Identify the orbit number from energy:
Energy formula: \(E_n = \dfrac{-13.6}{n^2} \text{ eV}\)
Given E = −3.4 eV:
$$n^2 = \frac{13.6}{3.4} = 4 \implies n = 2$$
So the electron is in the 2nd orbit (first excited state ✓)
Apply Bohr's radius formula:
$$r_n = n^2 \times a_0 = n^2 \times 0.529 \text{ Å}$$
For n = 2:
$$r_2 = 4 \times 0.529 \text{ Å} = 2.116 \text{ Å}$$
Convert to metres:
$$r_2 = 2.116 \text{ Å} = 2.116 \times 10^{-10} \text{ m} \approx \mathbf{2.1 \times 10^{-10} \text{ m}}$$
Answer: 2.1 × 10⁻¹⁰ m ✓
Alternative using PE method:
PE = \(-\dfrac{e^2}{4\pi\epsilon_0 r} = 2E_n\)
So \(r = \dfrac{-e^2}{4\pi\epsilon_0 \times 2E_n}\)
\(r = \dfrac{9\times10^9 \times (1.6\times10^{-19})^2}{2 \times 3.4 \times 1.6\times10^{-19}}\)
\(= \dfrac{9\times10^9 \times 2.56\times10^{-38}}{1.088\times10^{-18}} \approx 2.12 \times 10^{-10}\) m ✓
Niels Bohr proposed his atomic model in 1913 to explain the stability of atoms and the discrete spectral lines of hydrogen. His model was built on three revolutionary postulates that modified classical physics.
Postulate 1 — Stationary Orbits: Electrons revolve around the nucleus in certain special circular orbits called stationary orbits or allowed orbits. While in these orbits, the electron does not radiate energy despite being accelerated (this contradicts classical electromagnetism).
Postulate 2 — Quantisation of Angular Momentum: Only those orbits are allowed for which the angular momentum of the electron is an integral multiple of h/2π: L = mvr = nh/2π, where n = 1, 2, 3... is the principal quantum number.
Postulate 3 — Radiation Condition: An electron emits or absorbs a photon only when it jumps from one allowed orbit to another. The energy of the photon equals the difference in energies of the two orbits: hν = E_higher − E_lower.
\(r_n = n^2 a_0 = n^2 \times 0.529 \text{ Å}\)
\(E_n = \dfrac{-13.6}{n^2} \text{ eV}\)
\(v_n = \dfrac{v_1}{n} = \dfrac{2.18 \times 10^6}{n} \text{ m/s}\)
\(L_n = \dfrac{nh}{2\pi}\)
The energy levels of hydrogen form a series approaching zero from below. The ground state (n=1) has energy −13.6 eV, the first excited state (n=2) has −3.4 eV, the second excited state (n=3) has −1.51 eV, and so on. As n → ∞, E → 0 (ionised state).
📌 n=1: E = −13.6 eV, r = 0.529 Å (Ground state)
📌 n=2: E = −3.4 eV, r = 2.116 Å (1st excited state)
📌 n=3: E = −1.51 eV, r = 4.76 Å (2nd excited state)
📌 n=4: E = −0.85 eV, r = 8.46 Å (3rd excited state)
📌 n=∞: E = 0 eV (Ionised — electron free)
This is a very common source of confusion in NEET. The first excited state is n=2 (one level above ground state). The second excited state is n=3. The nth excited state corresponds to principal quantum number (n+1). So always add 1 to the excited state number to get the orbit number.
⚠️ "First excited state" means n = 2, NOT n = 1
⚠️ "Second excited state" means n = 3, NOT n = 2
⚠️ Ground state is n = 1 — it is NOT an excited state
For an electron in the nth orbit of hydrogen:
📌 Total Energy: \(E_n = -\dfrac{13.6}{n^2}\) eV (always negative)
📌 Kinetic Energy: \(KE_n = +\dfrac{13.6}{n^2}\) eV (always positive)
📌 Potential Energy: \(PE_n = -\dfrac{27.2}{n^2}\) eV (always negative)
📌 Relation: KE = −E and PE = 2E, so PE = −2KE
📌 Also: E = KE + PE → −KE + PE = E ✓
When electrons jump between energy levels, they emit photons whose wavelengths form distinct series. The Rydberg formula gives the wavelength:
\(\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)\)
📌 Lyman series: n₁=1, n₂=2,3,4... → Ultraviolet region
📌 Balmer series: n₁=2, n₂=3,4,5... → Visible region
📌 Paschen series: n₁=3, n₂=4,5,6... → Infrared region
📌 Brackett series: n₁=4 → Far infrared
📌 Pfund series: n₁=5 → Far infrared
Despite its success in explaining hydrogen spectra, Bohr's model has several limitations: (1) It cannot explain the spectra of multi-electron atoms. (2) It cannot explain the fine structure (splitting) of spectral lines. (3) It cannot explain the relative intensities of spectral lines. (4) It violates the Heisenberg uncertainty principle by specifying both the position and momentum of the electron precisely. (5) It cannot explain the behaviour of atoms in magnetic fields (Zeeman effect) and electric fields (Stark effect).
The modern quantum mechanical model (Schrödinger equation, orbitals) replaced Bohr's model and correctly explains all these phenomena. However, for NEET, Bohr's model is sufficient for hydrogen-like atoms.