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`:

`\frac12=e^(138k)\implies k=-(\ln2)/(138)`

Then after 250 days, there will be

`265e^(250k)\approx75.492`

grams left.

Answer 2

`\boxed{\text{75.5 g}}`

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( (1)/(2)\right)^(x)`

where x = the number of half-lives.

Data:

a₀ = 265 g

`t_{(1)/(2)} = \text{138 da}`

t = 250 da

Calculations:

(a) Calculate x

`x = (250)/(138) \approx 1.812`

(b) Calculate a

`a = 265\left( (1)/(2)\right)^(1.812)\\\\a = 265 * 0.2849 = \boxed{\textbf{75.5 g}}`