For light and heavy hydrogen (H and D) atoms, let $E_H$ and $E_D$ be the binding energies of their e

Question

For light and heavy hydrogen (H and D) atoms, let $E_H$ and $E_D$ be the binding energies of their electrons in the ground state, and $\lambda_H$ and $\lambda_D$ be the wavelengths of the first lines of the Lyman series. Given the ratio of proton mass to electron mass $m_p/m_e \approx 1836$ and deuteron mass $m_d \approx 2m_p$, which of the following statement(s) is/are correct? (Take $E_H \approx 13.6 	ext{ eV}$ and $\lambda_H \approx 1216 	ext{ \AA}$)

Professor PC Thomas Classes · Answer

$\mu = m_e M / (m_e+M)$. $\mu_D > \mu_H \implies E_D > E_H$ and $\lambda_D < \lambda_H$. $\Delta E \approx 13.6 (\frac{\mu_D-\mu_H}{m_e}) \approx 13.6 (\frac{1}{1836} - \frac{1}{3672}) \approx 3.7 	imes 10^{-3} 	ext{ eV}$. $\Delta \lambda \approx \lambda_H (\frac{\Delta E}{E}) \approx 1216 	imes \frac{1}{3672} \approx 0.33 	ext{ \AA}$. $R \propto \mu$, so $R_D/R_H = \mu_D/\mu_H \approx 1.00027$.