Is there an explanation for the unexpected increase in activity of Nuclide A?

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The discussion revolves around the unexpected increase in the activity of Nuclide A, which initially appears to contradict the principles of radioactive decay. The decay constant for Nuclide A is calculated, leading to an initial activity of 2.84 Bq, yet the activity later rises to 10,000 Bq. Participants suggest potential errors in the problem, such as incorrect half-lives or an underestimated initial number of nuclei. They introduce the concept of "Transient Equilibrium," where the decay of a parent nuclide leads to an increase in the activity of its daughter nuclide. The thread emphasizes the need for clarity in the problem's parameters to resolve the apparent contradiction.
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Homework Statement
Nuclide A decays to nuclide B. Initially, there are ##1.29 \times10^9## nuclei of A and after some time the activity of A is 10 000 Bq. If the half life of A and B is 10 years and 10 hours respectively, find activity of B
Relevant Equations
##A=\lambda N##

##t_{\frac 1 2}=\frac{ln~2}{\lambda}##
I found something I think does not make sense.

Decay constant of A:
$$\lambda_{A}=\frac{ln~2}{t_{\frac 1 2}}$$
$$=\frac{ln~2}{10\times 365 \times 24 \times 3600}$$
$$=2.2\times 10^{-9} / s$$

Initial activity of A = ##\lambda_{A} N_{\text{initial}}## = 2.2 x 10-9 x 1.29 x 109 = 2.84 Bq

Then after some time the activity becomes 10 000 Bq. How can the activity increase instead of decrease?

Is there something wrong with the question or something wrong with me?

Thanks
 
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songoku said:
Is there something wrong with the question or something wrong with me?
There's nothing wrong with you! The question appears to have one or more mistakes.

Some possibilities are:
- the half-lives of A and B are the wrong way round;
- ##1.29 \times10^9## is a very small number for a number of nuclei in this context; maybe it should be ##1.29 \times10^{19}## for example.
 
You could at least obtain the general form of the solution.
 
Thank you very much Steve4Physics and haruspex
 
songoku said:
Homework Statement:: Nuclide A decays to nuclide B. Initially, there are ##1.29 \times10^9## nuclei of A and after some time the activity of A is 10 000 Bq. If the half life of A and B is 10 years and 10 hours respectively, find activity of B
Relevant Equations:: ##A=\lambda N##

##t_{\frac 1 2}=\frac{ln~2}{\lambda}##

Then after some time the activity becomes 10 000 Bq. How can the activity increase instead of decrease?
One should research "Transient Equilibrium", where t1/2(parent) > t1/2(daughter), or λ(parent) < λ(daughter). This is observed for the natural decay series of radionuclides 232Th, 235U, 238U and others.

https://en.wikipedia.org/wiki/Transient_equilibrium
Also see related Secular equilibrium.
 
Astronuc said:
One should research "Transient Equilibrium", where t1/2(parent) > t1/2(daughter), or λ(parent) < λ(daughter). This is observed for the natural decay series of radionuclides 232Th, 235U, 238U and others.

https://en.wikipedia.org/wiki/Transient_equilibrium
Also see related Secular equilibrium.
At this stage it is not clear from @songoku's posts whether he has any difficulty in solving a correctly posed version of the question. The thread centres on the impossible combination of given facts.
 
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