In an alpha decay, the daughter nucleus produced is itself unstable and undergoes further decay. If the number of parent and daughter nuclei at time t are N p and N d respectively and their decay

Question

In an alpha decay, the daughter nucleus produced is itself unstable and undergoes further decay. If the number of parent and daughter nuclei at time t are N p ​ and N d ​ respectively and their decay constants are λ p ​ and λ d ​ respectively. What condition needs to be satisfied in order for N d ​ to remain constant?

TestHub · Accepted Answer

The rate of change of the number of daughter nuclei is given by the difference between its rate of formation and its rate of decay.

$\frac{d N _{d}}{d t} = λ_{p} N_{p} - λ_{d} N_{d}$

For the number of daughter nuclei $N_{d}$ to remain constant, its rate of change must be zero.

$\frac{d N _{d}}{d t} = 0$

$λ_{p} N_{p} - λ_{d} N_{d} = 0$

$λ_{p} N_{p} = λ_{d} N_{d}$

Answer: $N_{p} λ_{p} = N_{d} λ_{d}$

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