physics-class-12-unit-12-chapter-03-bohr-model-of-atom-ii-l-3-9-tpzsusqpjri

### <Success style="font-size:25px"> Bohr Model Of Atom Ii L-3 </Success>
### <primary style="font-size:24px"> The Rydberg Formula </primary>
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- $ \frac{1}{\lambda}=R y\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) $

- $ { Ry }=1.0973731568508(65) \times 10^7 m^{-1} $

- $n_1 n_2 \text { integers }$

- **Spectral series**

- Lyman: $n_1=1, n_2=2,3, \cdots$
- Balmer: $n_1=2, n_2=3,4 \cdots$
- Paschen: $n_1=2, n_2=3,4, \cdots$

- Brackett, Pfund, Humphreys, Strong and Hansen

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<footer style = "font-size: 18px">Bohr Model of Atom $\rightarrow$ Spectral Problems $\rightarrow$ <span style = "color:blue"> The Rydberg Formula </span> $\rightarrow$ Angular Momentum and Modes of Vibration $\rightarrow$ Derivation of the Bohr Formula </footer>

### <Success style="font-size:25px"> Bohr Model Of Atom Ii L-3 </Success>
### <primary style="font-size:24px"> Angular Momentum and Modes of Vibration </primary>
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- Only discrete orbits are allowed

- Quantization of Angular Momentum

- $m v_n r_n=n \hbar \equiv \frac{n h}{2 \pi}$

- <span style="color:green">Modes of vibration</span>

- **Analogy with vibrating strings revise.**

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<footer style = "font-size: 18px">Spectral Problems $\rightarrow$ The Rydberg Formula $\rightarrow$ <span style = "color:blue"> Angular Momentum and Modes of Vibration </span> $\rightarrow$ Derivation of the Bohr Formula $\rightarrow$ Derivation of the Bohr Formula </footer>

### <Success style="font-size:25px"> Bohr Model Of Atom Ii L-3 </Success>
### <primary style="font-size:24px"> Derivation of the Bohr Formula </primary>
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- $m v_n r_n=n \hbar \quad \hbar =\frac{h}{2 \pi} $

- $\frac{m v_n^2}{r_n}=\frac{k}{r^2} ; \quad k=\frac{z e^2}{4 \pi \epsilon_0} $

- $z=1 $

- $\frac{n^2 \hbar^2}{m r_n}=k $

- $r_n=\frac{n^2 \hbar^2}{m k}$

- $\frac{n^2}{r_n}=$constant

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<footer style = "font-size: 18px">The Rydberg Formula $\rightarrow$ Angular Momentum and Modes of Vibration $\rightarrow$ <span style = "color:blue"> Derivation of the Bohr Formula </span> $\rightarrow$ Derivation of the Bohr Formula $\rightarrow$ Derivation of the Bohr Formula </footer>

### <Success style="font-size:25px"> Bohr Model Of Atom Ii L-3 </Success>
### <primary style="font-size:24px"> Derivation of the Bohr Formula </primary>
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- $r_n=\frac{n^2 \hbar^2}{m k} \quad  $

- $V_n=-\frac{k}{r_n}=\frac{-m k^2}{n^2 \hbar^2} $

- $T_n=\frac{1}{2} m v_n^2$

- $m v_n r_n=n \hbar$

- $v_n^2=\frac{n^2 \hbar^2}{m^2 r_n^2} = \frac{1}{2} m \frac{n^2 \hbar^2}{m^2 r_n^2}$

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<footer style = "font-size: 18px">Angular Momentum and Modes of Vibration $\rightarrow$ Derivation of the Bohr Formula $\rightarrow$ <span style = "color:blue"> Derivation of the Bohr Formula </span> $\rightarrow$ Derivation of the Bohr Formula $\rightarrow$ Scaling Laws </footer>

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### <primary style="font-size:24px"> Franck-Hertz Experiment </primary>
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- Franck hertz experiment

* Additional evidence

- Scattering of electrons by mercury

- Franck hertz experiment

* Take a small quantity of mercury (in the vacuum tube) and evaporate

* Low density Hg vapour in the Vacum tube distributed all over.

