34. how can scientists determine the age of a rock sample using the half - life of a parent isotope?

Explanation:

Step1: Understand radioactive decay

Radioactive isotopes decay over time. The half - life ($t_{1/2}$) is the time it takes for half of the parent isotope to decay into the daughter isotope.

Step2: Use the decay formula

The formula for radioactive decay is $N = N_0(\frac{1}{2})^{\frac{t}{t_{1/2}}}$, where $N$ is the amount of parent isotope remaining, $N_0$ is the initial amount of parent isotope, $t$ is the time elapsed (age of the rock), and $t_{1/2}$ is the half - life of the parent isotope.

Step3: Measure isotope amounts

In a rock sample, scientists measure the current amount of the parent isotope ($N$) and estimate the initial amount ($N_0$) (for example, by assuming that initially there was no daughter isotope and using the current amounts of parent and daughter isotopes together to infer $N_0$).

Step4: Solve for age

Rearrange the decay formula to solve for $t$: $t=-t_{1/2}\frac{\ln(\frac{N}{N_0})}{\ln(2)}$. By substituting the measured values of $N$, $N_0$, and the known half - life $t_{1/2}$, they can calculate the age of the rock sample.

Answer:

Scientists measure the current amount of the parent isotope ($N$) and estimate the initial amount ($N_0$) in the rock sample. They use the known half - life ($t_{1/2}$) of the parent isotope and the radioactive decay formula $t=-t_{1/2}\frac{\ln(\frac{N}{N_0})}{\ln(2)}$ to calculate the age of the rock sample.