Limitations of Thomson’s atomic model

Electrostatic force between the nucleus and α–particle

Rutherford’s scattering formula

Limitations of Rutherford atomic model

Velocity of electron in an orbit

Time period of revolution of an electron

Kinetic energy of an electron in an orbit

Potential energy of an electron in an orbit

Total energy of an electron in an orbit

Wave number of an electron in an orbit

Line spectra and electron transitions between energy levels

de Broglie’s explanation of Bohr’s second postulate of quantisation

Limitations of Bohr’s atomic model

**CBSE NOTES CLASS 12 PHYSICS **

**CHAPTER 12 ATOMS**

**Dalton’s atomic theory**

All elements are consists of very small invisible particles, called atoms. Atoms of same element are exactly same and atoms of different element are different.

**Thomson’s atomic model**

Every atom is uniformly positive charged sphere of radius of the order of 10^{-10} m, in which entire mass is uniformly distributed and negative charged electrons are embedded randomly.

The atom as a whole is neutral.** **

**Limitations of Thomson’s atomic model**

1. It could not explain the origin of spectral series of hydrogen and other atoms.

2. It could not explain scattering of α–particles in the Rutherford’s experiment.

**Rutherford’s atomic model**

On the basis of α–particle scattering experiment, Rutherford made following observations.

The entire positive charge and almost entire mass of the atom is concentrated at its centre in a very tiny region of the order of 10^{-15} m, called nucleus.

He proposed the following model for the structure of atom.

- The negatively charged electrons revolve around the nucleus in different orbits.
- The total positive charge on the nucleus is equal to the total negative charge on electron. Therefore atom as a whole is neutral.
- The electrons revolve around the nucleus. The centripetal force required by electron for revolution is provided by the electrostatic force of attraction between the electrons and the nucleus.

**Electrostatic force between the nucleus and α–particle**

When the α–particle approaches a nucleus with atomic number Z (+ve charge Ze) the magicnitude of this **electrostatic force**

$$\mathrm{F}=\frac{1}{4\mathrm{\pi}{\mathrm{\varepsilon}}_{\mathrm{o}}}\frac{\left(2\mathrm{e}\right)\left(\mathrm{Z}\mathrm{e}\right)}{{\mathrm{r}}^{2}}$$

**Distance of closest approach**

$${\mathrm{r}}_{\mathrm{o}}\mathrm{}\mathrm{}=\mathrm{}\frac{1}{4\mathrm{\pi}{\mathrm{\epsilon}}_{\mathrm{o}}}\mathrm{}\frac{{2\mathrm{Z}\mathrm{e}}^{2}}{{\mathrm{E}}_{\mathrm{K}}}$$

where, E_{K} = kinetic energy of the α-particle.

**Impact parameter**

The impact parameter is the perpendicular distance of the initial velocity vector of the α-particle from the centre of the nucleus.

$$\mathrm{b}=\frac{1}{4\mathrm{\pi}{\mathrm{\epsilon}}_{\mathrm{o}}}\mathrm{}\frac{{\mathrm{Z}\mathrm{e}}^{2}\mathrm{cot}\frac{}{2}}{{\mathrm{E}}_{\mathrm{K}}}\mathrm{}$$

**Rutherford’s scattering formula**

$$\mathrm{N}\left(\right)=\mathrm{}\frac{{\mathrm{N}}_{\mathrm{i}}\mathrm{n}\mathrm{t}\mathrm{}{\mathrm{Z}}^{2}{\mathrm{e}}^{4}}{{\left(8\mathrm{\pi}{\mathrm{\epsilon}}_{\mathrm{o}}\right)}^{2}\mathrm{}{\mathrm{r}}^{2}{\mathrm{E}}^{2}\mathrm{}{\mathrm{s}\mathrm{i}\mathrm{n}}^{4}\frac{}{2}}$$

where,

N(θ) = number of α–particles with scattering angle θ,

N_{i} = total number of α-particles reaching the screen,

n = number of atoms per unit volume in the foil,

Z = atomic number of the foil,

E = kinetic energy of the alpha particles and

t = foil thickness

**Limitations of Rutherford’s atomic model**

**About the stability of atom -**According to Maxwell’s electromagnetic wave theory electron should emit energy in the form of electromagnetic wave during its orbital motion. Therefore radius of orbit of electron will decrease gradually and ultimately it will fall in the nucleus.

