Question

Polonium-21O The number of radioactive atoms remaining after \(t\) days in a sample of polonium- 210 that starts with \(y_{0}\) radioactive atoms is \(y=y_{0} e^{-0.005 t}\) (a) Find the element's half-life. (b) Your sample will not be useful to you after 95\(\%\) of the radioactive nuclei present on the day the sample arrives have disintegrated. For about how many days after the sample arrives will you be able to use the polonium?

Short answer

The half-life of polonium-210 is approximately \(138.63\) days, and a sample will be useful up to roughly \(299.57\) days.

Step-by-step solution

Calculate half-life

We are given the decay formula \(y = y_{0}e^{-0.005 t}\), where \(k = 0.005\). The formula for half-life is \(\frac{ln 2}{k}\). Substitute \(0.005\) into the formula to find the half-life. This gives: \(\frac{ln 2}{0.005} = 138.63\) days (rounded to two decimal places).

Calculate lifespan of a sample

We need to determine when 95% of the sample has decayed. This means 5% is left, so we use the decay formula \(y = y_{0}e^{-0.005 t}\) with \(y = 0.05 y_{0}\). Then solve for \(t\) by taking the natural logarithm of both sides and rearranging: \(\ln\left(\frac{y}{y_{0}}\right) = -0.005 t\), \(\ln(0.05) = -0.005 t\), which gives \(t \approx 299.57 \) days.

Conclusion

Hence, the half-life of polonium-210 is approximately \(138.63\) days, and a sample will be useful up to roughly \(299.57\) days.