Question

a sample of a radioactive metal starts with a mass of 400 grams. when a chemist examines it later, it has gone through 4 half - lives. how much metal remains?
100 grams
50 grams
25 grams
12.5 grams
there is not enough information to answer this question
a radioactive metal starts with a mass of 1024 grams. after some time, only 2 grams are left. how many days later did the chemist make her second measurement?
1 day
9 days
10 days
512 days
not enough information to solve this.

Explanation:

Step1: Calculate the remaining mass after 4 half - lives

The formula for the remaining mass \(m\) of a radioactive substance after \(n\) half - lives is \(m = m_0\times(\frac{1}{2})^n\), where \(m_0\) is the initial mass.
Given \(m_0 = 400\) grams and \(n = 4\), then \(m=400\times(\frac{1}{2})^4\).
First, calculate \((\frac{1}{2})^4=\frac{1}{16}\).
Then \(m = 400\times\frac{1}{16}=25\) grams.

Step2: Calculate the number of half - lives for the second problem

Using the formula \(m = m_0\times(\frac{1}{2})^n\), where \(m_0 = 1024\) grams and \(m = 2\) grams.
We have \(2=1024\times(\frac{1}{2})^n\), then \((\frac{1}{2})^n=\frac{2}{1024}=\frac{1}{512}\).
Since \(\frac{1}{512}=(\frac{1}{2})^9\), so \(n = 9\) half - lives. But we don't know the length of a half - life (no information about the time per half - life is given), so there is not enough information to calculate the number of days.

Answer:

For the first question: 25 grams.
For the second question: Not enough information to solve this.