Question

question 23
1 pts
if 150g sample of $^{14}$n decays to 18.75g in 21.6 seconds, what is its half - life?
7.2sec
14.4sec
10.8sec
21.6sec

Explanation:

Step1: Determine the number of half - lives

The radioactive decay formula is $N = N_0(\frac{1}{2})^n$, where $N$ is the final amount, $N_0$ is the initial amount, and $n$ is the number of half - lives. Given $N_0=150g$, $N = 18.75g$. We substitute these values into the formula: $18.75=150\times(\frac{1}{2})^n$. Then $\frac{18.75}{150}=(\frac{1}{2})^n$, and $\frac{18.75}{150}=\frac{1}{8}=(\frac{1}{2})^3$, so $n = 3$.

Step2: Calculate the half - life

We know that the time elapsed $t = 21.6$ seconds and the number of half - lives $n = 3$. The relationship between time elapsed $t$, number of half - lives $n$, and half - life $t_{1/2}$ is $t=n\times t_{1/2}$. We solve for $t_{1/2}$: $t_{1/2}=\frac{t}{n}$. Substituting $t = 21.6$ seconds and $n = 3$ into the formula, we get $t_{1/2}=\frac{21.6}{3}=7.2$ seconds.

Answer:

7.2 sec