**About the line spectrum -**Rutherford atomic model cannot explain atomic line spectrum.

**Bohr’s atomic model**

- An electron in an atom could revolve in certain stable orbits without the emission of radiant energy.
- The angular momentum of electron is an integer multiple of (h/2π)
$$\mathrm{m}\mathrm{v}\mathrm{r}\mathrm{}=\frac{\mathrm{n}\mathrm{h}}{2\mathrm{\pi}}$$

where n = 1, 2. 3,… called

**principle quantum number or orbit number**. - The radiation of energy occurs only when any electron jumps from one permitted orbit to another permitted orbit. Energy of emitted photon
$$\mathrm{\Delta}\mathrm{E}=\mathrm{h}\mathrm{\nu}={\mathrm{E}}_{2}\mathrm{}\u2013\mathrm{}{\mathrm{E}}_{1}$$

$$=-{\mathrm{R}}_{\mathrm{H}}\left(\frac{1}{{{\mathrm{n}}_{2}}^{2}}-\frac{1}{{{\mathrm{n}}_{1}}^{2}}\right)$$

where E

_{1}and E_{2}are energies of electron in orbits.$${\mathrm{R}}_{\mathrm{H}}\mathrm{}=\mathrm{}\mathrm{R}\times \mathrm{h}\mathrm{c}$$

$$\mathrm{R}\mathrm{}=\mathrm{}\mathrm{R}\mathrm{y}\mathrm{d}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{\u2019}\mathrm{s}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{}=\mathrm{}1.09678\mathrm{}\times {\mathrm{}10}^{7}\mathrm{}{\mathrm{m}}^{-1}$$

$$\mathrm{h}={\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}^{\mathrm{\text{'}}}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}=\mathrm{}6.63\mathrm{}\times \mathrm{}{10}^{-34}\mathrm{}{\mathrm{m}}^{2}\mathrm{}\mathrm{k}\mathrm{g}\mathrm{}\mathrm{}{\mathrm{s}}^{-1}$$

**Radius of orbit of electron** is given by

$$\mathrm{r}=\frac{4\mathrm{\pi}{\mathrm{\epsilon}}_{0}{\mathrm{n}}^{2}{\mathrm{h}}^{2}}{4{\mathrm{\pi}}^{2}\mathrm{m}\mathrm{Z}{\mathrm{e}}^{2}}=\frac{{\mathrm{\epsilon}}_{0}{\mathrm{n}}^{2}{\mathrm{h}}^{2}}{\mathrm{\pi}\mathrm{m}\mathrm{Z}{\mathrm{e}}^{2}}$$

where,

n = principle quantum number,

h = Planck’s constant,

m = mass of an electron,

Z = atomic number and

e = electronic charge.

Substitution of values of *h*, *m*, ε_{o} and *e *gives r_{1} = 5.29 × 10^{–11} m. This is called the **Bohr radius**, represented by the symbol *a*_{o}.

**Velocity of electron** in any orbit is given by

$$\mathrm{v}=\frac{{\mathrm{e}}^{2}\mathrm{Z}}{2{\mathrm{\epsilon}}_{0}\mathrm{n}\mathrm{h}}=\frac{\mathrm{c}}{137}.\frac{\mathrm{Z}}{\mathrm{n}}$$

$$\Rightarrow \mathrm{}\mathrm{v}\mathrm{}\propto \mathrm{}\frac{\mathrm{Z}}{\mathrm{n}}$$

**Frequency of electron** in any orbit is given by

$$\mathrm{f}=\frac{\mathrm{v}}{2\mathrm{\pi}\mathrm{r}}=\frac{\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}{4{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{n}}^{3}{\mathrm{h}}^{3}}$$

**Time period of an electron**

$$\mathrm{T}=\frac{1}{\mathrm{f}}=\frac{4{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{n}}^{3}{\mathrm{h}}^{3}}{\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}=\mathrm{}\frac{{\left(4\mathrm{\pi}{\mathrm{\epsilon}}_{0}\right)}^{2}{\mathrm{n}}^{3}{\mathrm{h}}^{3}}{4{\mathrm{\pi}}^{2}\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}$$

**Kinetic energy of an electron**

$$\mathrm{K}\mathrm{}=\frac{1}{2}\mathrm{m}{\mathrm{v}}^{2}=\frac{\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}{8{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{n}}^{2}{\mathrm{h}}^{2}}$$

**Potential energy of an electron**

$$\mathrm{U}\mathrm{}=-\frac{\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}{4{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{n}}^{2}{\mathrm{h}}^{2}}$$

**Total energy of an electron**

$${\mathrm{E}}_{\mathrm{n}}=\mathrm{K}+\mathrm{U}\mathrm{}=-\frac{\mathrm{m}{\mathrm{e}}^{4}{\mathrm{Z}}^{2}}{8{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{n}}^{2}{\mathrm{h}}^{2}}$$

$${\mathrm{E}}_{\mathrm{n}}=-13.6\frac{{\mathrm{Z}}^{2}}{{\mathrm{n}}^{2}}\mathrm{}\mathrm{e}\mathrm{V}$$

