A question about radioactive decay

[ubergewehr273]

Homework Statement

Two species of radioactive atoms are mixed in equal numbers. The disintegration constant of first species is $\lambda$ and that of second species is $\frac \lambda {3}$. After a long time, the mixture will behave as a species with mean life ________.

Homework Equations

$\lambda=\frac{0.693} {t_{1/2}}$
$t_{mean}=\frac{1}\lambda$

The Attempt at a Solution

What I did was to equate the mean life of mixture being the average of the mean lives of the individual species.
$$t_{mean}=\frac{\frac{1} \lambda + \frac{3}\lambda} {2}$$

However in this article https://en.wikipedia.org/wiki/Half-life it says that $\frac{1} {t{eff}} = \frac{1} {t_{1}} + \frac{1} {t_{2}}$ where $t_{eff}$,$t_{1}$ & $t_{2}$ are half lives of mixture, species A and B respectively.

 

[phyzguy]

Actually the question can be answered from the equations you gave without calculating a combined mean life. After a long time, which of the two species will dominate?

 

[ubergewehr273]

Species dominates as in amount of nuclei ? Then should be species B

 

[ubergewehr273]

Actually the question can be answered from the equations you gave without calculating a combined mean life.

How so?

 

[hilbert2]

If you have two quantities, $a(t)$ and $b(t)$, which decay exponentially:

$a(t) = Ae^{-k_a t}$, $b(t) = Be^{-k_b t}$,

then the ratio of the quantities, $\frac{a(t)}{b(t)} = \left(\frac{A}{B}\right)e^{(k_b - k_a )t}$, will approach either zero or infinity when $t\rightarrow\infty$, depending on whether $k_a > k_b$ or $k_b > k_a$.

Based on only this, you can deduce that if two mixed radioactive species have even a small difference in half-life, one of them will be the only significant species after a long enough time has passed.

 

[phyzguy]

Species dominates as in amount of nuclei ? Then should be species B

Correct. So after a long time, species B dominates and basically all of species A has decayed away. So it isn't really a mixture any more, it is only species B. So what then is the mean life?