Nuclear Chemistry — Hard Practice Quiz

A Chemistry cheat sheet for Nuclear Chemistry — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. An isotope has a half-life of $T$. If an initial sample of $N_0$ atoms decays for a time $t = 2T$, what fraction of the original mass remains?

   • $1/2$
   • $1/4$
   • $1/e$
   • $e^{-2}$

   Answer: $1/4$

2. If a radioactive sample decays to $1/e$ of its initial amount in time $t$, what is its half-life in terms of $t$?

   • $t/\ln(2)$
   • $t \ln(2)$
   • $t/0.693$
   • $t/e$

   Answer: $t \ln(2)$

3. Isotope A has a half-life of $T_A$. Isotope B has a half-life of $T_B = 2T_A$. If both start with the same number of atoms, after what time will the ratio of remaining atoms $N_{t,B}/N_{t,A}$ be $e$?

   • $T_A/\ln(2)$
   • $2T_A/\ln(2)$
   • $T_A \ln(2)$
   • $2T_A \ln(2)$

   Answer: $2T_A/\ln(2)$

4. A sample of a radioactive substance initially contains $X$ grams. After $Y$ years, $Z$ grams remain. What is the decay constant $k$ for this substance?

   • $\frac{1}{Y} \ln(\frac{Z}{X})$
   • $Y \ln(\frac{X}{Z})$
   • $\frac{1}{Y} \ln(\frac{X}{Z})$
   • $\frac{0.693}{Y} \ln(\frac{X}{Z})$

   Answer: $\frac{1}{Y} \ln(\frac{X}{Z})$

5. Consider two radioactive isotopes, X and Y, with half-lives $t_{1/2,X}$ and $t_{1/2,Y}$ respectively. If both isotopes undergo a decay process that converts the same mass defect $\Delta m$ per atom into energy, and we start with equal initial masses of X and Y, which isotope will release energy at a faster initial rate?

   • Isotope X, if $t_{1/2,X} > t_{1/2,Y}$
   • Isotope Y, if $t_{1/2,X} > t_{1/2,Y}$
   • Both release energy at the same initial rate.
   • It depends on the specific value of $\Delta m$.

   Answer: Isotope Y, if $t_{1/2,X} > t_{1/2,Y}$

6. A radioactive sample's activity drops to $1/8$ of its initial value in $12$ hours. What is the decay constant $k$ in units of $\text{hr}^{-1}$?

   • $\frac{\ln(2)}{12} \text{ hr}^{-1}$
   • $\frac{\ln(8)}{12} \text{ hr}^{-1}$
   • $\frac{0.693}{8} \text{ hr}^{-1}$
   • $\frac{1}{12} \text{ hr}^{-1}$

   Answer: $\frac{\ln(8)}{12} \text{ hr}^{-1}$

7. An ancient artifact is found to have $1/16$ of the original amount of Carbon-14. If the half-life of Carbon-14 is $5730$ years, how many years old is the artifact?

   • $5730 \text{ years}$
   • $11460 \text{ years}$
   • $22920 \text{ years}$
   • $28650 \text{ years}$

   Answer: $22920 \text{ years}$

8. Isotope A has a decay constant $k_A$. Isotope B has a decay constant $k_B = 2k_A$. If both start with the same initial number of atoms, after how many half-lives of Isotope A will Isotope B have decayed to $1/4$ of its initial amount?

   • $0.5$
   • $1$
   • $2$
   • $4$

   Answer: $1$

9. If a nuclear reaction releases energy $E$, and the speed of light is $c$, what is the equivalent mass defect $\Delta m$ in terms of $E$ and $c$? If $c$ were twice its actual value, how would the mass defect required to release the same energy $E$ change?

   • $\Delta m = E/c^2$; it would be $1/2$ of the original.
   • $\Delta m = E/c^2$; it would be $1/4$ of the original.
   • $\Delta m = Ec^2$; it would be $4$ times the original.
   • $\Delta m = Ec^2$; it would be $2$ times the original.

   Answer: $\Delta m = E/c^2$; it would be $1/4$ of the original.

10. A radioactive isotope has a half-life of $10 \text{ years}$. Each decay event results in a mass defect of $1.5 \times 10^{-28} \text{ kg}$. If an initial sample contains $N_0 = 1.0 \times 10^{24}$ atoms, what is the total energy released from the sample after $20 \text{ years}$? (Use $c = 3.0 \times 10^8 \text{ m/s}$)

    • $2.53 \times 10^{12} \text{ J}$
    • $5.06 \times 10^{12} \text{ J}$
    • $1.01 \times 10^{13} \text{ J}$
    • $1.52 \times 10^{13} \text{ J}$

    Answer: $1.01 \times 10^{13} \text{ J}$