Radioactive Decay: Proving Effective Half-Life of Nucleus

[roshan2004]

Homework Statement

A radioactive nucleus can decay by two different processes. The half life for the first processes is $$t_{1}$$ and that for the second is $$t_{2}$$. Show that the effective half life t of the nucleus is given by
$$\frac{1}{t}=\frac{1}{t_{1}}+\frac{1}{t_{2}}$$

Homework Equations

$$t=\frac{0.693}{\lambda }$$

The Attempt at a Solution

Tried to use $$\lambda =\lambda _{1}+\lambda _{2}$$, and got the answer but don't know why, how and is it the correct way to prove this?

 

[tiny-tim]

hi roshan2004!

(have a lambda: λ )

if one process is dA/dt = -λ₁A, and the other is dA/dt = -λ₂A,

then the total rate is dA/dt = -(λ₁A + λ₂A)

from which you can obtain the https://www.physicsforums.com/library.php?do=view_item&itemid=193" as you did

 

[roshan2004]

@tiny-tim Now you made me totally lost

 

[gneill]

@tiny-tim Now you made me totally lost

Suppose the two half-lives are τ1 and τ2. Let the initial amount of substance be A.

After time t, due to the first process you expect to see remaining:

A*2^-(t/ τ1)

But the second process has a go at the other stuff that didn't go by the first process. So the remaining amount becomes:

A*2^(t/ τ1)* 2^-(t/ τ2)

Now, n^a * n^b = n^(a + b). So do the obvious with the above.

 

[tiny-tim]

oooh, sorry

the decay equation is A = A₀e^-λt,

which is the same as dA/dt = -λA