Mathematica is to be able to use the help browser efficiently.

•Elementary calculations in Mathematica

•Addition and subtraction:

Execute the following cell

by clicking anywhere in the cell and then pressing the enter key (remember    to use the keypad).  Mathematica will respond with the results of the calculation in an output cell.

•Multiplication and division:

A space between numbers and/or variables represents multiplication, as does the * symbol (the star). For example, compare the two expressions (execute both cells

It is recommended that you always write a * for multiplication, it is    clearer and thus reduces the probability of undetected errors in your Mathematica commands.

The symbol, /, represents division.  For example:

The output is a rational number (evaluate the cell to verify this).  In Mathematica 3.0 and later versions, an alternative (fancy) way of writing mathemtacal    expressions was  introduced.  By using the BasicInput palette which you  find   under  "Palettes"  in the "File" menu, you can write the  expression  above  in the following manner

Verify that this gives the same answer as above.  

As you can see from this, Mathematica tries to keep the answer analytical, rather than a number with limited    significant figures as we would get from a typical computer program.

•Numbers raised to a power:

The caret, ^, indicates raising to a power.  For example:

gives 2 raised to the power of 64.  By using the palettes (a filled square with a smaller open square in the upper right corner) this can also be written as:

Neither of these two numbers need to be integers.  Try for example

•Some basic concepts

•Variables

Variables in Mathematica do not need to be declared as integers, real numbers, arrays, etc. (as in    Fortran or C).  When a value is assigned to a variable, it automatically    assumes the attribute of the assigned value.  For example, the statements

automatically declare the variable a as an integer, b as a real number, and c as a 2x2 matrix with integer entries.  You may choose any name you wish    for a variable, but try to make the names as descriptive as possible.  Most    likely, the names of the above variables are too general.  This is not very    important in short programs.  Predefined variables in Mathematica always have names that start with a capital letter. For example, the    number π is given by

The exponential function is Exp[argument], for example

Note that Mathematica does not give a numerical answer, it tries to keep the result exact as    far as possible (there is a way to force Mathematica to give a numerical answer, see below). It is good practice to choose    variables that start with a lower case letter so as to prevent accidental    conflict with variables that are predefined in Mathematica.

•Brackets

There are three types of brackets used in Mathematica. They have very different meaning and it is extremely important to learn    which type of bracket to use in various circumstances. The round brackets are used to group parts of an arithmetical expression together.  For    example, when a fraction is raised to a power

When a list of something is created, such as a list of numbers in an array    or a matrix, the items in the list are separated by a comma and the list    starts and ends with a curly bracket

A variable called "mylist" has been created in the cell above.  It contains three real numbers.  The entries in a list can also be lists.  For example the matrix "c" defined above could equally well have been defined by the expression

The square brackets envelope the argument(s) of functions in Mathematica.  See, for example, the exponential function discussed above.  A function    f(x) = x may be defined by the command

We will later deal with functions in detail.

One nice aspect of lists is that a they can be arguments for functions.  For example, the exponential of all the numbers in the list mylist can be calculated by simply typing

Square brackets are also used when a selected element of a list is needed.  For example, the second element in the list mylist is

•The semicolon

If a semicolon is placed at the end of a line in an input cell, Mathematica will not print the output from the command.  For example, compare

•Complex numbers

The square root of -1 is represented as I (or rather, I is defined in such    a way that ).  For example, try

You can divide one complex number into another using Mathematica, such as

Q:  What is the value of  ? (Calculate this using Mathematica.  You don't need to use the palettes). Richard Feynman, one of the    greatest physicists of the 20th century, called this result "the most    remarkable formula in math" (See the book:  'Surely You Must be Joking Mr.    Feynman').

•Some useful intrinsic functions and commands

There are hundreds of intrinsic functions, commands and object types at    your disposal in Mathematica. Their names begin with a capital letter (if the name is constructed by    combining two words the second part of the name also begins with a capital    letter), and their arguments are enclosed within square brackets.  Since    there are so many of these, only a small fraction can be shown here.  You can    find them all in the Help Browser.

•Intrinsic functions

You have already seen the exponential function. Some other examples of
intrinsic functions are

The first function is the Cos[] function, which evaluates the cosine of an angle (given in radians).  The    second function is the square root function. Then we have an example of a    function within another function.  The Sqrt[] gives the square root of the argument and IntegerPart[] returns the integer part of the square root.

