1 J J Thomson’s Model of an Atom:

According to JJ Thomson model of an atom, an atom is just like a watermelon in which seeds behave as electrons and reddish parts behave as  charged matter. Atom as a whole is electrically neutral.

Failure of JJ Thomson Model:

It could not explain the origin of spectral series of hydrogen atom.

It fails to explain the large angle scattering of hydrogen atom.

2 Rutherford α-particle Scattering Experiment ^imp:

Rutherford along with Geiger and Marsden performed a number of experiments on the scattering of α-particle due to atoms of thin gold foils. The experimental demonstration of their model is as below.

Experimental Setup:

A narrow beam of α particle obtained from a radioactive substance placed in a lead cavity is generated, which is allowed to fall on a thin gold foil, after passing through a narrow slit. After striking the gold foil, the beam is allowed to fall on Z screen and the pattern is seem with the help of movable detector. As shown in fig.

Observation:

Rutherford observed that

• Most of the α-particle passed through the gold foil un-deflected.
• Some of α-particle was deflected through small angle.
• A few (about 1in 8000) regained their path i.e. deflected at 180° angle.

Conclusion:

1. As most of α-particle passed un-deflected, which indicate that there is large empty space inside the atom
2. As α-particle is a  charge particle, it can deflect only by large  charge particle it means whole  charge of an atom lies at a single point which was called nucleus.
3. Using high speed α-particle we can find out the size of nucleus of order of
4. The nucleus is surrounded by electrons having equal  charge to charge of nucleus. Atom as a whole is electrically neutral.

3 Distance of Closest Approach: Estimation of Nuclear Size ^imp:

In Rutherford α-ray scattering experiment the α-particle stops for a moment and then retraces its path. Here the distance  is called the distance of closet approach.

At this distance the entire K.E of α-particle gets convert into electrostatic potential energy. Here the kinetic energy of α-particle is  and electrostatic P.E. of α-particle and nucleus is 

Now

⟹

Clearly the radius of the nucleus must be small then. 

In Rutherford experiment energy of α particle was 5.5MeV.

So

AlsoSo

So we can say that radius of the nucleus is of order of 

4 Impact Parameter ^imp:

It is defined as the perpendicular distance of the velocity vector of α-particle from centre of the nucleus when the path of α-particle is un-deflected.

Rutherford deduced a relation between impact parameter and scattering angle θ.

As

Or

• If impact parameter is large then repulsive force experienced by α particle is small and vice versa.
• For a head on collision the impact parameter

5 Rutherford Model of an Atom ^imp

On the basis α-ray scattering experiment Rutherford proposed a model of atom. The main points are:

1. Most of the mass and all  charge of an atom are concentrated at centre of the atom called nucleus.
2. The size of the nucleus  is extremely small as compared to size of the atom. So most of the space in a atom is empty.
3. The number of electrons in a atom is equal to  charge in the nucleus hence atom is electrically neutral.
4. Electron revolves around the nucleus in circular orbit the centripetal force required to revolve electron is provide by the columbium force of attraction between nucleus and electrons.

6 Limitations:

1. It fails to Explains the Stability of an Atom:

As according to Rutherford model of an atom electron revolve around the nucleus in circular orbits, but, but according to electromagnetic theory a acceleration charge particle must radiate energy. By doing this the path of e^– will become shorter and spiral and ultimately it will fall into the nucleus. 

(ii)In Rutherford model electron may revolve in orbits of any possible radii so it should emit continuous spectra. But an atom like H emits discrete spectra. Thus Rutherford model cannot explain spectra of H-atom.

7 Bohr’s Model of an Atom ^m.imp:

To explain stability and spectrum of an atom, Bohr applied Planck’s quantum theory of radiations to Rutherford model of atom. The main postulates of Bohr model are as given below:

1. An atom has a small  charged core where whole of mass of an atom is supposed to be concentrated. This core is called nucleus of the atom.
2. The electron revolves around the nucleus in definite energy orbits and does not radiate any energy during their rotation.
3. The electron can revolve only in those orbits, in which its angular momentum is an integral multiple of
4.        i.e   
5.         Electron radiates or absorbs energy when it moves from one orbit to another orbit.

