What is the formula when two exponential-decay processes take place simultaneously?

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Answer 1 · 2014-08-10T22:18:50

The formula is $N_t = N_0e^-((λ_1 + λ_2)t)$ .

For a single decay mode, the formula for decay rate is

$-(dN)/(dt) = λN$

If there are two different simultaneous decay processes, the total decay rate is the sum of the two processes.

$-(dN)/(dt) = λ_1N + λ_2N = (λ_1 + λ_2)N$

On integration, this becomes

$N_t = N_0e^-((λ_1 + λ_2)t)$

We can simplify this by defining a new total decay constant $λ_"c"$ :

$λ_"c" = λ_1 + λ_2$

Then the equation becomes

$N_t = N_0e^-(λ_"c"t)$