Question

{"given":"An electron in hydrogen transitions from an excited state (quantum number $n > 1$) to the ground state ($n = 1$). In Bohr's model: $$KE_n = \\frac{13.6}{n^2} \\text{ eV}, \\quad PE_n = -\\frac{27.2}{n^2} \\text{ eV}, \\quad E_n = -\\frac{13.6}{n^2} \\text{ eV}$$","key_observation":"As $n$ decreases (excited → ground state), $KE_n \\propto \\frac{1}{n^2}$ increases (positive), while $PE_n \\propto -\\frac{1}{n^2}$ becomes more negative (decreases), and total energy $E_n \\propto -\\frac{1}{n^2}$ also becomes more negative (decreases).","option_analysis":[{"label":"(A)","text":"Kinetic energy increases; potential and total energies decrease.","verdict":"correct","explanation":"When $n$ decreases from an excited state to the ground state, $KE = \\frac{13.6}{n^2}$ eV increases. Both $PE = -\\frac{27.2}{n^2}$ eV and $E = -\\frac{13.6}{n^2}$ eV become more negative, meaning they decrease."},{"label":"(B)","text":"Kinetic energy decreases; potential energy increases; total energy remains the same.","verdict":"incorrect","explanation":"Kinetic energy actually increases as $n$ decreases, and total energy also changes (becomes more negative); it does not remain constant."},{"label":"(C)","text":"Kinetic and total energies decrease; potential energy increases.","verdict":"incorrect","explanation":"Kinetic energy increases (not decreases) when transitioning to a lower energy level; potential energy becomes more negative, so it decreases, not increases."},{"label":"(D)","text":"Kinetic, potential, and total energies all decrease.","verdict":"incorrect","explanation":"Kinetic energy increases as $n$ decreases; only potential energy and total energy decrease (become more negative)."}],"answer":"(A)","formula_steps":[]}