Atoms and Nuclei Part 1

Given, initial amount of X and Y be N1 and N2.
Let half-life of X be tx and y be ty.
According to question, tx=‌

ty

2

=t
⇒‌‌tx=t and ty=2t
After 3 half-lives of Y,
3ty=6t
As we know that,
N=N0e−λt
where, N is the number of nuclei left undecayed.
and t1∕2‌‌=‌

0.693

λ

λ‌‌=‌

0.693

t1∕2

λ1‌‌=‌

0.693

t

‌ and ‌λ2=‌

0.693

2t

Since, after 3 half-lives of Y number of nuclei of both become equal.
∴ ∴‌N1e−λ16t=N2e−λ26t
⇒‌N1∕N2=e6t(−λ2+λ1)
⇒‌N1∕N2=e6t(‌

0.693

t

−‌

0.693

2t

)
‌‌=e0.693(‌

6t

t

−‌

6t

2t

)=e0.693×3=7.9∼eq8
‌

N1

N2

‌‌=‌

8

1