Question

the half life of a radio active metal is one hour.what would be the time required for 99.9% completion?

Answer 1

The reaction of radioactive transformation is the first order reaction. That's why the rate equation is

dN / dτ = λN

After integrating when τ= 0& --> N = N0 we have
tau; = τ --> N = N

ln(N/ N0) = - λτ &
or N = N0 x e- λτ& &

where N - the amount of atoms at time τ
N0 - he amount of atoms in the beginning of reaction
& lambda; - the coefficient, that depends on the kind of the radionuclide

The half-life time -& the time interval required for an amount of certain radioactive nuclei to decay to its half of original value &
The equation for the half life time is

tau;1/2& = ln 2/ λ= 0,69 / λ &
lambda; = 0,69 /& τ1/2 &
from& the task& tau;1/2 = 1 hour. That's why lambda; = 0,69 / 1 = 0,69 (hour^-1)
N/N0 = (1-0,999)N0/N0 =& exp(- λτ)
0,001 = exp (-0,69 x τ)
ln0,001 = -0,69 x τ
τ = ln0,001/(-0,69) = -6,9/(-0,69) = 10 hours

Answer:

The time required for 99,9% completion is 10 hours