Bohr Model of Hydrogen — Revision Notes

⚡ 30-Second Revision

Bohr's Postulates:

1. Stationary orbits (no radiation). 2. Quantized angular momentum: $L = mvr = n\frac{h}{2\pi}$. 3. Energy transitions: $h\nu = E_i - E_f$.

Radius: — $r_n = \frac{n^2h^2\epsilon_0}{\pi m Ze^2} = \frac{n^2}{Z} a_0$, where $a_0 \approx 0.529\,\text{Å}$. ($r_n \propto n^2/Z$)
Velocity: — $v_n = \frac{Ze^2}{2\epsilon_0 nh}$. ($v_n \propto Z/n$)
Energy: — $E_n = -\frac{m Z^2 e^4}{8\epsilon_0^2 n^2 h^2} = -13.6\frac{Z^2}{n^2}\,\text{eV}$. ($E_n \propto -Z^2/n^2$)
Rydberg Formula: — $\frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$, where $R_H \approx 1.097 \times 10^7\,\text{m}^{-1}$.
**Spectral Series (for H, $Z=1$):**

* Lyman: $n_f=1$ (UV) * Balmer: $n_f=2$ (Visible) * Paschen: $n_f=3$ (IR) * Brackett: $n_f=4$ (IR) * Pfund: $n_f=5$ (IR)

Energy Relationships: — $\text{KE} = -E$, $\text{PE} = 2E$, $\text{PE} = -2\text{KE}$.
Ionization Energy: — Energy to remove electron from ground state ($n=1$ to $n=\infty$). For H, $13.6\,\text{eV}$.

2-Minute Revision

The Bohr model provides a quantum explanation for the hydrogen atom. Its three core postulates are: electrons orbit in stable, non-radiating 'stationary orbits'; their angular momentum is quantized ($mvr = n h/2pi$); and energy is emitted or absorbed only during transitions between these orbits ($h u = E_i - E_f$).

This model successfully explains atomic stability and the discrete line spectra of hydrogen. Key derivations show that the radius of the $n$-th orbit $r_n$ is proportional to $n^2/Z$, the electron's velocity $v_n$ is proportional to $Z/n$, and its total energy $E_n$ is proportional to $-Z^2/n^2$.

For hydrogen, $E_n = -13.6/n^2,\text{eV}$. The Rydberg formula, rac{1}{lambda} = R_H Z^2 left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), predicts the wavelengths of spectral lines, grouped into series like Lyman ($n_f=1$, UV) and Balmer ($n_f=2$, visible).

Remember the relationships between kinetic, potential, and total energy: KE = -E, PE = 2E. While groundbreaking, the model's limitations include its inability to explain multi-electron atoms or the fine structure of spectral lines.

5-Minute Revision

The Bohr model, a quantum leap from Rutherford's planetary model, explains the hydrogen atom's stability and discrete spectrum. Its foundation rests on three postulates: 1) Electrons reside in specific 'stationary orbits' without radiating energy, each with a quantized energy.

2) The angular momentum of an electron in these orbits is quantized, $mvr = n h/2pi$, where $n$ is the principal quantum number. 3) Energy is emitted or absorbed as photons ($h u = E_i - E_f$) only when electrons transition between these allowed orbits.

From these postulates, we derive crucial formulas for hydrogen-like atoms (atomic number $Z$):

Radius of $n$-th orbit: — $r_n = \frac{n^2h^2epsilon_0}{pi m Ze^2} = \frac{n^2}{Z} a_0$, where $a_0 approx 0.529,\text{Å}$ (Bohr radius). Thus, $r_n propto n^2/Z$.

* *Example:* The radius of the second orbit ($n=2$) of hydrogen ($Z=1$) is $r_2 = 4a_0 approx 2.116,\text{Å}$.

Velocity of electron in $n$-th orbit: — $v_n = \frac{Ze^2}{2epsilon_0 nh}$. Thus, $v_n propto Z/n$.

* *Example:* The velocity in the first orbit ($n=1$) of hydrogen is $v_1 approx c/137$.

Total Energy of electron in $n$-th orbit: — $E_n = -\frac{m Z^2 e^4}{8epsilon_0^2 n^2 h^2} = -13.6 \frac{Z^2}{n^2},\text{eV}$. Thus, $E_n propto -Z^2/n^2$.

* *Example:* The energy of the ground state ($n=1$) of hydrogen is $E_1 = -13.6,\text{eV}$. For $n=2$, $E_2 = -13.6/4 = -3.4,\text{eV}$.

Spectral Series: Transitions to a common final state $n_f$ define a series. The Rydberg formula rac{1}{lambda} = R_H Z^2 left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) calculates the wavelength. For hydrogen ($Z=1$):

Lyman Series ($n_f=1$): — Transitions from $n_i=2,3,dots$. (Ultraviolet region)
Balmer Series ($n_f=2$): — Transitions from $n_i=3,4,dots$. (Visible region)
Paschen Series ($n_f=3$): — Transitions from $n_i=4,5,dots$. (Infrared region)

Energy Relationships: For an electron in a Bohr orbit, Kinetic Energy (KE), Potential Energy (PE), and Total Energy (E) are related as: $ext{KE} = -E$, $ext{PE} = 2E$, and $ext{PE} = -2\text{KE}$.

