# Technetium 99 is used for brain scans. If a laboratory receives a shipment of 200 g of this isotope, how much will remain after 24 hours? The half life of Technetium 99 is 6 hours.

Apr 24, 2018

12.5 g

#### Explanation:

The basic formula for half life equations is

$A = P {\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

A= amount remaining
P= original amount
t= elapsed time
h=half life time

so just plug in your amounts
P=200
t=24
h=6

$A = 200 {\left(\frac{1}{2}\right)}^{\frac{24}{6}}$

when you put it into a calculator, you get 12.5 grams

:)

Apr 24, 2018

Consider that radioactive decay follows first order kinetics. Thus, recall,

$\ln {\left[A\right]}_{\text{t}} = - k t + \ln {\left[A\right]}_{0}$, and by extension,

${t}_{\frac{1}{2}} = \ln \frac{2}{k}$

Now, let's derive the rate constant,

$\implies k = \ln \frac{2}{t} _ \left(\frac{1}{2}\right) \approx 0.116 {\text{h}}^{-} 1$

Given $t = 6 \text{h}$,

$\implies \ln \left(\frac{{\left[A\right]}_{\text{t}}}{{\left[A\right]}_{0}}\right) = - k t$

$\implies {\left[A\right]}_{\text{t" approx 12.5"g}}$

of technetium-99 will remain of the shipment.