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Atoms, Molecules and Nuclei

Origin of Spectral lines

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If En is the energy level in nth orbit and Ep is the

energy in the pth orbit (where n > p), the energy

radiated during the fall from nth to pth orbit is,

En – Ep = hν, ---- (1) as per Bohr’s third postulate,

where ν is the frequency of radiation.But

But, according to equation (1), En – Ep = hν

But , ν = c / λ

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Where is called Rydberg,s constant

R = 1.097 × 107 m-1

This formula gives the wavelength of the spectral line

emitted by the atom, when the excited electron jumps

from nth orbit to pth orbit.

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Series of Spectral lines

Form the formula,

1. Lyman Series

Thus, for Lyman series, p = 1 & n = 2, 3, 4, ... and

the wavelength in U.V. range.

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2. Balmer Series

for Balmer series, p = 2 & n = 3, 4, 5, ..& the

wavelength is given as

All the lines in this series are in visible range.

3. Paschen Series

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Paschen series, p = 3 & n = 4, 5, 6, & the

wavelength is given as

All the lines in this series are in mid infra red range.

Two more series are emitted when the excited

electron falls to 4th and 5th orbit from higher orbits.

These series are known as Brackett series and

pfund series, respectively, which are in infra red

and far infra red regions.

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MCQQ.11  If r1 is the first Bohr radius, then the radius of the third Bohr orbit is given by

          (a.11) 3r1     (b.11) 6r1     (c.11) 9r1     (d.11)

Q.12  If the orbital velocity of the electron in the first orbit of H-atom is 2.2×106m/s, then its

orbital velocity in the second orbit is given by

          (a.12) 1.1×106m/s          (b.12) 4.4×106m/s         (c.12)         (d.12)

1.1×103m/s

Q.14  The energy required to remove an electron in a hydrogen atom from the state

corresponding to n=10 is

          (a.14) 13.6eV        (b.14) 1.36eV        (c.14) 0.136eV      (d.14) 136eV

Q.24  What is the time taken by an electron to traverse the first Bohr orbit?

          (a.24) 0.50×10-16s          (b.24) 1.52×10-16s          (c.24) 2×10-16s     (d.24) 2.52×10-16s

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Continuous and characteristic X-rays

Coolidge tube Experiment. - It can be observed from

the nature of the graph that there is a threshold

wavelength or cutoff wavelength which is minimum,

below which no X-ray is emitted. The X-ray emitted can

be divided in two categories.

K, K wavelengths for which the intensity of X-rays is

very large. These X-rays are known as characteristic

X-rays. For remaining wavelengths intensity varies

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gradually and corresponding X-rays are called

continuous X-rays.

The origin of continuous X-rays and cutoff wavelength

can be explained by using the relation

.Thus depends upon accelerating voltage (V) applied

and not on the material of the target on which electron

are incident.

The wavelengths for characteristic X-rays may be used

to identify the element from which they originate. For a

particular material, Wavelengths have definite values.

These X-rays emitted are called characteristics X-rays.

The value of energies are different for different

materials.

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de Broglie’s hypothesis

1926 a French physicist de Broglie

He proposed that matter also has dual nature like light,

i.e. wave nature and particle nature. Moving particles of

matter like electrons, atoms, ions etc show wave like

properties under certain conditions. The waves

associated with moving particles are called Broglie

Waves or matter waves.

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where h is plank’s constant

The principle is also applicable to light, when it

behaves like a photon of energy hν. If m is the mass of

the photon, moving with velocity c (velocity of light),

according to Einstein’s equation

E = mc2, mc2 = hν

this is de Broglie’s relation for photon.

1. de Broglie wavelength can be calculated only for

subatomic particles like electrons or protons.

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2. Two velocities of the moving bodies, viz its linear

velocity and velocity of the matter wave associated

with it are different.

3. The energy carried by moving particles like

electrons is carried by virtue of its linear velocity

and not by its velocity of the matter wave.

Matter Waves

According to de Broglie, every moving particle is

associated with a wave of wavelength given by-

These waves are called matter waves.

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As v → 0, λ → ∞ and as v → ∞, λ → 0.

Matter waves travel faster than light. The velocity of

matter wave is not constant because it depends upon

the velocity of particle. The de Broglie wavelength is

independent of the charge of particle.

The concept of matter wave is introduced

mathematically in modern quantum physics. Actually

these waves are a new kind of waves, they are not

electromagnetic in nature and are proposed to locate

the position of moving particles. The intensity of wave

at a point represents the probability of the associated

particle being there.

de Broglie’s hypothesis of matter wave also solved the

puzzle in Bohr’s 2nd postulate : why should the angular

momentum of electron have only those values that are

integral multiple of ? He thought that motion of 13

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electron within atom is associated with standing wave

along the orbit as shown in Figure. We know about

standing waves in stretched strings, that only those

waves survive for which the distance travelled in round

trip between the ends is integral multiple of

wavelength.

Similarly, electron moving in nth orbit of radius rn, the

distance travelled in one trip is 2πrn that should be

integral multiple of wavelength.

By de Broglie hypothesis,

Substituting this value of ‘λ’ in above expression, 14

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we get

Or

[since mvn rn is angular momentum]

angular momentum =

This is quantum condition proposed by Bohr for

angular momentum of the electron in 2nd postulate.

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Wavelength of an electron

The de Broglie wavelength associated with an electron

can be calculated by using the relation λ = h / mv

If an electron of mass m and charge e is accelerated

by a potential difference V (volt), it will acquire energy

E given by E = eV = mv2/2

∴ m2v2 = 2meV

∴ mv = 2meV = momentum p of the electron

∴ λ = h / 2meV

Substituting the values of h.e and m we get

∴ λ = 12.27 / V

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Davisson and Germer Experiment

1927 Davisson and Germer

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Working

The beam of electrons is allowed to fall normally on the

crystal. The collector is rotated on the circular scale

and for different values of the scattering angle (θ), the

intensity (I) of the scattered electron beam is

measured, in terms of the galvanometer deflection.

Different observations are taken for different

accelerating voltages & curves showing the relation

between I and θ are plotted as follows.

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Conclusions

A bump begins to appear in the curve for 44V electron

beam onwards. As the voltage increases, it moves

upwards, reaches a maximum value at 54V and then

decreases gradually.

For 54V electron beam, strong maxima is observed at

θ = 50o. According to de Broglie’s theory, the

wavelength associated with a 54V electron beam is

given by

This is theoretical value of wavelength λ. In Devisson

Germer experiment, the sharp diffraction pattern is due

to first maxima due to constructive interference of

electrons scattered by the crystal.

According to Bragg’s law, 2d sin φ = nλ20

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In Devisson Germer experiment, φ = 65o (90 – θ/2)

and interplaner separation d for the nickel crystal is

0.91oA. For first maxima, n = 1. Substituting these

values we gat λ = 1.66oA which completely agrees

with the theoretical value. Thus, electron beam behave

as waves (X-rays because of wavelengths in that

range) and possesses wave characteristics.

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