Ionization Energy: The energy required to remove an electron from the ground state ($n=1$) to infinity ($n=infty$). For hydrogen, it's $0 - (-13.6,\text{eV}) = 13.6,\text{eV}$.

Limitations: The model fails for multi-electron atoms, cannot explain the fine structure of spectral lines, or the Zeeman/Stark effects. It's a semi-classical model, a stepping stone to full quantum mechanics.

Prelims Revision Notes

Bohr Model of Hydrogen: NEET Revision Notes

1. Bohr's Postulates (Key Principles):

* Stationary Orbits: Electrons revolve in specific, stable, non-radiating orbits (stationary states) without emitting energy. Each orbit has a definite, quantized energy. * Quantization of Angular Momentum: Angular momentum ($L$) of an electron in a stationary orbit is quantized: $L = mvr = n\frac{h}{2\pi}$, where $n=1, 2, 3, \dots$ (principal quantum number).

* Energy Transitions: Electrons emit or absorb a photon when transitioning between orbits. Photon energy $h\nu = E_i - E_f$.

2. Derived Quantities for Hydrogen-like Atoms (Atomic Number $Z$):

* **Radius of $n$-th orbit ($r_n$):** * Formula: $r_n = \frac{n^2h^2\epsilon_0}{\pi m Ze^2}$ * Proportionality: $r_n \propto \frac{n^2}{Z}$ * For Hydrogen ($Z=1$), $r_n = n^2 a_0$, where $a_0 = 0.529\,\text{Å}$ (Bohr radius).

* **Velocity of electron in $n$-th orbit ($v_n$):** * Formula: $v_n = \frac{Ze^2}{2\epsilon_0 nh}$ * Proportionality: $v_n \propto \frac{Z}{n}$ * $v_1$ for H is approx. $c/137$ (fine-structure constant).

* **Total Energy of electron in $n$-th orbit ($E_n$):** * Formula: $E_n = -\frac{m Z^2 e^4}{8\epsilon_0^2 n^2 h^2}$ * Proportionality: $E_n \propto -\frac{Z^2}{n^2}$ * For Hydrogen ($Z=1$): $E_n = -\frac{13.

6}{n^2}\,\text{eV}$. * Ground state ($n=1$):$E_1 = -13.6\,\text{eV}$. * First excited state ($n=2$):$E_2 = -3.4\,\text{eV}$. * Second excited state ($n=3$):$E_3 = -1.51\,\text{eV}$.

3. Spectral Series (Rydberg Formula):

* $\frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$, where $R_H \approx 1.097 \times 10^7\,\text{m}^{-1}$ (Rydberg constant). * **For Hydrogen ($Z=1$):** * Lyman Series: $n_f=1$, $n_i=2,3,4,\dots$ (Ultraviolet region) * Balmer Series: $n_f=2$, $n_i=3,4,5,\dots$ (Visible region) * Paschen Series: $n_f=3$, $n_i=4,5,6,\dots$ (Infrared region) * Brackett Series: $n_f=4$, $n_i=5,6,7,\dots$ (Infrared region) * Pfund Series: $n_f=5$, $n_i=6,7,8,\dots$ (Infrared region) * Longest wavelength: Smallest energy difference (smallest $n_i$ for a given $n_f$).

* Shortest wavelength (series limit): Largest energy difference ($n_i=\infty$ for a given $n_f$).

4. Energy Relationships (Virial Theorem for Coulombic Systems):

* Kinetic Energy (KE): $\text{KE} = -E$ * Potential Energy (PE): $\text{PE} = 2E$ * $\text{PE} = -2\text{KE}$

5. Ionization and Excitation Energy:

* Ionization Energy: Energy required to remove an electron from its ground state ($n=1$) to $n=\infty$. For H, $13.6\,\text{eV}$. * Excitation Energy: Energy required to move an electron from a lower energy state to a higher energy state (e.g., $n=1$ to $n=2$).

6. Limitations of Bohr Model:

* Only applicable to hydrogen and hydrogen-like ions (single electron systems). * Fails to explain spectra of multi-electron atoms. * Cannot explain the fine structure of spectral lines. * Cannot explain the Zeeman effect (splitting of lines in magnetic field) or Stark effect (in electric field). * Violates Heisenberg's Uncertainty Principle (assumes definite orbits).

Vyyuha Quick Recall

Bohr's Postulates Really Value Energy Spectra:

Bohr's Postulates: (1) Stationary orbits, (2) Quantized Angular Momentum ($L=n h/2pi$), (3) Energy Transitions ($h

u = E_i - E_f$).

Radius: $r_n \propto n^2/Z$ (R for Radius, R for $n^2$).
Velocity: $v_n \propto Z/n$ (V for Velocity, V for $1/n$).
Energy: $E_n \propto -Z^2/n^2$ (E for Energy, E for $-1/n^2$).
Spectra: Lyman ($n_f=1$), Balmer ($n_f=2$), Paschen ($n_f=3$) - Lovely Boys Play. (UV, Visible, IR)