Question #24536

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

Expert's answer

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

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

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