Question

samarium-146 has a half-life of 103.5 million years. after 1.035 billion years, how much samarium-146 will remain from a 205-g sample?
○ 0.200 g
○ 0.400 g
○ 20.5 g
○ 103 g

Explanation:

Step1: Convert time units

First, convert 1.035 billion years to million years. Since 1 billion = 1000 million, 1.035 billion years = \(1.035\times1000 = 1035\) million years.

Step2: Calculate number of half - lives

The half - life of Samarium - 146 is 103.5 million years. The number of half - lives \(n\) is given by the formula \(n=\frac{\text{total time}}{\text{half - life}}\). So \(n=\frac{1035}{103.5}=10\).

Step3: Use the radioactive decay formula

The formula for radioactive decay is \(N = N_0\times(\frac{1}{2})^n\), where \(N_0\) is the initial amount, \(n\) is the number of half - lives, and \(N\) is the remaining amount. Here, \(N_0 = 205\space g\) and \(n = 10\). So \(N=205\times(\frac{1}{2})^{10}\).
We know that \((\frac{1}{2})^{10}=\frac{1}{1024}\approx0.0009766\). Then \(N = 205\times0.0009766\approx0.200\space g\).

Answer:

0.200 g