The energy of the electron in any given state for the hydrogen atom is \( -13.6/n^{2}\,\mathrm{eV} \). Identify the correct statements among the following.
(A) the negative sign in \( -13.6\,\mathrm{eV} \) signifies that the electron is bound to the nucleus.
(B) the negative sign in \( -13.6\,\mathrm{eV} \) signifies that the electron requires \( 13.6\,\mathrm{eV} \) energy to be separated from the nucleus.
(C) \( 3.4\,\mathrm{eV} \) energy is required to take the electron from the ground to the first excited state.
(D) to find the energy of third excited state, the value of \( n \) is 4.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
In the Bohr model of hydrogen the energy of the electron in the \( n \)th orbit is
\( \displaystyle E_n = -\frac{13.6}{n^{2}}\,\mathrm{eV} \)
\( E_1 = -13.6\,\mathrm{eV} \)
\( E_2 = -3.4\,\mathrm{eV} \)
(A) the negative sign in \( -13.6\,\mathrm{eV} \) signifies that the electron is bound to the nucleus. — Correct. A negative total energy means the electron cannot escape on its own, so it is bound.
(B) the negative sign in \( -13.6\,\mathrm{eV} \) signifies that the electron requires \( 13.6\,\mathrm{eV} \) energy to be separated from the nucleus. — Correct. To raise the ground-state energy from \( -13.6 \) eV to zero, 13.6 eV must be supplied; this is the ionisation energy of hydrogen.
(C) \( 3.4\,\mathrm{eV} \) energy is required to take the electron from the ground to the first excited state. — Incorrect. The first excited state is \( n = 2 \), so the energy needed is the difference of the two levels:
\( \displaystyle E_2 - E_1 = -3.4 - (-13.6) = 10.2\,\mathrm{eV} \)
3.4 eV is the energy needed to remove the electron from the first excited state.
(D) to find the energy of third excited state, the value of \( n \) is 4. — Correct. The ground state is \( n = 1 \), so the first, second and third excited states are \( n = 2, 3, 4 \).
NCERT (2026-27):
Bohr Model of the Hydrogen Atom – “… The negative sign of the total energy of an electron moving in an orbit means that the electron is bound with the nucleus.”
Energy Levels – “… The energy of this state (n = 1), E1 is –13.6 eV. Therefore, the minimum energy required to free the electron from the ground state of the hydrogen atom is 13.6 eV.”
Energy Levels – “… It means that the energy required to excite an electron in hydrogen atom to its first excited state, is an energy equal to E2 – E1 = –3.40 eV – (–13.6) eV = 10.2 eV.”