Having been operated long enough, the ionixer producing `n_(1) = 3.5.10^(9) cm^(-3). S^(-1)` of one pairs per unit volume of air per unit time was switched off. Assuming that the only process tending to reduce the numnber of ions in air is thier recombination with coefficient `r = 1.67.10^(-5)cm^(3)//s`, find how soon after the ioner concentration decreases `eta = 2.0` times.

Explanation

Initiallly `n = n_(0) = sqrt(n_(1)//r)`
Since we can assume that the long exposure to the ionizer has caused equilibrium to be set up. After the ionzer is switched off,
`(dn)/(dt) = -r n^(2)`
or, `r dt = (d n)/(n^(2))`, or, `rt = (1)/(n) +` constant
But `n = n_(0)` at `t = 0`, so `rt = (1)/(n) (1)/(n_(0))`
The concentratiion will decrease by a factor `eta` when
`r t_(0) = (1)/(n_(0)//eta) - (1)/(n_(0)) = (eta - 1)/(n_(0))`
or, `t_(0) = (eta - 1)/(sqrt(r dotn_(i))) = 13 ms`.