Question

niobium-91 has a half-life of 680 years. after 2,040 years, how much niobium-91 will remain from a 300.0-g sample?
○ 3 g
○ 18.75 g
○ 37.5 g
○ 100.0 g

Explanation:

Step1: Determine the number of half - lives

The half - life of niobium - 91 is \(t_{1/2}=680\) years. The total time elapsed \(t = 2040\) years.
The number of half - lives \(n=\frac{t}{t_{1/2}}=\frac{2040}{680}=3\).

Step2: 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 of the substance, \(n\) is the number of half - lives, and \(N\) is the remaining amount.
We know that \(N_0 = 300.0\) g and \(n = 3\).
Substitute these values into the formula: \(N=300.0\times(\frac{1}{2})^3\).
First, calculate \((\frac{1}{2})^3=\frac{1}{8}\).
Then, \(N = 300.0\times\frac{1}{8}=37.5\times\frac{1}{2}=18.75\) g.

Answer:

18.75 g