MTH 455 Homework #5

1. Use Maple to make a plot (on an appropriate timescale) 
of [A], [B], and [C] versus time (all 3 on the same axes) 
for the irreversible reaction A + B -> C with 
k = 7 x 106 and initial conditions [A]=1, [B]=0.35,[C]=0.

2. Show that there is only one physically meaningful (i.e. non-negative)
solution of the equilibrium condition given at the bottom of p3 of Text 3A
for the reversible reaction A + B <-> C.

3. EXTRA CREDIT. Sometimes chemical kineticists will write a reaction
as 

                3 A -> B                          (1).

But in reality there aren't any reactions that require
simultaneous collisions of 3 molecules the way this appears to imply.
The more likely reality is that there is an intermediate, say X, and
what really happens is 

          2 A <-> X,  and     A + X -> B.        (2) 

Under what circumstances would the simpler scheme (1) give approximately
the same dynamical predictions as the more honest scheme (2)?

4. For the Schlogl model with k=1, use Maple to make a quantitatively 
accurate picture of (an appropriate portion of) the region 
in the ab parameter plane where multiple equilibria coexist.
How to do it: 
(i) Write down the equilibrium (steady-state) condition. 
(ii) Observe that this is a cubic in x, and that varying b just
shifts the cubic vertically. Therefore, for (a,k) values for which
the cubic has extrema, b can be adjusted to obtain multiple equilibria
(intersections of the cubic with the x-axis). By differentiating
the cubic with respect to x, and using the quadratic formula,
find what the extreme points are (values of x), and 
a condition on a and k which says when the cubic does have extrema.
(iii) Find the values of b that correspond to the two extreme points.
These depend on a and k. With k=1, we can plot these values of b 
versus a. Use the "view=[]" plot option to exclude any portions of the
curves that have b<0. 
To give you something to compare with, here's my plot: 

The region with multiple steady states is the lower right.