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<footer style = "font-size: 18px">How Realistic is the Bohr model? $\rightarrow$ Franck-Hertz Experiment $\rightarrow$ <span style = "color:blue"> Franck-Hertz Experiment </span> $\rightarrow$ Energy Range $\rightarrow$ Particle Interaction </footer>

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### <primary style="font-size:24px"> Franck Hertz Experiment </primary>
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- Additional evidence

- Scattering of electrons by mercury

* Energy of scattered $e^{-}$ in zero

- $7-4.9 = 2.1 eV$

* Cathorde $e^{-}$ energy in $7eV$

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<footer style = "font-size: 18px">Energy Range $\rightarrow$ Particle Interaction $\rightarrow$ <span style = "color:blue"> Franck Hertz Experiment </span> $\rightarrow$ Scattering of Electrons by Mercury $\rightarrow$ Scattering of Electrons by Mercury </footer>

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### <primary style="font-size:24px"> Scattering of Electrons by Mercury </primary>
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- $E_{in} \quad 10.8 \mathrm{eV}$

- Final energy is $10.8 - 9.8 = 1.0 eV$

- $\frac{4 \cdot 9 \times 3}{14.7 \mathrm{eV}}$

- Waited: Excited vapour emits light of energy $4.9 eV$

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<footer style = "font-size: 18px">Franck Hertz Experiment $\rightarrow$ Scattering of Electrons by Mercury $\rightarrow$ <span style = "color:blue"> Scattering of Electrons by Mercury </span> $\rightarrow$ Franck Hertz Experiment $\rightarrow$ Vibrating Molecules </footer>

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### <primary style="font-size:24px"> Franck Hertz Experiment </primary>
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- Additional evidence

- Scattering of electrons by mercury

* Modern day Experiment

- <span style="color:green">Neon $\rightarrow$ already a gas atoms</span>

* Emitted radiation is in the visible range

- Franck hertz experiment

-  Additional evidence

- Scattering of electrons by neon

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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e05c80ec-8543-4e07-72a9-09e37ed95100/public" width="80%" >

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<footer style = "font-size: 18px">Scattering of Electrons by Mercury $\rightarrow$ Scattering of Electrons by Mercury $\rightarrow$ <span style = "color:blue"> Franck Hertz Experiment </span> $\rightarrow$ Vibrating Molecules $\rightarrow$ Vibrating Molecules </footer>

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### <primary style="font-size:24px"> Vibrating Molecules </primary>
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- Example:- Van-der Waals

- $\frac{A}{r^6} - \frac{B}{r^7}$

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<footer style = "font-size: 18px">Scattering of Electrons by Mercury $\rightarrow$ Franck Hertz Experiment $\rightarrow$ <span style = "color:blue"> Vibrating Molecules </span> $\rightarrow$ Vibrating Molecules $\rightarrow$ Taylor Expansion </footer>

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### <primary style="font-size:24px"> Harmonic oscillator potential </primary>
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- $\rho=|r-R| .$

- ${V=V_0+\frac{1}{2} K P^2}$

* Harmonic oscillator potential

- $K \rightarrow$ Spring constant

* Apply the Bohr model hypothesis to SHO

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<footer style = "font-size: 18px">Vibrating Molecules $\rightarrow$ Taylor Expansion $\rightarrow$ <span style = "color:blue"> Harmonic oscillator potential </span> $\rightarrow$ Harmonic oscillator potential $\rightarrow$ Velocity </footer>

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### <primary style="font-size:24px"> Harmonic oscillator potential </primary>
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- $E =\frac{1}{2} m v^2+\frac{1}{2} k x^2 = \frac{1}{2} m v^2+\frac{1}{2} m \omega^2 x^2 $

- $F =-k$

- $E =\frac{1}{2} m \vec{v}^2+\frac{1}{2} k \vec{r}^2 =\frac{1}{2} m \vec{v}^2+\frac{1}{2} m \omega^2 \vec{r}^2 $

- $\vec{F} =-k \vec{r}$ Hooke's law in three dimension

- **Circular orbits + Bohr quantization.**

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<footer style = "font-size: 18px">Taylor Expansion $\rightarrow$ Harmonic oscillator potential $\rightarrow$ <span style = "color:blue"> Harmonic oscillator potential </span> $\rightarrow$ Velocity $\rightarrow$ Velocity </footer>