Question

1. potassium - 40 has a half - life of 1.25 billion years. what percent of a sample will be left after 5 billion years?

Explanation:

Step1: Calculate the number of half - lives

The formula to find the number of half - lives $n$ is $n=\frac{t}{T_{1/2}}$, where $t$ is the time elapsed and $T_{1/2}$ is the half - life. Given $t = 5$ billion years and $T_{1/2}=1.25$ billion years. So, $n=\frac{5}{1.25}=4$.

Step2: Use the radioactive decay formula

The amount of a radioactive substance remaining $N$ after $n$ half - lives, starting from an initial amount $N_0$, is given by $N = N_0\times(\frac{1}{2})^n$. We want to find the fraction of the sample remaining, which is $\frac{N}{N_0}=(\frac{1}{2})^n$. Substituting $n = 4$ into the formula, we get $(\frac{1}{2})^4=\frac{1}{16}$.

Step3: Convert the fraction to a percentage

To convert $\frac{1}{16}$ to a percentage, we use the formula $\text{Percentage}=\frac{1}{16}\times100\%$. So, $\frac{1}{16}\times100\% = 6.25\%$.

Answer:

$6.25\%$