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

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)##