where N A and N B are the number of nuclei and lambda A and lamba B are the deca

Question

where NA and NB are the number of nuclei and lambda A and lamba B are the decay constants for A and B respectively. The equations represent decrease of A due to decay into B and the change in the amount of B due to both decay into C and decay from A.

a) write the differential equations as a matrix equation of the form N'=KN where K is a matrix. Calculate the eigenvalues of the matrix K.

b)For the case lambdaA = 0.02 and lmbda B = 0.1 and for the initial conditions NA(0) =1000, NB(0) = 500, find and graph NA(t) and NB(t) on the same graph

c)for the case in b graph a phase parametric plot with NB on the vertical axis and NA on the horizontal axis.

d) The total decay rate is defined with

(for case b) find and graph Rtot(t)

Explanation / Answer

a) K = {[lambda A, 0], [LambdaA, - lambdaB]}

b) both are the exponential graphs with the asymtote with the final value

for b it increases initially then decreases and then turns to zero

c) i.e when the dNb/dt turns to zero at that time the transition in graph occurs i.e Na/Nb = lambda b/lambda a

d) the graph for this equation is again an exponential graph with decreasing slope and asymtope at 0

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