Atoms - Class 12 Engineering Physics - Extra Questions

$$T=\cfrac { 2\pi r }{ v } =\cfrac { 2\pi \times 5\times { 10 }^{ -11 } }{ \pi \times { 10 }^{ 6 } } =10\times { 10 }^{ -17 }sec$$

So, $$T=10$$

(a)  $$v$$ is the orbital speed of the electron in the hydrogen atom in level $$n$$ .

Thus, $$r_0 \propto \dfrac{1}{E_k}$$

Hence, when the kinetic energy of alpha particle is doubled, the closet distance $$r_0$$ will be halved.

Speed of electron in $$2$$ nd orbit $$v = 1.09\times 10^6  m/s$$

Frequency of revolution of an electron in that orbit   $$\nu = \dfrac{v}{2\pi r}$$

$$\therefore$$ $$\nu = \dfrac{1.09\times 10^{6}}{2(3.142)(2.14\times 10^{-10})}$$

$$\implies$$ $$\nu = 8.1\times 10^{14}$$ Hz

Gyromagnetic ratio $$\gamma = 8.8\times 10^{10}C/kg$$

Magnetic moment   $$\mu = e\nu (\pi r^2)$$

$$\therefore$$ $$\mu = (1.6\times 10^{-19}) \times (9\times 10^{15}) \times \pi \times (0.53\times 10^{-10})^2$$

$$\implies$$ $$\mu = 12.7\times 10^{-24}$$ $$A m^2$$

Orbital angular momentum $$L = \dfrac{\mu}{\gamma}$$

$$\therefore$$   $$L = \dfrac{12.7\times 10^{-24}}{8.8 \times 10^{10}} = 1.44 \times 10^{-34}$$ $$kg m^3 /s$$

Concept of Photon:

In $$1990$$ , Max Planck proposed his Quantum Theory of radiation. According to this theory, the energy of an electromagnetic wave is not continuously distributed over the wavefront (like the energy of water waves), instead of an electromagnetic wave travels in the form of discrete packets or bundles of energy called Quanta. One quanta of light radiation is called a photon, which travels with the speed of light. 

We can summarize the important properties of photon or the photon picture of the electromagnetic radiation as follows :

1. In the interaction of radiation with matter, radiation behaves as it is made up of particles, called photons.

2. All the photons are emitted by any source which are travelled with speed of light in a free space $$[c = 3 \times 10^8 \,m/sec]$$  

3.Each photon has a definite energy depending upon the frequency y of the incident radiation and not on its intensity. 

Energy of each photon, $$E = hv = \dfrac{hc}{\lambda}$$  

Where $$h$$ is Planck’s constant and its value is $$6.63 \times10^{-34}\, Js$$ or $$4.1 × 10^{-15}\, eVs$$   

4. A photon may collide with a material particle. The total energy and the total momentum remain conserved in such a collision. However, the number of photons may not be conserved in a collision. The photon may be absorbed or a new photon may be created.

5. Photons are electrically neutral and are not deflected by electric and magnetic fields.

6. If the intensity of light of a given wavelength is increased, there is an increase in the number of photons emitted by the incident radiations on a given area in a given time. But the energy of each photon remains the same.

7. The rest mass of a photon is consider to be zero.

The 'series limit' refers to the 'shortest wavelength'.

$$f= k \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$$

Here, $$k$$ is a constant.

For the series limit of Lyman series,

$$n_1=1$$

$$n_2 = \infty$$

For the series limit of Balmer series,

$$n_1=2$$

$$n_2 = \infty$$

Thus, the difference in the frequencies of the series of Lyman series and Balmer series is equal to the frequency of the first line of the Lyman series.

Series Limit of Balmer series

$$n=1$$

$$\dfrac{1}{\lambda}= R \left( \dfrac{1}{n^2} \right) = R$$

Given: $$\dfrac{1}{R}= 912 \overset{\circ}{A}$$

Series Limit of Balmer series

$$n=2$$

$$\dfrac{1}{\lambda}= R \left( \dfrac{1}{n^2} \right) = \dfrac{R}{4}$$

$$\dfrac{1}{\lambda}= R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$$

The electrostatic force between electron and proton at a distance r is given by, $$F=\dfrac{dU}{dr}=\dfrac{ke^2}{r^4}$$

According to Bohr's first postulate, $$\dfrac{mv^2}{r}=F=\dfrac{ke^2}{r^4}   ...(1)$$