# Answer to Question #24536 in Physical Chemistry for ramya sarvani

Question #24536
the half life of a radio active metal is one hour.what would be the time required for 99.9% completion?
1
2013-02-19T08:55:24-0500
The reaction of radioactive transformation is the first order reaction. That&#039;s why the rate equation is

dN / d&tau; = &lambda;N

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

ln(N/ N0) = - &lambda;&tau; &
or N = N0 x e- &lambda;&tau;& &

where N - the amount of atoms at time &tau;
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/ &lambda;= 0,69 / &lambda; &
lambda; = 0,69 /& &tau;1/2 &
from& the task& tau;1/2 = 1 hour. That&#039;s why lambda; = 0,69 / 1 = 0,69 (hour^-1)
N/N0 = (1-0,999)N0/N0 =& exp(- &lambda;&tau;)
0,001 = exp (-0,69 x &tau;)
ln0,001 = -0,69 x &tau;
&tau; = ln0,001/(-0,69) = -6,9/(-0,69) = 10 hours

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

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