Consider the
following reactions, dear to the hearts of many a
physical chemist:
A  B
If the forward
and reverse reactions are both first-order with
rate constants kf and kr
respectively, then we can write the rates of
formation of A and B as:
The applet below
allows you to model the system by numerical
integration of these two differential equations.
The intial values of A and B are entered as well
as the formulas for the time derivatives which
are denoted A' and B' in the applet. For easy
equation entering, kf and kr
are entered on separate lines and assigned time
derivatives of 0 since they are both constant
with respect to time. Click the
"Calculate" button to see the time
behavior of [A] and [B]. Instructions for using
the applet are given toward the bottom of this
page.
Exercise:
a) What is the
ratio [B]/[A] as the system
comes to equilibrium at t = 20? Move the mouse
over the plot to read the (x,y) coordinates. What
is the ratio
kf/kr?
Adjust the intial values of [A] and[B] and
recalculate [B]/[A] as the
system comes to equilibrium.
b) Change kf
to 0.1 and kr to 0.2. Then repeat the
calculations for [B]/[A]
and kf/kr.
c) The ratio [B]/[A]
is the equilibrium constant, K, for the reaction.
What relationship seems to be true between the
equilibrium constant and the rate constants, kf
and kr?
Kinetics Modeling Instructions
The Kinetics
Modeling applet integrates systems of ordinary
differential equations that represent chemical
reaction kinetics and graphs the results. The
applet integrates with respect to the variable
"t" (often representing
"time"). Since any number of variables
may be entered into the applet, it can be used to
model systems with an arbitrary number of
reactions. For each species, the user should
enter an intial value (value for t = 0) and a
derivative with respect to "t".
Constants can be entered as 'variables' as long
as their derivatives are set equal to 0.
Input
AN IMPORTANT
NOTE: When you refresh or reload this page, all
input fields are reset.
Values entered
in the "Name:" and "Title:"
fields are for identification and do not affect
the numerical integration. The applet uses a
Runge-Kutta fixed-step method for the numerical
integration between the time entered in the
"Time Start:" field and the time
entered in the "Time End:" field. The
applet performs the integration using the
integral number of steps entered in the
"Number of Steps:" field. The applet
will also plot this many points. Make the value
high enough for accuracy, but as low as possible
for speed.
Variable names,
initial values, and time derivatives must be
entered in the large text area on the left.
Variable names can be of any length, but must
start with a letter. Capitalization is
significant ("a" is different from
"A"). The variable "t" is
reserved to represent time. The format for
entering variables is to enter one variable on
each line. Start the line with the variable name,
enter an equals sign (=) and then enter the
initial value of the variable. On the same line
then enter a comma (,), the variable name with a
prime ('), another equals sign, and then a
formula for the time derivative of the variable.
The formula can contain the any of following:
Operators:
+
for addition
- for subtraction
* for multiplication
/ for division
^ for power
Parentheses:
( )
Constants:
any number, e.g. 1.23, -6, 5.55E-5
Variables:
any string of letters and numbers,
beginning with a letter and having no
spaces
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Valid Functions: ABS - absolute
value, ABS(x) = |x|
SQR - square, SQR(x) = x2
EXP - exponential function, EXP(x) = ex
LN - natural logarithm, LN(x) = loge(x)
LOG10 - base 10 logarithm, LOG10(x) = log10(x)
SQRT - square root
SIN - sine
COS - cosine
TAN - tangent
ATN - arc tangent in the range of -pi/2
to pi/2
INT - nearest integer
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Controls
Pressing the
"Calculate" buttons starts the
integration and performs the plotting. The
"Update Plot" button will replot the
results of the integration. This can be used to
look at the behavior of different variables
without redoing the integration.
Output
The text area
below the plot gives feedback on any problems
with the input values and variables, or with
problems during the integration.
The values to be
plotted on the graph are controlled by the
"vertical" and "horizontal"
fields. The "vertical" field will
accept any number of defined, non-constant
variables, each separated by a comma. The
horizontal field will accept only one defined,
non-constant variable including "t".
©
2002 BPReid
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