   If a electron in one orbit absorb amount of energy then it jumps to higher energy orbits. i.e

Or

 

 

8 Bohr’s Theory of H-Atom:

(1)Radius of H-atom

As H-atom have nucleus charge  and electronic charge. Then coulomb’s force of attraction between the nucleus and the electron is given by

This coulomb ion force provides the necessary centripetal force to the electron to revolve in circular orbits around the nucleus. i.e

or

Now according to Bohr’s postulate of H-atom

Using in eq. (i) we get

or

⟹

But

⟹

Thus radius of an orbit is directly proportional to the square of principle quantum number of the orbit.

The radius of the inner most orbits  in the hydrogen atom is called Bohr’s radius 

as

(ii) Speed of Electron in an Orbit:

As we know from eq. (ii) that

Using eq. (iii) we get

Orwhere is a constant

Thus speed of electron in an orbit is inversely proportional to the principle quantum number.

(iii)Energy of a electron in an orbit:

The total energy of an electron in an orbit is equal to the sum of K.E and P.E, i.e.

From eq. (i) we get

And P.E=Potential of e^– in a orbit  charge on electron

=

Using in eq. (v) we get

= 

Multiplying and dividing R.H.S by  we get

Where 

Or

Thus total energy is inversely proportion to square of principle quantum number.

Here  sign indicate that the total energy of e^– in an orbit is due to attractive force between e^– and nucleus.

9 Spectral Series of H-Atom:

Before studying spectral series of H-atom we will study absorption and emission spectra.

1.             Emission Spectra:

When an electron jumps from a higher energy level to lower energy level, then it emits energy in form of radiations of certain wavelength called emission spectra. 

The amount of energy emitted by electron is

.

2.  Absorption Spectra:

When an element is heated to a high temperature then it absorb energy and emit certain spectra called absorption spectra. In this case electron absorb  amount of energy and excite to higher energy level as

10 Spectral Series of H-atom ^imp:

 

 

 

 

 

As electron move from higher energy orbit to lower energy orbit then it emits line spectra of frequency as

Or

Or 

Or 

Where  is called Rydberg constant.

Now the origin of the various spectral series in the H-atom can be explained as flow.

1. Lyman Series:

If an electron energy level to  then electron emits Lyman series in ultraviolet region.

This series is given by

 Where 

2. Balmer Series:

If a electron jumps from higher energy level to  level then electron emits Balmer Series I visible region. 

This series is given by

 Where

3. Paschen Series:

If an electron jumps from higher energy level to n=3 level then electron emits Paschen series in infrared region. This series is given by 

 Where

4. Brakett Series:

If a electron jumps from higher energy level to  level then it emits Brakett series in infrared region. This series is given by

Where

5. P fund Series:

If an electron jumps from higher energy level to n=5 level then it emits P fund series in infrared region. This series is given by

 Where

Bohr explained successfully the series of Lyman, Balmer and Paschen series and predicted Brackett and P fund series which were latterly explained by Brakett and P fund.

Q. Find the wavelength of the electron orbiting in the first excited state in hydrogen atom. 

Ans.  The energy of nth state of the hydrogen atom is given

 as; 

 So, in first excited state, n = 2

Therefore:

    So 

11 Energy Level Diagram For H-atoms  ^imp:

It is a diagram in which the energies of different stationary states of an atom are represented by parallel horizontal lines, drawn according to some suitable energy scale.

Energy level for H-atom:

According to Bohr’s theory, the total energy of electron in  orbit is 

Or 

Now energy of electron in first orbit  is 

Again energy of electron in second orbit is

Again for 

And

And 

Clearly an electron has certain definite energy while revolving in a orbit. This is called quantization of energy of electron in a atom.

In case of H-atom energy state of maximum energy  is called ground state and other  are called excited states.

12 Limitations of Bohr’s Theory:

• Bohr’s theory explained certain spectral lines of H-atom but it fails to explain fine features of Hydrogen spectrum.
• It could not explain the spectral series of other atoms except Hydrogen.
• According to Bohr Theory, the orbits of electrons are circular only but elliptical orbits are also possible.
• As electron also have wave nature so exact position of electron cannot be explained by Bohr’s Theory.
• It does not tell anything about the relative intensities of spectral series.
• It does not explain the further splitting of spectral series in magnetic field (Zeeman Effect) or in analectic field (Stark effect).

