Question

1. a geologist analyzes a volcanic ash layer using uranium - lead (u - pb) dating. the results show that a quarter of the original uranium - 238 remains, and the rest has decayed into lead - 206. the half - life of uranium - 238 is 4.5 billion years. based on this data, what is the approximate age of the volcanic ash layer? 0 0 billion years 2.25 billion years 9.0 billion years 4.5 billion years

Explanation:

Step1: Understand half - life concept

The amount of a radioactive substance $N$ at time $t$ is given by $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N_0$ is the initial amount, $T_{1/2}$ is the half - life, and $t$ is the time elapsed. We know that $\frac{N}{N_0}=\frac{1}{4}$.

Step2: Substitute values into formula

Since $\frac{N}{N_0}=\frac{1}{4}=(\frac{1}{2})^2$ and $\frac{N}{N_0}=(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, and $T_{1/2}=4.5$ billion years. Then $(\frac{1}{2})^{\frac{t}{4.5}}=(\frac{1}{2})^2$.

Step3: Solve for $t$

Equating the exponents, we get $\frac{t}{4.5}=2$. So $t = 2\times4.5=9.0$ billion years.

Answer:

9.0 billion years