Question

a sample of zircon contains 25% of its original uranium - 238 (u - 238) remaining. the half - life of u - 238 is 4.5 billion years. 1. how many half - lives have passed? type your answer... 2. what is the approximate age of the sample (in billions of years)? type your answer...

Explanation:

Step1: Use the radioactive - decay formula

The amount of a radioactive substance $N$ at time $t$ is given by $N = N_0(\frac{1}{2})^n$, where $N_0$ is the initial amount, $n$ is the number of half - lives. We know that $N = 0.25N_0$. Substituting into the formula: $0.25N_0=N_0(\frac{1}{2})^n$.

Step2: Solve for $n$

Divide both sides of the equation $0.25N_0=N_0(\frac{1}{2})^n$ by $N_0$ (since $N_0
eq0$), we get $0.25 = (\frac{1}{2})^n$. Since $0.25=\frac{1}{4}=(\frac{1}{2})^2$, then $n = 2$.

Step3: Calculate the age of the sample

The half - life of U - 238 is $T_{1/2}=4.5$ billion years. The age of the sample $t=nT_{1/2}$. Substituting $n = 2$ and $T_{1/2}=4.5$ billion years, we get $t=2\times4.5 = 9$ billion years.

Answer:

1. 2
2. 9