Energy Levels Of Hydrogen

Question

Consider a hydrogen atom whose electron has been raised to the first excited state corresponding to principal quantum number two. What is the total energy of the electron in this state?

RankGuru · Accepted Answer

The bound energy levels of hydrogen are given by $E_n = -\dfrac{13.6}{n^2}$ eV, where the negative sign signifies that the electron is trapped in the potential well of the proton. The first excited state corresponds to $n = 2$, so $E_2 = -\dfrac{13.6}{2^2} = -\dfrac{13.6}{4} = -3.4$ eV. The value -13.6 eV is the ground-state energy for $n = 1$ and therefore does not apply to the first excited state. The value -1.51 eV corresponds to $n = 3$, the second excited state, not the requested level. The value -6.8 eV incorrectly divides by 2 instead of $2^2 = 4$, ignoring the inverse-square dependence on $n$. This is the standard NCERT energy-level formula derived from the Bohr model. A plausibility check confirms the answer: excited states must lie higher (less negative) than the ground state, and indeed -3.4 eV is greater than -13.6 eV, which is exactly what we expect for an electron promoted away from the nucleus.

Energy Levels Of Hydrogen

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