Question

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

Answer 1

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)`