Protactinium-234 has a half life of 1.17 minutes. How long does it take for a 10 mg sample to decay to 2 mg?

Dec 6, 2015

Answer:

Solve ${\left(\frac{1}{2}\right)}^{\frac{t}{1.17}} = \frac{2}{10}$ by taking logs to find time $2$ minutes $43$ seconds.

Explanation:

We need to solve ${\left(\frac{1}{2}\right)}^{\frac{t}{1.17}} = \frac{2}{10} = \frac{1}{5}$

Taking logs of both sides:

$\left(\frac{t}{1.17}\right) \log \left(\frac{1}{2}\right) = \log \left(\frac{1}{5}\right)$

So:

$\frac{t}{1.17} = \frac{\log \left(\frac{1}{5}\right)}{\log \left(\frac{1}{2}\right)} = \frac{- \log \left(5\right)}{- \log \left(2\right)}$

So $t = 1.17 \cdot \frac{\log \left(5\right)}{\log \left(2\right)} \approx 1.17 \cdot \frac{0.69897}{0.30103} \approx 2.71666$ minutes

or about $2$ minutes $43$ seconds.