Question

Polonium-210 is a radioactive substance with a half life of 138 days. If a nuclear facility is handling 265 grams of polonium-210, then how many grams of polonium-210 will be left after 250 days.

Answer 1

The half life is 138 days, meaning exactly half of the material will decay after this time. Use this information to find the decay factor $k$:

$\dfrac12=e^{138k}\implies k=-\dfrac{\ln2}{138}$

Then after 250 days, there will be

$265e^{250k}\approx75.492$

grams left.

Answer 2

Answer:

$\boxed{\text{75.5 g}}$

Step-by-step explanation:

The amount of substance decreases by 50 % every half-life.

Thus, 50 % of the substance remains at the end of each half-life.

The exponential function is

$a = a_{0}\left( \dfrac{1}{2}\right)^{x}$

where x = the number of half-lives.

Data:

a₀ = 265 g

$t_{\frac{1}{2}} = \text{138 da}$

t = 250 da

Calculations:

(a) Calculate x

$x = \dfrac{250}{138} \approx 1.812$

(b) Calculate a

$a = 265\left( \dfrac{1}{2}\right)^{1.812}\\\\a = 265 \times 0.2849 = \boxed{\textbf{75.5 g}}$