Question

{"given":"A hypothetical atom with a proton and a particle of mass $m' = 2m_e$ (double the electron mass) and charge $-e$. The Rydberg constant for hydrogen is $R$. Find the longest wavelength photon emitted during transitions to the first excited state ($n = 2$).","key_observation":"In Bohr's model, the energy of the $n$th orbit scales linearly with the particle mass: $E_n \\propto m$. Doubling the mass doubles all energies, so the effective Rydberg constant becomes $R' = 2R$. The longest wavelength (minimum energy) transition to $n = 2$ is from $n = 3$.","option_analysis":[{"label":"(A)","text":"$\\dfrac{9}{5R}$","verdict":"incorrect","explanation":"This does not correspond to any valid transition in this hypothetical atom. The $n=3 \\to n=2$ transition gives $\\lambda = \\dfrac{18}{5R}$, not $\\dfrac{9}{5R}$."},{"label":"(B)","text":"$\\dfrac{36}{5R}$","verdict":"incorrect","explanation":"This would correspond to $R' = R$ (normal hydrogen), but here the effective Rydberg constant is $R' = 2R$, so this value is off by a factor of 2."},{"label":"(C)","text":"$\\dfrac{18}{5R}$","verdict":"correct","explanation":"With $R' = 2R$, the transition $n=3 \\to n=2$ gives $\\dfrac{1}{\\lambda} = 2R\\left(\\dfrac{1}{4} - \\dfrac{1}{9}\\right) = 2R \\cdot \\dfrac{5}{36} = \\dfrac{5R}{18}$, so $\\lambda = \\dfrac{18}{5R}$."},{"label":"(D)","text":"$\\dfrac{4}{R}$","verdict":"incorrect","explanation":"This value does not match the $n=3 \\to n=2$ transition with $R' = 2R$; it would require an unrealistic transition energy."}],"answer":"(C)","formula_steps":[]}