Chemistry Electromagnetic Radiation and its Dual Behaviour, Spectrum and Bohr's Model for `H`-atom

Topics to be covered :

`●` Dual Behaviour of EM Radiation
`●` Atomic Spectra and its Type
`●` Line Spectrum of Hydrogen
`●` Spectral Lines for Atomic Hydrogen
`●` Bohr's Model for Hydrogen Atom
`●` Bohr's Explanation of Line Spectrum of Hydrogen

Dual Behaviour of EM Radiation :

Particle nature could explain black body radiation and photoelectric effect but could not explain phenomenon of interference and
diffraction.

So, it was accepted that light process both particles and wave - like properties i.e light has dual behaviour.

Some microscopic particles like electron also exhibit wave-particle duality.

Evidence for the Quantized Electronic Energy Levels : Atomic Spectra

Some definitions are given below :

Spectrum :

The splitting of a beam of light into radiations of different wavelengths or frequencies after passing through a prism or diffraction grating is called dispersion and pattern of radiation observed after the dispersion is called spectrum.

Continuous Spectrum :

When white light is allowed to pass through a prism, it gets resolved into several colours. The spectrum is a rainbow of colours i.e. there is no sharp boundary between two colours.

Discontinuous or Line Spectrum or Atomic Spectra :

When gases or vapours of a chemical substances are heated in an electric arc or in a bunsen flame, light is emitted. If the ray of this light is passed through a prism, a line spectrum is produced. The lines are seperated from each other by dark spaces.

The pattern of lines are unique for each element. Hydrogen spectrum is an example of line spectrum.

Emission Spectrum :

The spectrum of radiation emitted by a substance that has absorbed energy is called an emission spectrum.

Atoms, molecules, or ions that have absorved radiation are said to be excited.

Absorption Spectrum :

A continuum of radiation is passed through a sample which absorbs radiation of certain wavelength. The missing wavelength which corresponds to the radiation absorbed by the matter, leave dark spaces in the bright continuous spectrum. An absorption spectrum is like the photographic negative of an emission spectrum.

Spectroscopy :

The study of emission or absorption spectra is called spectroscopy.

`=>` The spectrum of the visible light, as discussed above, was continuous as all wavelengths (red to violet) of the visible light are represented in the spectra.

`=>` The emission spectra of atoms in the gas phase, on the other hand, do not show a continuous spread of wavelength from red to violet, rather they emit light only at specific wavelengths with dark spaces between them. Such spectra are called line spectra or atomic spectra because the emitted radiation is identified by the appearance of bright lines in the spectra

`

Line Spectrum of Hydrogen :

When an electron discharge is passed through hydrogen gas at low pressure, a bluish light is emmited. When a ray of this light is passed through a prism, a discontinuous line spectrum of several isolated sharp line is obtained. The hydrogen atom consists of several lines and these lines lie is visible, UV and IR regions.

Wavelength of Spectrum Line :

It is given by

`1/lamda = RZ^2 [ (1/n_1^2 - 1/n_2^2)]`

where R = Rydberg constant `= 109678 cm^(-1)`

`n_1 = 1, 2, 3 ........`

`n_2 = (n_1 +1), (n_1 + 2), (n_1 + 3).....................`

The Spectral Lines for Atomic Hydrogen :

The first five series of line that correspond to `n_1 = 1, 2, 3,4, 5` are given in Table `2.3` and the Lyman, Balmer and Paschen series of transition for H - atom is show in Fig.2.11.

Development which leads to the Bohr's Model of Atom :

(i) Dual character of the electromagnetic radiation
(ii) Experimental results regarding atomic spectra

Bohr's model for Hydrogen Atom :

Neils Bohr (1913) explained quantitatively the general feature of H-atom structure and its spectrum.

Postulates of Bohr's Model :

(i) The atom has a nucleus which contains protons and neutrons.

(ii) The electron in the H-atom revolve around the nucleus in a circular path of fixed radius and energy called orbits.

Force of attraction towards nucleus = centrifugal force

(iii)The electron can revolve only on those orbits whose angular momentum is an integral multiple of `h/(2 pi)` i.e.

`mvr = n h/(2 pi)`

where ` m =` mass of electron
`v` = velocity of electron
`r` = radius of the orbit
`n = 1,2,3.....` number of the orbit

Angular momentum can have quantized value only. These quantized circular orbits are called stationary orbit.

(iv) The energy of electron in the orbit does not change with time. Such a state is called ground or normal state.

In ground state, potential energy of electron is minimum.

(v) Each stationary orbit is associated with a definite amount of energy.

Greater the distance of the orbit from the nucleus, more shall be the energy associated with it.

(vi) The emission or absorption of energy in the from of radiation occurs only when an electrons jumps from one
stationary orbit to another.

` Delta E = E _text(high) - E_text(low) = h nu`

`nu= (Delta E)/h = ( E_("high")- E_("low"))/h`

This expression is commonly known as Bohr's frequency rule.