•Commands

Here we look at four commands, Clear[], Expand[], Factor[], and N[].

The command, Clear[var1,var2,...], clears the values assigned (if any) to the variables var1, var2,....  It is very important to clear variables at the beginning of each    program you write in Mathematica.  Many mistakes result from variables having different values than    assumed (sometimes the variable should not have a value assigned, but Mathematica remembers an assignment that was made earlier. Computers have in some    sense too good a memory!). It is good practice to include all the variables    you use in a Clear command at the beginning, adding to the list as you go on    writing the Mathematica program.  This includes the names of functions (not the intrinsic ones,    though, only the ones you define). For example, compare

and

The command Expand[expr] expands the expression that appears in the argument and Factor[expr] does the opposite, it factors the argument.  For example

Q: Try combining Factor and Expand, for example in    Factor[Expand[(1+x+x^2)^3]].  What is the net result?

The N[] command gives a numerical approximation to the expression in the argument    to a given number of significant figures.  N[expr, n] gives the expression expr to n significant digits.  For example

gives the constant π to 10-digits.  The default for N[expr] is to give the 6-digit approximation to expr.  You can attach //N to an expression to force matematica to evaluate it to 6-digits.

Q:   What is the value of Eulers constant, e, to 20   digits? (Note, the  symbol for the Euler constant is E, since all intrinsic   variable have a name  that starts with a capital letter)

•Plotting data points and curve fitting

A list of data points (x,y) can be graphed and fitted easily using Mathematica.  First the data is stored in a matrix (it is a good idea to clear the    variable first, just in case)

Now the data can easily be manipulated by referring to the variable called    data.  The ListPlot function can be used to graph the points

Mathematica chooses the range of the horizontal and vertical axis automatically if it    is not specified.  The first data point then lands right on the vertical   axis  and may not be noticed.  It is often better to specify the range   explicitly  by using the  
           PlotRange -> {{xmin,xmax},{ymin,ymax}}  
option in  the plotting function.  Also, for future reference it is   convenient to give  this graph a name, say 'dataplot'.  The plotting command   then becomes

The points appear as dots on the screen.  It is possible to connect the    points by straight line segments by adding the  PlotJoined -> True  option.  There are many options that can be added to the plotting    command. The various options need to be separated by a comma in the argument    list of the plotting function.  

Q:   Modify the plot to include straight line   segments between the data  points.

In many cases one would like to obtain a fit of a smooth curve to data    points.  Linear least squares fits are obtained in Mathematica by using the Fit function.  For example, the data can be fitted with a    straight line with the following command

The expression {1,x} here specifies that the form of the function that is to be fitted to the data is a constant plus a linear term in x (i.e. straight line).  The results are given the name 'f' for future reference.  For example, the straight line can be plotted with  

The composite plot of the original data points and the fit can be superimposed by

If the data is given as a list of x values and a separate list of y values, the matrix of (x,y) pairs can be created with the Table command. For example, if the list of x values is

and the list of y values is

then the matrix of (x,y) pairs can be generated by (the argument {i,4} says that the dummy variable i should run from 1 up to 4)

Q:    Below is data on the equilibrium constant for the formation   of  ammonium from hydrogen and nitrogen,  
                  + 3 -> 2
as a function of temperature.  The task is to extract   ΔH  and ΔS from the data assuming these   quantities are independent  of temperature. Recall that the Gibbs free energy   is related to the  equilibrium constant through
                 ΔG =  ΔH - TΔS = -RT ln(Keq)
  Assuming ΔH  and ΔS are independent of T, a plot   of ln(Keq) vs. 1/T should  give a straight line with a slope of -  ΔH/RT and an intercept of  ΔS/R.  An estimate of   ΔH and ΔS can  therefore be obtained by fitting a   straight line to the data points.
Measurements of the equilibrium constant   were made at the following temperature values
   T  =    300,   400,  450,    500,  550,   600
and the observed value of the equilibrium constant (in the   right order)  was
   Keq =  0.00434, 0.000163, 0.0000449, 0.0000145, 0.0000054, 0.0000023
The goal is now to fit a straight line through the points   ln(Keq) vs.  1/T. Create a variable that is a list of all the temperature values (use copy  and paste from the edit menu to enter the data into an   input cell). Then calculate the inverse temperature (you can take the inverse of the list).  Similarly, create a   variable with all the  values of the equilibrium constant and calculate the   natural logarithm of the  values. Then use the Table command to create the   matrix of (1/T, ln(Keq))  pairs. Then make a plot of the data and the least   squares line on the same  graph .  What is the slope of the line (the slope   is related to the enthalpy  change in the reaction)? Does this correspond to   a positive or negative  enthalpy change as ammonia is formed?

