Physics homework on Radioactivity

[zorro]

Homework Statement

A radioactive material decays by simultaneous emission of two particles with respective half-lives t1 and t2. If the material decays by the emission of the particle with half-life t2 only, then the time in year after which 1/4 th of the material remains is. (Given t1=1620 years and mean-life T of the sample = 540 years)

The Attempt at a Solution

None.

Is there any relation between the mean life of the sample T and half-lives t1 and t2?

 

[cupid.callin]

Yes there is:

t_{average} = \frac{1}{\lambda}

t_{half life} = \frac{0.693}{\lambda}

so you can find the relation

 

[zorro]

That is the relation for one decay alone. Here we have two half lives t1 and t2 of two decays.

 

[cupid.callin]

You can find it like this:

activity of first rxn = A₁
activity of second rxn = A₂

so net activity, A = A₁ + A₂
(pretty obvious as activity is measure of decay and net decay is just sum of 2 decays)

λN = λ₁N + λ₂N

solve and substitute for λ, λ₁, λ₂ in terms of T, t₁, t₂

 

[zorro]

We don't get the correct answer that way. Backsolving from the correct answer, t2 should be 810 years. But using your method it comes out to be 487 years.

 

[cupid.callin]

i guess the method is correct because i have used it in many questions and it gives a correct answer