Texas A&M University
Artie McFerrin Department of Chemical Engineering
Using Mathcad to solve non-linear equations
CHEN 354: Fall 2009
Prof. Perla B. Balbuena
Mathcad is a mathematical software easy to use. Among the capabilities of Mathcad are:
Solving non-linear equations, graphing functions in two or three dimensions, solving differential
equations, performing symbolic calculations, etc.
The objective of this quick guide is to provide you the basic knowledge to use Mathcad in
order to solve systems of non-linear equations. First, we go through the basic operations in
Mathcad, such as defining variables and evaluating functions. Then, we solve a vapor-liquid
equilibrium problem involving a system of non-linear equations.
Mathcad basics
The Mathcad window is shown in the following figure. All the expressions and objects are
inserted in the workspace
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T.A. Fuman Zhao | zhaofuman@tamu.edu
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CHEN 354: Fall 2009
Prof. Perla B. Balbuena
The “Math toolbar” is useful to get quick access to the different capabilities of Mathcad.
Defining variables and functions
To define a variable, just place the cursor in the workspace and type its name followed by
the “Definition symbol”, located in the “Evaluation toolbar” or typing the “:” key. For example, define
P equal to 101.33.
If you want to know the numerical value of a previously defined variable, place the cursor
below its definition, type the name of the variable followed by the “Evaluate numerically symbol”
whose shortcut is the “=” key.
Defining functions is simple. Type the name of the function followed by the name of its
independent variables enclosed in parenthesis. Then use the “Definition symbol” and create the
correct expression aided by “Space bar”.
You can evaluate the functions for given values of their independent variables.
Prepared by Rafael Callejas Tovar
T.A. Fuman Zhao | zhaofuman@tamu.edu
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CHEN 354: Fall 2009
Prof. Perla B. Balbuena
Useful tips
If you want to use Greek symbols to name the variables, use the “Greek symbol toolbar”. If
you know the equivalent letter in the Latin alphabet, type the letter then “CTRL+g”. For example, to
obtain gamma type “g” then “CTRL+g”
To insert a textbox just type the “double quote key” and the region will become a text
region. All the characters inside the textbox will be treated as text. When selecting a math region
the cursor is blue, in a text region the cursor is red.
The Given… Find solution block
The “Given… Find block” is the tool that allows us to solve non-linear equations. The
following is an example of its implementation. It is required to provide initial approximations for
the unknowns. The block starts with the word “Given” written as a math expression; below it we
place all the equations we want to solve. These equations may contain previously defined functions,
is important to use the “Boolean equal symbol” to write the equations (“Ctrl+=”).
Prepared by Rafael Callejas Tovar
T.A. Fuman Zhao | zhaofuman@tamu.edu
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CHEN 354: Fall 2009
Prof. Perla B. Balbuena
The “Given… Find block” ends with the “Find” command. Its arguments are the unknowns
that we want to calculate. The result of the “Find” command may be assigned to a variable or
function. In the example shown above the results are stored in the function “Solution(y1)”. This is
useful when we want to solve the system for several values of y1.
In the example shown above, we have a two-equations/two-unknonws system. Therefore,
the solution porvided by the “Find” command will be a two-rows/one-column vector. We can split
assign each vector element to separate functions as follows.
To make reference to the elements of a vector use the subscript, “[“ key. In Mathcad the
array subscripts begin at zero.
Making plots
To make plots, first define a “range variable”, in general, it will be in the x-axis of the 2Dplot. A range variable takes an initial value and add it a constant increment until a final value. To
define it, name the variable, use the definition symbol, put the initial value, type a comma “,”, put the
increment and type a semicolon “;”, then type the final value
Prepared by Rafael Callejas Tovar
T.A. Fuman Zhao | zhaofuman@tamu.edu
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CHEN 354: Fall 2009
Prof. Perla B. Balbuena
Insert a 2D-plot template (“Shift+2”), fill the blanks accordingly to the functions you want to
graph. You can format the plot by selecting it and right click > Format…
Example: Create T-x-y and x-y diagrams
Solve the following problem using Mathcad. For the system methanol(1)/methyl acetate(2),
the following equations provide a reasonable correlation for the activity coefficients:
 1  exp  Ax22 
 2  exp  Ax12 
where
A  2.771  0.00523T
In addition, the following Antoine equations provide vapor pressures:

P1Sat
3643.310 
 exp 16.59158 

kPa
T
K  33.424 


P2Sat
2665.54 
 exp 14.25326 

kPa
T
K
 53.424 

Assuming the validity of
yi P  xi  i Pi Sat
i  1, 2,
,N
Calculate T and  xi  , for P  101.33 kPa and y1  0.40 . Also, prepare T-x-y and x-y diagrams
at this pressure.
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T.A. Fuman Zhao | zhaofuman@tamu.edu
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