bohr's theory

7/28/2019 Bohr's Theory

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FOUR QUANTUM NUMBERS

1. PRINCIPAL QUANTUM NUMBER

2. ORBITAL QUANTUM NUMBER

3. SPIN ANGULAR QUANTUM NUMBER

4. MAGNETIC QUANTUM NUMBER

PAULIS EXCLUSION PRINCIPLE

Ref: Perspective of Modern Physics By Beiser

Modern Physics By Theraja

Physics of atom By Rajyam


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The observation of line spectra from rarified

gases in electrical discharge tubes could also notbe explained by classical physics.

The only way to explainline spectra was toassume that the electronscould occupy stationarystates in the atom, andthat radiation would onlybe emitted upon atransition between twostationary states at

different energies

ATOMIC LINE SPECTRA

V

H2 DISCHARGE COLLIMATORS DIFFRAACTION SCREEN

TUBE GRATING


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BOHR MODEL OF THE ATOM

Bohr set out to explain

location of spectral lines of hydrogen (e.g., Balmer

series)

Johann Balmer measured the wavelengths of this

series of lines from atomic hydrogen and found that theyfit the relation

1/ = const ( 1/4 - 1/n2).(1)where n = 3, 4, 5, 6.


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SPECTRAL SERIES OF H ATOM

1/ = R ( 1/n2- 1/m2)R being the Rydberg const for infinite nuclear mass

1. Lyman Series (uv region), n=1, m=2, 3, 4

2. Balmer Series (visible), n=2, m=3, 4, 5 3. Paschen series (IR) n=3, m= 4, 5, 6 4. Brackett Series (IR) n=4, m=5, 6, 7 5. Pfund Series (IR) n=5, m= 6, 7, 8


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BOHRS POSTULATES

In 1913, Niels Bohr proposed a model which is

based on the following three postulates :

1. The nucleus is considered to be a point masslocated at the center of the atom, while the

electrons move in circular orbits around it

The electrostatic attraction between the positivenucleus and negative electron keeps the electron

in orbit.

)2..(..........

4

12

22

r

e

r

mv

o


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2. Only those circular orbits of electrons arepermissible for which the angular momentum of

the atom is an integral multiple of . h/2

These permissible orbits are 'steady' or 'stationary'..

)3......(..........2/

....,3,2,1;..

h

nnmvrei


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3. Electrons in 'stationary'orbits do not radiate theirenergy; only when the

electron jumps from a oneorbit to another, thedifference in the energy of

the orbits is absorbed oremitted depending onwhether the transitioncorresponds to excitation

or deexcitation. Emissionof energy is in the form ofa photon:

E2E1 = E = h .(4)


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Derivation of Quantized Energy States

The circular orbits postulated for the

electron requires a balance between the

centripetal [(4

o)

-1q1q2/r

2

]

and centrifugal(mv2/r) forces.

(4o)-1q1q2/r2 = (kZe)e/r2=mv2/r (1)

where for convenience we usek =(4o)-1= 8.99 x109 N m2/C2.


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Recall that the angular momentum is mvr, and so

Bohr's postulate about angular momentum is

mvr = nh/2 (2)Combining equation (1) with (2), we can solve forr, the

radius of the orbit, as

m(kZe)e = m2v2r2 /r = n2h2/42ror

r = n2(h2/42me2kZ) = on2h2/ mZe2 ...........(3)= a

1n2

where a1=oh2/ mZe2 =5.29 x 10-11 m is the Bohrradius for the ground state of the hydrogen atom.


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This equation says that electrons are allowed only

in orbits of certain (quantized) radii, as n = 1, 2, 3... .

Associated with these quantized orbits is aquantized energy. Total energy is the sum of kinetic

energy and potential energy,E = K + U.

We will take the zero of potential energy at

infinity. The potential energy difference is just the

negative of work, or force times the displacement, so the

potential energy of an electron inside an atom is

U = dU = kZe2/r2dr =-kZe2/rso that the total energy is K.E. + P.E.

E =1/2 mv2-kZe2/r


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and using equations (1), (2) and (3)

above, after some simple mathematics,we find the quantized energy

En= -(22me4k2Z2)/n2h2 = -m Z2 e4 /8o 2h2n2

Note: The total energy is negative, or

the electron is bound, and that as n ->infinity,E->0. The numbern is called

the principal quantum number.


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The quantity (22me4k2Z2)/h2 is a constant that

depends only on fundamental physical quantities,

R' =(22me4k2)/h2 = 13.6 eV.The electron-volt (eV) is a unit of energy (the

energy an electron would gain after acceleration through

a potential of 1 volt). 1 eV = 1.602 x 10-19

J.Thus the total energy equation becomes extremely

simple:

En=-R' Z2/n2.En=-13.6/n2 eV for H atom.


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THE BOHR MODEL OF THE HYDROGEN ATOM (Z = 1)

GAVE US THE FOLLOWING RESULTS:

The angular momentum, l, was postulated to bequantized according to

l = mvr= nh/2. where n is the principle quantum

number.The radii of allowed electron orbits were also quantized

according

r = aon2 = n2h2/(42ke2m)where ao= 5.29 x 10

-11m is theBohr radius.


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The energies of the orbits were likewise quantized

according to

En=-R'[1/n2]whereR' = 13.6 eV is a constant.

The difference between energy levels (orbits)

corresponds to a wavelength

1/ab = ab/c = (Eb- Ea)/ch=RH[1/na2-1/nb2]whereR

H= 10.96776 mm-1 is the sameRydberg

constantthat was measured experimentally by

Rydberg.


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Here is a scale model of the orbits of the hydrogen

atom, with the radii getting further apart according to n2:


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However, despite the orbits getting ever farther apart

in space, they get closer togetherin energyaccording

to 1/n2. Any energy level diagram is shown below:


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K G A

V Vo

Hg

CONFIRMATION OF BOHRS THEORY

The Franck-Hertz experiment is one of theclassic demonstrations of the quantization ofatomic energy levels. Electrons emitted by thefilament are accelerated through mercury vapor.When the accelerating voltage reaches V1, theelectrons have just enough energy to excite the

mercury atoms from the ground state to thelowest excited state. Thus, many of theelectrons lose their energy and cannot reach thecollector;

this is signaled by anabrupt drop in collectorcurrent as theaccelerating voltage isincreased past V1.

Fi h li i d f ll


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Figure shows qualitative trend of collector currentvs. accelerating voltage. As the acceleratingvoltage is increased beyond V1, electrons which

have been brought to rest as a result of excitingmercury atoms are again accelerated until theycan produce another excitation. Thus, a secondpeak occurs at V2, where ideally one expects V2 -

V1 = V1.

However, contactpotentials in the system

displace the first peakat V1 from its expectedvalue, and V1 itself isnot a good measure of

the excitation potential.


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The sequence of electron acceleration and atomicexcitation may continue as the accelerating

voltage is further increased so that a series ofpeaks may be observed.

Breakdown of the gas in the tube will occur whenthe accelerating voltage becomes too high. Onsetof breakdown may be delayed by increase invapor pressure (higher oven temperature) ordecrease in number of electrons (lower filament

current).


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bohr's theory

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