Here students can locate TS Inter 2nd Year Physics Notes 13th Lesson Atoms to prepare for their exam.

## TS Inter 2nd Year Physics Notes 13th Lesson Atoms

→ J.J. Thomson thought that the positive charge of the atom is uniformly distributed through out the volume of the atom and the negatively charged electrons are embedded in it like seeds in a watermelon.

→ According to Rutherford the entire positive charge and most of the mass of the atom is concentrated in a small volume called nucleus. Electrons are revolving around the nucleus at some distance just as planets revolve around the sun.

→ Alpha particle scattering experiments on gold foil showed that size of nucleus is about 10^{-14} to 10^{-15} m and size of atom is nearly 10^{-10} m.

→ Most of the atom is empty space. The electrons would be moving in certain orbits with some distance from nucleus just like planets around the sun.

→ Alpha particle scattering experiment :

Magnitude of force between α – particle and gold nuclie is F = \(\frac{1}{4 \pi \epsilon_o} \frac{(2 \mathrm{e})(\mathrm{Ze})}{\mathrm{r}^2}\) Where ‘r’ is the distance between α – particle and nucleus.

The magnitude and direction of force changes continuously as it approaches the nucleus.

→ Impact Parameter: It is the perpendicular distance of the initial velocity vector of a particle from centre of nucleus.

In case of head on collision impact parameter is minimum and α – particle rebounds back (θ = π). For a large impact parameter α – particle goes undeviated. The chance of head on collision is very small. It in turn suggested that mass of atom is much concentrated in a small volume.

→ Bohr postulates : Bohr model of hydrogen atom consists of three main postulates.

- Electrons in an atom could revolve in certain permitted stable orbits. Electrons revolving in these stable orbits do not emit or radiate any energy.
- The stable orbits are those whose orbital angular momentum is an integral multiple of h/ 2π.

i. e., L = nh / 2π where n = 1, 2, 3, ……………… etc. (an integer.)

These stable orbits are also called as non – radiating orbits. - An electron may take a transition between non-radiating orbits. When electron transition takes place a photon of energy equals to the energy difference between initial and final states will be radiated.

E = hv = E_{i}– E_{j}

→ Bohr radius (a_{0}): According to Bohr theory radius of the orbit, r = \(\frac{\mathrm{n}^2 \mathrm{~h}^2 \epsilon_0}{\pi \mathrm{me}^2}\) when n = 1. It is called first orbit. Radius of 1st orbit r_{1} = \(\frac{h^2 \epsilon_0}{\pi \mathrm{me}^2}\) = 5.29 × 10^{-11} m. This is called Bohr orbit a_{0}.

a_{0} = \(\frac{h^2 \epsilon_0}{\pi \mathrm{me}^2}\) = 5.29 × 10^{-11} m = 0.529 Å

→ Energy of orbit: From Bohr theory energy of the orbit E = –\(\frac{m e^4}{8 n^2 h^2 \epsilon_o^2}\)

Where – ve sign indicates the force of attraction between electron and nucleus.

For 1st orbit n = 1.

Its energy E_{1} = -2.18 × 10^{-18} J or

E_{1} = – 13.6 eV.

For all other orbits their energy E = \(\frac{13.6}{n^2}\) eV

Note: The energy of an atom is least (i.e., it has maximum – ve value) when electron is revolving with n = 1 orbit. This energy state (n = 1) is called lowest state of the atom or ground state. For ground state of hydrogen atom E = – 13.6 eV.

→ Spectral series: From Bohr model electrons are permitted to transit between the energy levels while doing so they will absorb or release the exact amount of energy difference of the initial and final states.

∴ Energy absorbed or released E = hv = E_{i} – E_{f}

E = hv = \(\frac{\mathrm{hc}}{\lambda}=\frac{m \mathrm{e}^4}{8 \varepsilon_{\mathrm{o}} \mathrm{h}^2}\left[\frac{1}{\mathrm{n}_{\mathrm{i}}^2}-\frac{1}{\mathrm{n}_{\mathrm{f}}^2}\right]\) or

\(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_{\mathrm{i}}^2}-\frac{1}{\mathrm{n}_{\mathrm{f}}^2}\right]\) where R is Rydberg’s constant R = 1.03 × 10^{7} / m

→ Lyman series: When electrons are jumping on to the first orbit from higher energy levels then that series of spectral lines emitted are called ”lyman series”.

In Lyman series \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{1^2}-\frac{1}{n^2}\right]\) where n = 2, 3, …… etc

→ Balmer series : When electrons are jumping on to the second orbit from higher levels then that series of spectral lines are called “Balmer series”.

For Balmer series \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{2^2}-\frac{1}{\mathrm{n}^2}\right]\) where n = 3,4,………. Spectral lines of Balmer series are in visible region.

→ Paschen series : When electrons are jump¬ing on to the 3rd orbit from higher energy levels then that series of spectral lines are called “Paschen series”.

For Paschen series \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{3^2}-\frac{1}{\mathrm{n}^2}\right]\)

n = 4, 5………….. These spectral lines are in near infrared region.

→ Brackett series: When electrons are jump¬ing on to the 4th orbit from higher levels then that series of spectral lines are called “Brackett series”.

