Bohr's Model — Revision Notes

⚡ 30-Second Revision

Key formulas and concepts for Bohr's Model:

Postulates: — Stationary orbits, quantized angular momentum ($L = n\frac{h}{2pi}$), energy transitions ($Delta E = h

u$).

Radius: — $r_n = 0.529 \frac{n^2}{Z} \text{ Å}$ ($r_n propto n^2/Z$)
Velocity: — $v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}$ ($v_n propto Z/n$)
Energy: — $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$ ($E_n propto Z^2/n^2$)
Rydberg Formula: — rac{1}{lambda} = R_H Z^2 left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)
Spectral Series: — Lyman ($n_1=1$, UV), Balmer ($n_1=2$, Visible), Paschen ($n_1=3$, IR).
Limitations: — Fails for multi-electron atoms, Zeeman/Stark effect, fine structure.

2-Minute Revision

Bohr's model was a quantum leap from Rutherford's, addressing atomic stability and line spectra for hydrogen-like atoms. Its core idea is that electrons orbit in specific, non-radiating 'stationary states' where their angular momentum is quantized ($mvr = n\frac{h}{2pi}$).

This quantization leads to discrete values for orbital radii ($r_n propto n^2/Z$), electron velocities ($v_n propto Z/n$), and most importantly, energy levels ($E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$).

When an electron jumps between these energy levels, it absorbs or emits a photon whose energy equals the energy difference ($Delta E = h u$). This explains the characteristic line spectra, with the Rydberg formula predicting wavelengths for series like Lyman (UV, $n_1=1$), Balmer (Visible, $n_1=2$), and Paschen (IR, $n_1=3$).

Remember, the model is limited to single-electron species and cannot explain phenomena like the Zeeman effect or the spectra of complex atoms. Focus on applying the formulas and understanding their proportionality for NEET questions.

5-Minute Revision

Bohr's model, a semi-classical quantum model, provided the first successful explanation for the hydrogen atom's stability and its unique line spectrum. It overcame Rutherford's model's shortcomings by introducing three key postulates: 1) Electrons exist in stable, non-radiating 'stationary orbits' with definite energy.

2) The angular momentum of an electron in these orbits is quantized, $mvr = n\frac{h}{2pi}$, where $n$ is the principal quantum number. 3) Energy is absorbed or emitted only during transitions between these discrete energy levels, with the photon energy $Delta E = h u$.

Radius of $n^{th}$ orbit: — $r_n = 0.529 \frac{n^2}{Z} \text{ Å}$. Note $r_n propto n^2/Z$.
Velocity of electron in $n^{th}$ orbit: — $v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}$. Note $v_n propto Z/n$.
Energy of electron in $n^{th}$ orbit: — $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$. Note $E_n propto Z^2/n^2$. The negative sign indicates a bound state.

These formulas are critical for NEET. For example, to find the radius of the second orbit of $He^+$ ($Z=2, n=2$): $r_2 = 0.529 \frac{2^2}{2} = 0.529 \times 2 = 1.058 \text{ Å}$.

The model also successfully explained the hydrogen spectrum using the Rydberg formula: rac{1}{lambda} = R_H Z^2 left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), where $R_H$ is the Rydberg constant. Different spectral series correspond to different final states ($n_1$): Lyman ($n_1=1$, UV), Balmer ($n_1=2$, Visible), Paschen ($n_1=3$, IR). For instance, the shortest wavelength in the Balmer series ($n_1=2$) occurs when $n_2 \to infty$.

Limitations are equally important: Bohr's model only works for single-electron species, fails to explain the splitting of spectral lines in magnetic (Zeeman effect) or electric (Stark effect) fields, and doesn't account for the fine structure of spectral lines or the wave nature of electrons. Despite these, it was a crucial stepping stone to modern quantum mechanics.

Prelims Revision Notes

Bohr's Model (1913) for Hydrogen and Hydrogen-like Species ($He^+, Li^{2+}$)

I. Postulates:

1
Stationary Orbits: — Electrons revolve in specific, non-radiating circular orbits (stationary states) with fixed energy.
2
Quantized Angular Momentum: — Angular momentum ($L$) is an integral multiple of $rac{h}{2pi}$: $L = mvr = n\frac{h}{2pi}$, where $n=1, 2, 3, ...$ (principal quantum number).
3
Energy Transitions: — Energy is absorbed/emitted only when an electron jumps between orbits. $Delta E = E_{final} - E_{initial} = h

u$.

II. Key Formulas & Proportionalities (for $n^{th}$ orbit, atomic number $Z$):

Radius ($r_n$): — $r_n = 0.529 \frac{n^2}{Z} \text{ Å}$.

* $r_n propto n^2/Z$. * Bohr radius ($a_0$) for H, $n=1$: $0.529 \text{ Å}$.

Velocity ($v_n$): — $v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}$.

* $v_n propto Z/n$.

Energy ($E_n$): — $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$.

* $E_n propto Z^2/n^2$. * Negative sign indicates electron is bound. * Ground state ($n=1$) is most negative (lowest energy). * $E_n \to 0$ as $n \to infty$ (ionization).

Kinetic Energy (KE): — $KE_n = -E_n = 13.6 \frac{Z^2}{n^2} \text{ eV}$.
Potential Energy (PE): — $PE_n = 2E_n = -27.2 \frac{Z^2}{n^2} \text{ eV}$.

* $PE = -2 \times KE$.

III. Atomic Spectra (Rydberg Formula):

rac{1}{lambda} = R_H Z^2 left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), where $n_2 > n_1$.
$R_H = 1.097 \times 10^7 \text{ m}^{-1}$ (Rydberg constant).
**Spectral Series for Hydrogen ($Z=1$):**

* Lyman Series: $n_1=1$, $n_2=2,3,4,...$ (Ultraviolet region). * Balmer Series: $n_1=2$, $n_2=3,4,5,...$ (Visible region). * Paschen Series: $n_1=3$, $n_2=4,5,6,...$ (Infrared region). * Brackett Series: $n_1=4$, $n_2=5,6,7,...$ (Infrared region). * Pfund Series: $n_1=5$, $n_2=6,7,8,...$ (Infrared region).

IV. Limitations of Bohr's Model:

Applicable only to single-electron species (H, $He^+$, $Li^{2+}$).
Fails to explain spectra of multi-electron atoms.
Could not explain the splitting of spectral lines in magnetic (Zeeman effect) or electric (Stark effect) fields.
Could not explain the fine structure of spectral lines.
Did not account for the wave nature of electrons (de Broglie) or Heisenberg's Uncertainty Principle.
Could not explain the relative intensities of spectral lines.

V. Ionization Energy: Energy required to remove an electron from its ground state ($n=1$) to $n=infty$. Numerically equal to $-E_1$.

Vyyuha Quick Recall

To remember the spectral series and their regions: Lovely Boys Play Baseball Professionally. (Lyman - UV, Balmer - Visible, Paschen - IR, Brackett - IR, Pfund - IR). Also, for the final orbit $n_1$: Look Before Passing Ball Properly (1, 2, 3, 4, 5).