The α particles emitted by radium have energies of 4.795 and 4.611 MeV. What is the wavelength of th — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

Step 1 - Find the Energy of the Gamma Ray The energy of the $\gamma$ ray photon equals the difference in kinetic energies of the two alpha particles emitted (corresponding to two nuclear energy levels): $$E_\gamma = 4.795\ \text{MeV} - 4.611\ \text{MeV} = 0.184\ \text{MeV}$$ Step 2 - Convert Energy to Joules $$E_\gamma = 0.184 \times 10^6 \times 1.602 \times 10^{-19}\ \text{J} = 2.948 \times 10^{-14}\ \text{J}$$ Step 3 - Calculate Wavelength Using $E = hc/\lambda$ $$\lambda = \frac{hc}{E_\gamma} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{2.948 \times 10^{-14}}$$ $$= \frac{1.988 \times 10^{-25}}{2.948 \times 10^{-14}} = 6.74 \times 10^{-12}\ \text{m} = \boxed{6.74\ \text{pm}}$$ Step 4 - Evaluate Options - **(A) 6.74 pm**: Matches our calculation. **Correct.** - **(B) 9.87 pm**: Corresponds to $E = 0.184/1.5 \approx 0.123$ MeV. Incorrect. - **(C) 3.37 pm**: Corresponds to $E \approx 0.368$ MeV. Incorrect. - **(D) 4.58 pm**: Incorrect. $$\boxed{\text{Answer: A — }\lambda = 6.74\ \text{pm}}$$