Question

A small quantity of solution containing the radionuclide $N{a}^{24}$ (Half life $15$ hours), of activity $1.0$ microcurie, is injected into the blood of a patient. A sample of the blood of volume $1c{m}^3$ taken after $5$ hours shown an activity of $296$ disintegration per minute. Determine the total volume of blood in the patients body. Assume that the solution radioactive solution mixes uniform in the blood. ($1CI=3.7\times {10}^{10}$ disintegration per second)

Answer 1

The activity varies according to the equation,
$A\left(t\right)=A_0{e}^{-\lambda t}$
or, $A\left(t\right)=A_0{\left(\frac{1}{2}\right)}^{t/T_{1/2}}$
In the problem, $A(t=5hr)=296$ disintegrations/min,
$\frac{296}{60}=A_0{\left(\frac{1}{2}\right)}^{5/15}$
or, $A_0=\frac{296}{60}\times 2^{1/3}=6.22$ disintegrations/sec.
$=$ the initial activity of $1cc$. of blood.
If the total volume of blood is $Vcc$.,
$6.22V=3.7\times {10}^{10}\times {10}^{-6}\Rightarrow V=5.9\ell$.