13 Excitation and Ionization Potential:

Excitation Energy:

The amount of energy required by an electron to jump from ground state to any other energy state is called excitation energy.

First excitation energy of 

Second excitation energy of 

Ionization Energy: 

The amount of energy required to remove an election from an isolated gasses atom is called ionization energy.

Ionization energy of 

Excitation Potential:

The amount of potential require to jump a electron from lower energy level to higher energy level is called excitation potential.

First excitation potential of 

Second excitation potential of 

Ionization potential:

The amount of potential require to just eject a electron from isolated gases atom is called ionization potential.

Ionization potential of 

SOME IMPORTANT MCQ:-

1.The Lyman series of hydrogen spectrum lies in the region

(a)Infrared     (b) Visible    (c) Ultraviolet (d) Of x rays

2.Which one of the series of hydrogen spectrum is in the visible region

(a)Lyman series  (b)Balmer series  (c)Paschen series  (d)Bracket series 

3.When a hydrogen atom is raised from the ground state to an excited state

(a)P.E. increases and K.E. decreases (b) P.E. decreases and K.E. increases 

(c)Both kinetic energy and potential energy increase (d) Both K.E. and P.E. decrease 

4. in Bohr model of the hydrogen atom, the lowest orbit corresponds to

(a)Infinite energy  (b) The maximum energy  (c) The minimum energy (d) Zero energy 

5.The minimum energy required to excite a hydrogen atom from its ground state is 

(a)13.6 eV  (b)6 . 13 eV   (c) 3.4 eV   (d) 10.2 eV

6.Which one of these is non-divisible?

(a)Nucleus  (b) Photon (c) Proton   (d) Atom 

7. to explain his theory, Bohr used 

(a)Conservation of linear momentum    (b) Conservation of angular momentum

(c)Conservation of quantum frequency   (d) Conservation of energy 

8. in Rutherford scattering experiment, what will be the correct angle for α scattering for an impact parameter b = 0 

(a)

9.The splitting of line into groups under the effect of electric or magnetic field is called

(a)Zeeman’s effect (b) Bohr’s effect  (c) Heisenberg’s effect  (d) Magnetic effect 

10.Minimum excitation potential of Bohr’s first orbit in hydrogen atom is 

(a)13.6 V  (b) 3.4 V   (c) 10.2 V (d) 3.6 V

11.Energy of electron in a orbit of H-atom is        www.vartmaaninstitute.com

(a)Positive   (b) Negative (c) Zero   (d) nothing can be said 

12.The concept of stationary orbits was proposed by

(a)Neil Bohr (b) J.J. Thomson  (c) Ruther ford   (d) I. Newton

13.Who discovered spin quantum number?

(a) Unlen-beck and Goud-smit  (b) Nell’s Bohr   (c) Zeeman  (d) Sommer-field 

14.In hydrogen atom which quantity is integral multiple of 

(a)Angular momentum (b) Angular velocity  (c) Angular acceleration  (d) Momentum

15.The order of the size of nucleus and Bohr radius of an atom respectively are

(a)  (b)   (c) (d) 

16.Which of the following is quantized according to Bohr’s theory of hydrogen atom?

(a)Linear momentum of electron  (b) Angular momentum of electron

(c)Linear velocity of electron (d) Angular velocity of electron 

17.The kinetic energy of an electron revolving around a nucleus will be

(a)Four times of P.E. (b)Double of P.E.  (c)Equal to P.E.  (d)Half of its P.E

18.Which of the following particles are constituents of the nucleus?

(a)Protons and electrons  (b) Protons and neutrons

(c)Neutrons and electrons  (d) Neutrons and positrons 

19.The neutron was discovered by

(a)Marie Curie (b) Pierre Curie  (c) James Chadwick (d) Rutherford 

20.Nuclear binding energy is equivalent to

(a)Mass of proton (b) Mass of neutron (c) Mass of nucleus  (d) Mass defect of nucleus

21.Size of nucleus is of the order of 

(a) m    (b)m (c) m   (d)m 

22 From a newly formed radioactive substance (Half-life 2 hours), the intensity of radiation is 64 times the permissible safe level. The minimum time after which work can be done safely from this source is 

(a)6 hours  (b) 12 hours   (c) 24 hours    (d) 128 hours 

Solution: (b) By using  so time