Where, 1 eV = 1.6 × 10^{–19} J,

We can calculate,

E_{1} = -13.6 eV,

E_{2} = -3.4 eV,

E_{3} = -1.5 eV,

E_{∞} = 0

**Wave number of a radiation**

$$\frac{1}{\mathrm{\lambda}}=\mathrm{R}{\mathrm{Z}}^{2}\left(\frac{1}{{{\mathrm{n}}_{2}}^{2}}-\frac{1}{{{\mathrm{n}}_{1}}^{2}}\right)$$

Where Rydberg’s constant,

$$\mathrm{R}=\frac{\mathrm{m}{\mathrm{e}}^{4}}{8{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{h}}^{3}\mathrm{c}}=1.09678\mathrm{}\times {\mathrm{}10}^{7}\mathrm{}{\mathrm{m}}^{-1}\mathrm{}\mathrm{}\mathrm{}$$

**Quantum numbers**

Quantum numbers are the numbers required to completely specify the state of the electrons.

There are four quantum number associated with each electron,

- Principle quantum number (n) can have value 1, 2, … ∞
- Orbital angular momentum quantum number
*l*can have value 0,1, 2, … ,(n – 1). - Magnetic quantum number (m
_{e}) which can have values –*l*to*l*. - Magnetic spin angular momentum quantum number (m
_{s}) which can have only two value ± $\frac{1}{2}$.

**Hydrogen spectrum series**

Each element emits a spectrum of radiation, which is characteristic of the element itself. The spectrum consists of a set of isolated parallel lines and is called the **line spectrum.**

Hydrogen spectrum contains five series

**Lyman Series - **When electron jumps from n = 2, 3, 4, … to n = 1 orbit, a line of Lyman series is obtained. This series lies in **ultra violet region.**

**Balmer Series - **When electron jumps from n = 3, 4, 5,… to n = 2 orbit, a line of Balmer series is obtained. This series lies in **visual region.**

**Paschen Series -**When electron jumps from n = 4, 5, 6,… to n = 3 orbit, a line of Paschen series is obtained. This series lies in **infrared region.**

**Brackett Series -**When electron jumps from n = 5,6, 7…. to n = 4 orbit, a line of Brackett series is obtained. This series lies in **infrared region.**

**Pfund Series -**When electron jumps from n = 6,7,8, … to n = 5 orbit, a line of Pfund series is obtained. This series lies in **infrared region.**

Emission lines in the spectrum of hydrogen

**Line spectra and electron transitions between energy levels**

Line spectra and electron transitions between energy levels

**de Broglie’s explanation of Bohr’s second postulate of quantisation**

The second postulate in the Bohr’s atomic model states that the angular momentum of the electron orbiting around the nucleus is quantised (that is, *L= nh/2*π; *n *= 1, 2, 3 …). Why should the angular momentum have only those values that are integral multiples of *h/2*π?

This can be explained using de Broglie’s hypothesis. According to his hypothesis, material particles, such as electrons, also have a wave nature. He argued that the electron in its circular orbit, as proposed by Bohr, must be seen as a particle wave. Just like the waves travelling on a string, particle waves too can lead to standing waves under resonant conditions. When a string is plucked, a vast number of wavelengths are excited. However only those wavelengths survive which have nodes at the ends and form the standing wave in the string. For an electron moving in *n*th circular orbit of radius *rn*, the total distance is the circumference of the orbit, 2π*rn*.

Standing waves are formed when the total distance travelled by a wave is one wavelength, two wavelengths, or any integral number of wavelengths. Waves with other wavelengths interfere with themselves upon reflection and their amplitudes quickly drop to zero. Thus,

$$2\pi {r}_{n}=n\lambda ,n=1,2,3...$$

Now λ *= h/p*, where *p *is the magnitude of the electron’s momentum. If the speed of the electron is much less than the speed of light, the momentum is mv_{n}. Thus,

$$\lambda =\frac{h}{m{v}_{n}}$$

Combining these equations, we have,

$$2\lambda {r}_{n}=\frac{nh}{m{v}_{n}}$$

$$\Rightarrow m{v}_{n}{r}_{n}=\frac{nh}{2\pi}$$

This is the quantum condition proposed by Bohr for the angular momentum of the electron.

This equation is the basis of explaining the discrete orbits and energy levels in hydrogen atom. Thus de Broglie hypothesis provided an explanation for Bohr’s second postulate for the quantisation of angular momentum of the orbiting electron.

**Limitations of Bohr’s atomic model**

- The Bohr model is applicable to hydrogenic atoms. It cannot be extended atoms with more than one electron.
- While the Bohr’s model correctly predicts the frequencies of the light emitted by hydrogenic atoms, the model is unable to explain the relative intensities of the frequencies in the spectrum.
- Bohr’s model is also unable to explain splitting of spectral lines.