Solve Nuclear Physics Decay Problem: A&B Half-Life

[SUDOnym]

Problem is:

have a mother and daughter sample, $$A$$ and $$B$$ respectively. both are radioactive. The number of daughter nuclei at time t is given by (*):

$$n(t)=\frac{N_{0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}[e^{-\lambda_{A}t}-e^{-\lambda_{B}t}]$$

where N_0 is number of mother nuclei at t=0 and n(t) is number of daughter nuclei at time t.

A has $$\tau_{\frac{1}{2}}=23minutes$$ and B has $$\tau_{\frac{1}{2}}=23days$$.

A is beta only emitter. B emits gamma and Beta. If A has been made and purified and 11.5minutes after this, the sample emits 1000 gammas/second and some time later the sample again emits 1000 gammas/second - how much time has elapsed?

My Thoughts:
I don't know how to handle this problem for the following reason: To find the rate of gammas being emitter, simply differentiate the equation I showed above (*) to get:

$$\frac{dn}{dt}=\frac{N_{0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}[-\lambda_{A}e^{-\lambda_{A}t}+\lambda_{B}e^{-\lambda_{B}t}]$$

(note can find the lambda_A and lambda_B as we know the half-llife and can also solve (*) for N_0).

The time dependence of the above equation is a negative exponential... so to solve for t, do some rearranging, and take the natural log... but this will be a linear equation... ie. there will only be one value of t for any dn/dt so it is not clear to me how at 11.5mins can have 1000 gammas/second and then again some time later can also have 1000 gammas/second.

What to do?

 

[Nate810]

The best approach is to calculate the total number of daughter nuclei at t=11.5mins, which can be done by plugging in the values of N_0, \lambda_A, \lambda_B, and t into equation (*). Then, you can use this initial value to determine how many daughter nuclei will decay in a given amount of time. For example, if you want to know how many gammas/second are emitted after 11.5minutes have elapsed, you can calculate the rate of decay (dn/dt) of the daughter nuclei over a given time interval. To do this, you can use the equation:\frac{dn}{dt}=\frac{N_{0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}[-\lambda_{A}e^{-\lambda_{A}t}+\lambda_{B}e^{-\lambda_{B}t}]where N_0, \lambda_A, and \lambda_B are as before, and t is the amount of time that has elapsed since 11.5mins.This equation can then be used to calculate the rate of gamma emission at any given point in time after 11.5 minutes have elapsed. By plugging in different values of t, you can determine the amount of time that must elapse in order for the sample to emit 1000 gammas/second again.