Important point from Bohr's theory regrading H-atom :

(i) The stationary states numbered ` n = 1,2 ,3 ........`

These integral numbers are known as principle quantum number

(ii) The radii of the stationary states are given as

`r_n = n^2a_0`

where `a_0 = 52.9` pm

`a_0 ` is also called Bohr orbit for first stationary state.

(iii) The energy of stationary state is given as

` E_n = - R_H ( 1/n^2)`, `n = 1, 2, 3,....`

where ` R_H =` Rydberg constant

and ` R_H = 2.18 xx 10^(-18) J`.

`ast` Why `E_n` is negative?

This mean that the energy of the electron in the atom is lower than the free electron at rest. The most negative energy value is given by ` n = 1` which correspond to the most stable orbit and is called ground state .

(iv) When the electron is free from the influence of nucleus, the energy is taken as zero. The electron in this situation is associated with the stationary state of Principal Quantum number `= n = oo` and is called as ionized hydrogen atom. When the electron is attracted by the nucleus and is present in orbit `n`, the energy is emitted and its energy is lowered. That is the reason for the presence of negative sign in equation and depicts its stability relative to the reference state of zero energy and `n = oo`.

(v) Bohr's theory can be applied to species containing one electron only.

e.g . ` He^+ , Li^(2 +) , Be^(3 +)` etc.

The energy expression is given as

`E_n = - 2.18 xx 10^(-18) ( Z^2/n^2) J`

and radii is given as

` r_n = 52.9 ( n^2/Z) ` pm

where ` Z = ` atomic number

(vi) The velocity of electrons moving in the orbit can also be calculated.

Q 2682234137

Calculate the energy associated with the first orbit of `He^+ `. What is the radius of this orbit?



Solution:

`E_n = - ((2.18xx10^(-18) J)Z^2)/n^2 J (text(atom))^(-1)`

For `He^(+), n = 1, Z = 2`

`E_1 = -((2.18xx10^(-18) J)(2^2))/1^2 = -8.72xx10^(-18)J`

The radius of the orbit is given by

`r_n = ((0.0529nm)n^2)/Z`

Since `n = 1`, and `Z = 2`

`r_n = ((0.0529nm)1^2)/2 = 0.02645 nm`

Explanation of Line Spectrum of Hydrogen :

Hydrogen spectrum can be explained quantitatively using Bohr's model. The energy gap between the two orbit is given as

`Delta E = E_f - E_i`

Using equation , ` E_n = ( (-R_H)/n^2 )`

` Delta E = ( (-R_H)/n_f^2 ) - ( - (R_H)/n_i^2)`

` n_i =` initial orbit

` n_f =` final orbit

` Delta E = R_H ( 1/n_i^2 - 1/n_f^2 ) = 2.18 xx 10^(-18) ( 1/n_i^2 - 1/n_f^2 ) J`

Now ` Delta E = h nu`

So , ` nu = ( Delta E)/h = R_H/h (1/n_i^2 - 1/n_f^2)`

` nu = 3.29 xx 10^(15) (1/n_i^2 - 1/n_f^2) Hz`

In terms of wave number `(bar nu)`

` bar nu = nu/c = R_H/(hc) ( 1/n_i^2 - 1/n_f^2 )`

` = 1.09677 xx 10^7 ( 1/n_i^2 - 1/n_f^2 ) m^(-1)`

Case I : Absorption Spectrum

when `n_f > n_i`

So, `Delta E = +ve`

Case II : Emission Spectrum

when `n_f < n_i`

So, ` Delta E = -ve`

Each spectral line, whether in absorption or emisison spectrum, can be associated to the particular transition in hydrogen atom. The intensity or brightness of spectral lines depends on the number of photons of same wavelength or frequency absorbed or emitted.
Q 2652234134

What are the frequency and wavelength of a photon emitted during a transition from n = 5 state to the n = 2 state in the hydrogen atom?

Solution:

Since `n_i = 5` and `n_f = 2,` this transition gives rise to a spectral line in the visible region of the Balmer series.

`DeltaE = 2.18xx10^(-18)J [1/5^2-1/2^2]`

`= -4.58xx10^(-19)J`
It is an emission energy
The frequency of the photon (taking energy in terms of magnitude) is given by

`nu = (DeltaE)/h`

` = (4.58xx10^(-19) J)/(6.626xx10^(-34) Js)`

` = 6.91xx10^(14) Hz`

`lamda = c/nu = (3.00xx10^8 ms^(-1))/(6.91xx10^(14)Hz) =434nm`

Limitations of Bohr's Model :

(i) It does not explain the spectra of multi - electron atoms.

(ii) It fails to account for the finer details (doublet i.e. two closed spaced lines) of the H-atom spectrum observed by advanced spectroscopic technology.

(iii) It does not explain the splitting of spectral lines into a group of finer lines under the influence of magnetic field (Zeeman effect) and elecrtic field (Stark effect).

(iv) Bohr theory is not in agreement with Heisenberg's uncertainty principle.

(v) Spectrum of isotpes of hydrogen was expected to be same by bohr but experimentally found to be different. It was due to the non-stationary nucleus.

(vi) quantisation of angular momentum could not be justified i.e. why ` mvr = nh//2pi`.

 
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