Question

1. how long will it take for 16 grams of ra-226 to break down until only 1.0 grams remain? ra-226 has a half-life of 1600 years.

Explanation:

Step1: Determine the number of half - lives

The formula for radioactive decay is $N = N_0(\frac{1}{2})^n$, where $N$ is the final amount, $N_0$ is the initial amount, and $n$ is the number of half - lives.
We know that $N_0 = 16$ g, $N = 1.0$ g.
Substitute these values into the formula: $1.0=16\times(\frac{1}{2})^n$
Divide both sides by 16: $\frac{1.0}{16}=(\frac{1}{2})^n$
$\frac{1}{16}=(\frac{1}{2})^n$
Since $\frac{1}{16}=(\frac{1}{2})^4$, we can see that $n = 4$.

Step2: Calculate the total time

The half - life ($t_{1/2}$) of Ra - 226 is 1600 years. The total time $t$ is given by the formula $t=n\times t_{1/2}$
We know that $n = 4$ and $t_{1/2}=1600$ years.
So $t = 4\times1600$ years.

Answer:

6400 years