Question

question 14 (mandatory) (1 point) a 150 g sample of protactinium - 233 has a half - life of 27 days. how long will it take for this sample to decay to 100 g? a) about 18 days b) about 25 days c) about 14 days d) about 16 days

Explanation:

Step1: Determine decay amount

The initial mass is 150 g and it decays to 100 g. The amount of decay is $150 - 100=50$ g.

Step2: Use half - life formula

The radioactive decay formula is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life. Here, $N_0 = 150$, $N = 100$, and $T_{1/2}=27$ days. So, $100 = 150(\frac{1}{2})^{\frac{t}{27}}$. Then $\frac{100}{150}=(\frac{1}{2})^{\frac{t}{27}}$, or $\frac{2}{3}=(\frac{1}{2})^{\frac{t}{27}}$.
Taking the natural logarithm of both sides: $\ln(\frac{2}{3})=\frac{t}{27}\ln(\frac{1}{2})$.
Solve for $t$: $t = 27\times\frac{\ln(\frac{2}{3})}{\ln(\frac{1}{2})}$.
$\ln(\frac{2}{3})\approx\ln(2)-\ln(3)\approx0.693 - 1.099=- 0.406$, $\ln(\frac{1}{2})=-\ln(2)\approx - 0.693$.
$t = 27\times\frac{-0.406}{-0.693}\approx16$ days.

Answer:

d) about 16 days