Question

question 22 of 30 how much of the original amount of an isotope is present after a period of two half-lives? a. half of the original amount b. four times the original amount c. twice the original amount d. one-fourth of the original amount

Explanation:

Step1: Understand half - life concept

A half - life is the time it takes for half of a radioactive isotope to decay. Let the original amount of the isotope be \(N_0\). After one half - life, the amount of the isotope remaining, \(N_1\), is given by \(N_1=\frac{1}{2}N_0\) (because half of the original amount decays).

Step2: Calculate amount after two half - lives

After the second half - life, the amount of the isotope remaining, \(N_2\), is half of the amount that was present after the first half - life. Since \(N_1 = \frac{1}{2}N_0\), then \(N_2=\frac{1}{2}\times N_1\). Substituting \(N_1=\frac{1}{2}N_0\) into the equation for \(N_2\), we get \(N_2=\frac{1}{2}\times\frac{1}{2}N_0=\frac{1}{4}N_0\). So after two half - lives, the amount of the isotope present is one - fourth of the original amount.

Answer:

D. One - fourth of the original amount