Question

if half of a radioactive isotope decays in one hour, how much of the original isotope will remain after three hours?
all of it
half of it
one quarter of it
one eighth of it
none of it

if you start with 100 grams of carbon - 14 atoms, how many grams will you have in 5,730 years?
100 g
75 g
50 g
25 g
0 g

if you start with 100 grams of hydrogen - 3, how many grams will you have after 24.6 years?
100 g
75 g
50 g
25 g
0 g

Explanation:

Step1: Understand 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.

Step2: For the first question

Given the half - life \(t_{1/2}=1\) hour. After \(t = 3\) hours, \(n=\frac{t}{t_{1/2}}=\frac{3}{1}=3\). Let \(N_0\) be the initial amount of the isotope. Then \(N = N_0\times(\frac{1}{2})^3=\frac{N_0}{8}\)

Step3: For the second question

The half - life of carbon - 14, \(t_{1/2}=5730\) years. After \(t = 5730\) years, \(n=\frac{t}{t_{1/2}}=\frac{5730}{5730}=1\). If \(N_0 = 100\)g, then \(N=100\times(\frac{1}{2})^1 = 50\)g

Step4: For the third question

The half - life of hydrogen - 3 (\(t_{1/2}=12.3\) years). After \(t = 24.6\) years, \(n=\frac{t}{t_{1/2}}=\frac{24.6}{12.3}=2\). If \(N_0 = 100\)g, then \(N=100\times(\frac{1}{2})^2=25\)g

Answer:

First question: One eighth of it
Second question: \(50\)g
Third question: \(25\)g