A source contains two phosphorous radio nuclides 3215P(T1/2=14.3d) and 3315P(T1/2=25.3d). Initially, 10% of the decays come from 3215P. How long one must wait until 90% to do so?
Let the number of atoms of 3215P be N1 and atoms of 3315P be N2.The decay constants of two atoms be λ1 and λ2 respectively.
The initial activity of 3315P is 19 times that of 3215P.
Therefore, we can write the decay constants as:
N1λ1=N2λ29 ___(i)
Let after time t the activity of 3315P be 9 times that of 3215P
N1λ1e−λ1t=9N2λ2e−λ2t ___(ii)
Now. Divide equation (ii) by (i) and taking the natural log of both sides we get
−λ1t=ln 81−λ2t
t=ln 81λ2−λ1
where λ2=0.048/day and λ1=0.027/day
t=4.3940.048−0.027=209.2 days
t comes out to be 209.2 days