Atoms and Nuclei

When 8% of nuclei decays, then remaining nuclei are 92% from
As, N=N0e−λt
Here, N=0.92N0
So, 0.92N0=N0e−λt1
When, 77% nuclei decay remaining number of nuclei are 23%.
So, ‌‌N=0.23N0
0.23N0=N0e−λt2
By dividing in equation (i) by equation (ii)
‌

0.92

0.23

=‌

e−λt1

e−λt2

=eλ(t2−t1)
So, eλ(t2−t1)=4
By taking natural log of both sides
λ(t2−t1)‌=ln‌4
‌ and ‌t2−t1‌=12‌ minutes ‌
12λ‌=2‌ln‌2
λ‌=‌

ln‌2

6

and relation between half life and decay constant is

λ=‌

ln‌2

t1∕2

So, ‌

ln‌2

6

=‌

ln‌2

t1∕2

and t1∕2=6 minutes