•Some important operators (=, :=, ==)

The equality sign,  =, is an assignment operator.  It is used to assign a value to a variable.  The symbol  :=  is also an assignment operator, but the expression is not evaluated until the results are requested.  It is called delayed assignment. Double equality sign,  = =, is a logical operator and the result is True or False.  To see the difference between assignment (=) and delayed assignment (:=), go through the following example

First x is assigned a value

Then the operation of squaring x is defined in two different ways

Q:  What is the value of z and p at this point (you   can check by simply typing z and then enter, and similarly for p)?

Now, change the value of x to, for example, 5

Q:  What is the value of z and p now?

Q:   Describe in words the difference between how the = and :=   operators  function in Mathematica.

•Defining your own functions

A function, such as  f(x,y)= Sin(y) can be defined in the following way:

Note the underscores that appear after the variables in the function    definition. Underscores are only used on the left hand side and they are only    used in the definition of a function, not when it is used later on.  It is    very important to remember this. You can evaluate the function at x=1, y=π /2 by simply typing

Q:   When the acid-base equilibrium of a buffer solution is   treated  exactly (see the text book by Oxtoby, section 10.8), the hydronium   ion concentration ends up  being expressed as a root of a third order   polynomial
       f(x) = x^3 +  (cb+ka)*x^2 - (kw+ca*ka)*x - ka*kw
where ca   is the initial concentration of  the acid and cb is the initial  concentration  of the base. Define a function  f[x] which gives the hydronium  concentration  according to the expression  above. First evaluate the  function for a couple  of values of x (to make sure  the definition statement  worked) and then plot  it for a solution of HCN with  the values:  ca=1.00*, cb=0.0, ka=6.17* (remember to also assign the autodissociation constant of water, kw, its value).  Plot the function and read of  roughly  where the three roots are (it is important to specify a reasonable   range for  x (while leaving the range on the vertical axis automatic) in this   plot.  Realize that x has a magnitude much less than one).

•Other things to know about

FindRoot:   to find the root(s) of a function.

Mathematica has a numerical procedure for finding roots of functions using a   modified  Newton-Raphson method. It is necessary to give an initial guess to   get the  procedure started. The following example illustrates how FindRoot  is  used.

Here the value 1.2 is an initial guess for the root.    It is used as a  starting value in an iterative procedure.

Q:    Find the positive root of the third order polynomial for   the hydronium ion concentration in a HCN solution discussed in the previous    section.  Use the approximate estimate of the root from your graph as an    initial guess for FindRoot.  Mathematica routines work best when the numbers it works with are on the   order of  unity.  Here the numbers are very small, so it is necessary to   increase the  number of significant figures that Mathematica works with and reduce the tolerance used to determine when   convergence is  reached.  This can be accomplished by adding
         AccuracyGoal->30,  WorkingPrecision->35
to the list of arguments in the   FindRoot statement  (within the square brackets).

D:   to calculate the derivative of a function.

Mathematica can evaluate derivatives analytically. For example, the   derivative of the  function g[x] defined above is

Q:  Derive the derivative of the third degree   polynomial for the hydronium  ion concentration in a buffer, defined in the   section above.  Clear all  constants first to obtain an analytical   expression.

•Integration

•Integration

Indefinite integration is performed symbolically when possible. For   example, to integrate with respect to the variable x

A definite integral can also be evaluated.  When the integration range is from b to c, the command should be

Mathematica can also carry out the integration numerically. This is particularly    important when analytical results of the integral cannot be obtained. The    name of commands that carry out numerical calculations often start with N,    for example NIntegrate[].

Q:    Practice doing integration and differentiation by solving the following definite integrals and sums using Mathematica; Do the definite integrals first analytically then numerically and compare results.  Under what conditions are the definite integrals defined.
1.

2.
3.  
4.
5.

6.)
7.)
8.) Also evaluate the sum numerically by making a table of length N and summing the elements of the table. How big must N be to get the series to converge to the 10th decimal place.  

Converted by Mathematica  (August 29, 2002)