Question

select all the correct answers. the isotope radium - 226 decays into radon - 222, with a half - life of around 1,600 years. if a rock contained 6 grams of radium - 226 when it reached its closure temperature but only 0.375 grams when it was discovered, which two statements about the rock are true? the rock reached its closure temperature 6,400 years ago. the rock reached its closure temperature 4,800 years ago. the rock had 2.625 grams of radon - 222 1,600 years ago. when the rock was discovered, it had 5.625 grams of radon - 222. when the rock was discovered, it had 3.375 grams of radon - 222.

Explanation:

Step1: Calculate the number of half - lives

Use the formula $N = N_0(\frac{1}{2})^n$, where $N$ is the final amount of the radioactive isotope, $N_0$ is the initial amount, and $n$ is the number of half - lives. Given $N_0 = 6$ grams, $N=0.375$ grams. So, $0.375 = 6\times(\frac{1}{2})^n$. Then $(\frac{1}{2})^n=\frac{0.375}{6}=\frac{1}{16}$, and $n = 4$.

Step2: Calculate the time since closure temperature

Since the half - life $T_{1/2}=1600$ years and $n = 4$, the time $t=n\times T_{1/2}=4\times1600 = 6400$ years. So the rock reached its closure temperature 6400 years ago.

Step3: Calculate the amount of radon - 222

The initial amount of radium - 226 was 6 grams. When the rock was discovered, the amount of radium - 226 was 0.375 grams. The amount of radon - 222 is $6 - 0.375=5.625$ grams.

Answer:

The rock reached its closure temperature 6,400 years ago.
When the rock was discovered, it had 5.625 grams of radon - 222.