Derive the expressions:
left parenthesis straight i right parenthesis space increment straight E space equals space 2.18 space cross times space 10 to the power of negative 18 end exponent straight J space open square brackets fraction numerator 1 over denominator straight n subscript 1 superscript 2 end fraction minus space fraction numerator 1 over denominator straight n subscript 2 superscript 2 end fraction close square brackets
left parenthesis ii right parenthesis space straight v with bar on top space equals 109677 space open square brackets fraction numerator 1 over denominator straight n subscript 1 superscript 2 end fraction minus fraction numerator 1 over denominator straight n subscript 2 superscript 2 end fraction close square brackets cm to the power of negative 1 end exponent


(i) Suppose the electron is in excited state with n = n2. During emission, the electron drops to a lower energy state with n = n1. The difference between the energies of the initial and final state is given by ∆E.

Since this transition results in the emission of a photon of frequency v with energy hv. We can write

Each spectral line in the emission spectrum corresponds to a particular transition in a hydrogen atom.
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How does Bohr's model of atom explain the atomic spectra of hydrogen and hydrogenlike particles?


There is only one electron in hydrogen atom but the hydrogen spectrum consists of a large number of lines in various regions of radiations namely ultraviolet, visible and infra-red. Bohr’s atomic theory provides a satisfactory explanation for the emission of atomic spectra of atoms containing one electron i.e. H, He+ , Li2+ etc.

When an electric discharge is passed through a tube containing hydrogen gas at low pressure, hydrogen molecules dissociate to form hydrogen atoms. These hydrogen atoms absorb energy and the electrons in them are promoted to higher energy levels from their ground state (n = 1). Since in a sample of hydrogen, there are large number of atoms, the electrons in different atoms absorb different amounts (quanta or photons) of energies and are accordingly promoted to different energy states (2, 3, 4, 5....)

All these excited states are metastable states. The electrons cannot remain in these forever. They soon radiate energy and return back to the ground state (n = 1) and others n = 2, n=3, n=4 etc. The electron may return to the lower states in one or more jumps. These transitions emit radiations of different frequencies or wave numbers and produce different lines in the hydrogen spectrum.

The difference in energy between two energy levels is related to the frequency of the radiation emitted as:

        
or            



where E2 and E1 represent the energies of the higher and lower energy levels respectively. ∆E is the difference in their energies, v is the frequency and h are Planck’s constant.

Thus, every line in the hydrogen spectrum corresponds to a particular drop from some higher to some lower energy level as shown.
The lines in the Lyman series are obtained when electrons drop from higher energy levels (i.e n = 2, 3, 4 etc.) to the first energy level (i.e. n =1). These lines fall in the ultraviolet region.



The lines in the Balmer series are obtained when electrons drop from higher energy levels (i.e. n = 3. 4, 5, 6 etc.) to the second energy level (i.e. n =2). These lines fall in the visible region.
Similarly, lines in Paschen, Brackett and Pfund series are obtained when electrons drop from higher energy levels to the third (n =3), fourth (n=4) and 5th energy level (n = 5) respectively. These lines fall in the infra-red region. In short,
Lyman series From n = 2, 3, 4, 5,6 .... to n = 1
Balmer series From n = 3,4, 5, 6 .... to n = 2
Paschen series From n = 4, 5,6..... to n = 3
Brackett series From n = 5,6, 7..... to n = 4
Pfund series From n = 6,7....... to n = 5

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When electromagnetic radiation of wavelength 300 nm falls on the surface of sodium, electrons, are emitted with a kinetic energy of 1.68 X 105 J mol-1. What is the minimum energy needed to remove an electron from sodium? What is the maximum wavelength that will cause a photoelectron to be emitted? 

The energy (E) associated with a 300 nm photon is given by
           E = hv
  
The minium energy needed to remove a mole of electrons from sodium
                              
The minimum energy for one electron
 

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Explain briefly the photoelectric effect.


The phenomenon of ejection of electrons from the surface of a metal when the light of suitable frequency strikes it is called photoelectric effect and the emitted electrons are called photoelectrons.

For each metal, there is a characteristic minimum frequency called threshold frequency below which the photoelectric effect does not occur. For the photoelectric effect to occur, the striking photon should have frequency more than that of the threshold frequency. If a photon of frequency v strikes a metal atom whose threshold energy is v0, then photoelectrons will be emitted only if v > v0. Since the striking photon has energy equal to hv and minimum energy required to eject electron is hv(called wave function W0), then the excess of energy i.e. hv - hv0 or h(v - v0) will be imparted to the ejected electron as kinetic energy.
Hence, 
K.E. of ejected electron  
                               
where me is the mass of electron and v be its velocity.

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Give essential features of Bohr's model of atom.


In 1913, Neil Bohr proposed his model of the atom which was based on Planck’s quantum theory. The essential features of Bohr’s model of atom are:

1. An atom contains a heavy positively charged nucleus situated at the centre, with electrons revolving around the nucleus in fixed circular paths.


2. These circular paths are called orbits, shells energy levels or stationary states in which electron can revolve around the nucleus without emitting radiations. So far as an electron revolves in a certain orbit, its energy remains constant.

3. These different energy levels are designated by numbers 1, 2, 3, 4 etc. or letters K, L, M, N etc. starting from the nucleus. The greater the distance of the energy level from the nucleus more is the energy associated with it.

4. The electrons in an atom can revolve only in those orbits in which the angular momentum (mvr) of the electron is a whole number (n) multiple of a constant .
          
where m = mass of electron,
v = velocity of electron
r = radius of orbit.
h = Planck's constant,
n = whole number
This postulate indicates that the angular momentum of an electron moving in a circular orbit is quantisied. The angular momentum can be


5.  When an electron jumps from one stationary state to another, the difference of energy (∆E) between two states (E1 and E2) is emitted or absorbed, as radiation of frequency (v) given by the equation
∆E = E2- E1 = hv
If an electron jumps from higher energy state to a lower energy state, energy is emitted. Energy is absorbed by an electron when it jumps from a lower energy state to a higher energy state.
It is obvious that the electron cannot radiate energy if no energy level is available. That is why atoms do not collapse.
From Bohr model, one can calculate the energy En of an electron in an orbit n. This is given by the expression,



Further, one can also calculate the radius of each circular orbit from the expression
rn = 0.529Å x n2 where n= 1,2,3......
The radius of the first orbit r1, called Bohr’s radius (n = 1) is 0.529A (or 52.9 pm).
Bohr model is also applicable to ions such as He+, Li2+ etc. For such cases,


                             
and                      
where Z is the atomic number and has values of 2 and 3 for He+ and Li2+ respectively.

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