For Brackett series \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{4^2}-\frac{1}{n^2}\right]\) where n = 5, 6, …………….

Brackett series are in middle infrared region.

→ Pfund series: When electrons are jumping on to the 5th orbit from higher energy levels then that series of spectral lines are called

“pfund series”.

For pfund series \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{5^2}-\frac{1}{\mathrm{n}^2}\right]\)

where n = 6, 7, ………… These spectral lines are in far infrared region.

Note : In spectral lines \(\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_{\mathrm{i}}^2}-\frac{1}{\mathrm{n}_{\mathrm{f}}^2}\right]\)R

But \(\frac{1}{\lambda}\) = v/c

Frequency of spectral line v = Rc\(\left[\frac{1}{n_i^2}-\frac{1}{n_f^2}\right]\)

→ Ionisation potential : It is the amount of minimum energy required to release an ele-ctron from the outer most orbit of the nucleus.

From Bohr’s model energy of the orbit is the ionisation energy of electron in that orbit.

Ex: Energy of 1st orbit in hydrogen is 13.6 eV.

Practically ionisation potential of hydro-gen is 13.6 eV.

Note : The success of Bohr atom model is in the prediction of ionisation energy of orbits.

→ Force between ‘a’ particle and positively charged nucleus

F = \(\frac{1}{4 \pi \varepsilon_o} \frac{2 \mathrm{e}(\mathrm{Ze})}{\mathrm{r} 2}=\frac{\mathrm{Ze}^2}{2 \pi \varepsilon_0 r^2}\)

→ Kinetic energy of α – particle,

K = \(\frac{2 Z \mathrm{e}^2}{4 \pi \varepsilon_{\mathrm{o}} \mathrm{d}}=\frac{Z \mathrm{e}^2}{2 \pi \varepsilon_{\mathrm{o}} \mathrm{d}}\)

→ Distance of closest approach d = \(\frac{\mathrm{Ze}^2}{2 \pi \varepsilon_0 \mathrm{k}}\)

→ For an electron moving in the orbit of hydrogen atom = \(\frac{m v^2}{r}=\frac{1}{4 \pi \varepsilon_o} \frac{Z^2}{r^2}\)

→ For an atom of atomic number ‘Z’, \(\frac{\mathrm{mv}^2}{\mathrm{r}}=\frac{1}{4 \pi \varepsilon_{\mathrm{o}}} \frac{\mathrm{Ze}^2}{\mathrm{r}^2}\)

→ Relation between orbit radius and velocity in hydrogen atom is r = e^{2} / 4πε_{0}

mv^{2} = \(\frac{1}{4 \pi \varepsilon_o}\) where k = \(\frac{1}{4 \pi \varepsilon_o}\) = 9 × 10^{9}

→ In hydrogen atom .

(1) Kinetic energy K = \(\frac{1}{2}\)mv^{2}

(ii) Potential energy U = \(\frac{\mathrm{e}^2}{8 \pi \varepsilon_o \mathrm{r}}=\frac{m \mathrm{e}^4}{8 \mathrm{n}^2 \mathrm{~h}^2 \varepsilon_o^2}\)

= \(-\frac{e^2}{4 \pi \varepsilon_o r}\) (-ve sigh for force of attraction)

(iii) Total energy E = K + U

= \(-\frac{e^2}{8 \pi \varepsilon_o r}=\frac{-m e^4}{8 n^2 h^2 \varepsilon_o^2}\)

(iv) Velocity of electron in orbit v = e/\(\sqrt{4 \pi \varepsilon_o \mathrm{mr}}\)

→ Spectral series: Wavelengths of spectral series are given by \(\frac{1}{\lambda}=\mathrm{R}\left(\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right)\)

where n_{1} and n_{2} are the number of orbits between which electron transition takes place.

Energy radiated In transition E = hv = E_{2} – E_{1}

In Bohr atom model.

Angular momentum of orbital L = mvr = \(\frac{\mathrm{nh}}{2 \pi}\)

radius of nth orbit r_{n} = \(\frac{\mathrm{nh}}{2 \pi}\)

Velocity of electron in nth orbit

V_{n} = e/\(\sqrt{4 \pi \varepsilon_o m r_n}\)

or v_{n} = \(\frac{1}{\mathrm{n}} \frac{\mathrm{e}^2}{4 \pi \varepsilon_{\mathrm{o}}} \frac{1}{(\mathrm{~h} / 2 \pi)}\)

or

r_{n} = \(\frac{\mathrm{n}^2 \mathrm{~h}^2 \varepsilon_0}{\pi \mathrm{me}^4}\)

Bohr radius a_{0} = \(\frac{h^2 \varepsilon_o}{\pi m e^4}\) = 5.29 × 10^{-11} m

Energy of nth orbit

E_{n} = \(\frac{-\mathrm{me}^4}{8 \mathrm{n}^2 \mathrm{~h}^2 \varepsilon_{\mathrm{o}}^2}=\frac{-2.18 \times 10^{-18}}{\mathrm{n}^2}\)J = \(\frac{-13.6}{n^2}\)eV

Rydberg’s constant R = \(\frac{m \mathrm{e}^4}{8 \varepsilon_o^2 h^3 \mathrm{c}}\)

= 1.03 × 10^{7} m^{-1}