Question

a sample contains 16.75 g of the radioisotope u-238 and 50.25 g of its daughter isotope, th-232. how long did it take for decay to take place if one half-life of u-238 is 23 million years?
○ 46 million years
○ 69 million years
○ 92 million years
○ 115 million years

Explanation:

Step1: Find total initial mass of U-236

The total initial mass of U - 236 is the sum of the remaining U - 236 and the mass of Th - 232 (since Th - 232 is the daughter isotope from U - 236 decay). So, initial mass \( m_0=16.75\ g + 50.25\ g=67\ g\).

Step2: Determine the fraction of remaining U - 236

The remaining mass of U - 236 is \( m = 16.75\ g\). The fraction of remaining U - 236 is \( \frac{m}{m_0}=\frac{16.75}{67}=0.25\).

Step3: Relate fraction to number of half - lives

We know that for radioactive decay, the fraction of remaining substance \( N/N_0=(1/2)^n\), where \( n\) is the number of half - lives. We have \( N/N_0 = 0.25=\frac{1}{4}=(\frac{1}{2})^2\). So, \( n = 2\) half - lives? Wait, no, wait. Wait, \( 0.25=\frac{1}{4}=(\frac{1}{2})^2\)? Wait, no, \( (\frac{1}{2})^2=\frac{1}{4}\), but wait, let's recalculate the fraction. Wait, initial mass is \( 67\ g\), remaining is \( 16.75\ g\). \( 16.75\div67 = 0.25\)? Wait, \( 67\times0.25 = 16.75\), yes. But wait, \( (\frac{1}{2})^n=0.25\), so \( n = 2\)? Wait, no, wait, \( (\frac{1}{2})^2=\frac{1}{4}=0.25\), but wait, maybe I made a mistake. Wait, let's think again. Wait, the daughter isotope is Th - 232, which comes from U - 236 decay. So the initial amount of U - 236 is the current U - 236 plus the U - 236 that decayed into Th - 232. The mass of Th - 232 is \( 50.25\ g\), and the mass of U - 236 is \( 16.75\ g\). The molar mass of U - 236 and Th - 232: since we are dealing with mass in grams and the decay is a 1:1 (assuming alpha decay, U - 236 decays to Th - 232 by emitting an alpha particle, so the number of moles of U - 236 decayed is equal to the number of moles of Th - 232 formed). But since we are dealing with mass, and the mass of U - 236 is 236 g/mol and Th - 232 is 232 g/mol, but maybe for the purpose of half - life calculation, we can consider the mass ratio. Wait, maybe a simpler way: the total initial mass of U - 236 is \( 16.75 + 50.25=67\ g\). The remaining mass is \( 16.75\ g\). So the fraction remaining is \( \frac{16.75}{67}=\frac{1}{4}\). We know that after \( n\) half - lives, the fraction remaining is \( (\frac{1}{2})^n\). So \( (\frac{1}{2})^n=\frac{1}{4}\), which means \( n = 2\)? Wait, no, \( (\frac{1}{2})^2=\frac{1}{4}\), so \( n = 2\) half - lives? But wait, the half - life is 23 million years. Wait, but \( \frac{1}{4}\) remaining means two half - lives? Wait, no, first half - life: 67 g becomes 33.5 g, second half - life: 33.5 g becomes 16.75 g. Oh! Right! So first half - life: 67 -> 33.5 (after 23 million years), second half - life: 33.5 -> 16.75 (after another 23 million years). So total time is \( 2\times23 = 46\) million years? Wait, but that contradicts my earlier thought. Wait, let's do it step by step.

Initial mass \( m_0 = 67\ g\)

After 1 half - life (\( t = 23\) million years), \( m_1=\frac{m_0}{2}=\frac{67}{2}=33.5\ g\)

After 2 half - lives (\( t = 2\times23 = 46\) million years), \( m_2=\frac{m_1}{2}=\frac{33.5}{2}=16.75\ g\), which matches the remaining mass of U - 236. So the number of half - lives is 2, and the total time is \( 2\times23 = 46\) million years? Wait, but the options include 46, 69, 92, 115. Wait, maybe I made a mistake in the initial mass. Wait, the daughter isotope is Th - 232, which has a molar mass of 232 g/mol, and U - 236 has 236 g/mol. So the number of moles of Th - 232 is \( n_{Th}=\frac{50.25\ g}{232\ g/mol}\approx0.2166\ mol\)

The number of moles of remaining U - 236 is \( n_{U}=\frac{16.75\ g}{236\ g/mol}\approx0.071\ mol\)

The initial number of moles of U - 236 is \( n_{0}=n_{U}+n_{Th}\approx0.071 + 0.2166 = 0.2876\ mol\)

The fraction remaining is \( \frac{n_{U}}{n_{0}}\approx\frac{0.071}{0.2876}\approx0.247\approx0.25=\frac{1}{4}\), which is the same as before. So the number of half - lives is 2, so time is \( 2\times23 = 46\) million years. Wait, but let's check again. Wait, if the initial mass is 67 g, after first half - life (23 Myr) it's 33.5 g, after second (46 Myr) it's 16.75 g. So the answer should be 46 million years.

Answer:

46 million years