Question

9. Thallium-208 has a half-life of 3.053 min. How long will it take for 120 g of it to decay

Answer 1

Answer:

12.213 minutes will be taken for 120 g-Thalium-208 to decay to 75 grams.

Explanation:

Radioactive isotopes decay exponentially in time, the mass of the isotope ($m(t)$), in grams, is described by the formula in time ($t$), in minutes:

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$ (1)

Where:

$m_{o}$ - Initial mass of the isotope, in grams.

$\tau$ - Time constant, in minutes.

In addition, the time constant associated with the isotope decay can be described in terms of half-life ($t_{1/2}$), in minutes:

$\tau = \frac{t_{1/2}}{\ln 2}$ (2)

If we know that $m(t) = 7.5\,g$, $m_{o} = 120\,g$ and $t_{1/2} = 3.053\,min$, then the time taken by the isotope is:

$\tau = \frac{t_{1/2}}{\ln 2}$

$\tau = \frac{3.053\,min}{\ln 2}$

$\tau \approx 4.405\,min$

$t = -\tau \cdot \ln \frac{m(t)}{m_{o}}$

$t = -(4.405\,min)\cdot \ln \left(\frac{7.5\,g}{120\,g} \right)$

$t \approx 12.213\,min$

12.213 minutes will be taken for 120 g-Thalium-208 to decay to 75 grams.