Vidyarthi Academy

Results of Cathode ray experiment

1. The cathode rays start from cathode and move towards the anode.


2. These rays themselves are not visible but their behaviour can be observed with the help of certain kind of materials (fluorescent or phosphorescent) which glow when hit by them.


3. In the absence of electrical or magnetic field, these rays travel in straight lines


4. In the presence of electrical or magnetic field, the behaviour of cathode rays is similar to that expected from negatively charged particles, suggesting that the cathode rays consist of negatively charged particles, called electrons.


5. The characteristics of cathode rays (electrons) do not depend upon the material of electrodes and the nature of the gas present in the cathode ray tube. Thus, we can conclude that electrons are basic constituent of all the atoms.

Charge to Mass (e/m) Ratio of Electron

Thomson measured the ratio of electrical charge (e) to the mass of electron (m_e) by using cathode ray tube and applying electrical and magnetic field perpendicular to each other as well as to the path of electrons Deviation of the particles from their path in the presence of electrical or magnetic field depends upon,

1. The magnitude of the negative charge on the particle, greater the magnitude of the charge on the particle, greater is the interaction with the electric or magnetic field and thus greater is the deflection.


2. The mass of the particle - lighter the particle, greater the deflection.


3. The strength of the electrical or magnetic field - stronger the electrical or magnetic field, greater will be the deflection.

• When only electric field is applied, the electrons deviate from their path and hit the cathode ray tube at point A.


• When only magnetic field is applied, electron strikes the cathode ray tube at point C.


• We can balance the electrical and magnetic field strength, so that the electron strikes the original position, that is, at point B.


• By balancing the different type of forces, it was found that,

  e/m_e = 1.758820 × 10¹¹ C kg^–1

Where m_e is the mass of the electron in kg and e is the magnitude of the charge on the electron in coulomb (C).

Millikan’s Oil Drop Method and Charge on the Electron

Oil droplets in the form of mist, produced by the atomiser, were allowed to enter through a tiny hole in the upper plate of electrical condenser. The air inside the chamber was ionized by passing a beam of X-rays through it. The electrical charge on these oil droplets was acquired by collisions with gaseous ions.

The fall of these charged oil droplets can be retarded, accelerated or made stationary depending upon the charge on the droplets and the polarity and strength of the voltage applied to the plate.

In the chamber, the forces acting on oil drop are gravitational force, electrostatic force due to electrical field and a viscous drag force when the oil drop is moving.

It was concluded that the magnitude of electrical charge, q, on the droplets is always an integral multiple of the electrical charge, e, that is, q = n e, where n = 1, 2, 3... .

He found the charge on the electron to be e = –1.6 × 10^–19 C.

The mass of the electron (m_e) was determined by combining these results with Thomson’s value of e/m_e ratio.

$${\mathrm{m}}_{\mathrm{e}}=\frac{\mathrm{e}}{\mathrm{e}/{\mathrm{m}}_{\mathrm{e}}}=\mathrm{ }9.1094\times {10}^{–31}\mathrm{ }\mathrm{k}\mathrm{g}\mathrm{ }$$

Canal Rays and Discovery of Protons

Electrical discharge carried out in the modified cathode ray tube, containing gas, led to the discovery of particles carrying positive charge, also known as canal rays.

Characteristics of positively charged particles

1. The positively charged particles depend upon the nature of gas present in the cathode ray tube. These are simply the positively charged gaseous ions.


2. The charge to mass ratio of the particles is found to depend on the gas from which these originate.


3. Some of the positively charged particles carry a multiple of the fundamental unit of electrical charge.


4. The behaviour of these particles in the magnetic or electrical field is opposite to that observed for electron or cathode rays.

The smallest and lightest positive ion was obtained from hydrogen and was called proton.

Neutrons were discovered by James Chadwick by bombarding a thin sheet of beryllium by α- particles. They are electrically neutral particles having a mass slightly greater than that of the protons.

Some Terms Related to Atom

Nucleons: The protons and neutrons present in the nucleus are collectively known as nucleons.

Atomic number (Z) = number of protons in the nucleus

= number of electrons in a neutral atom

The total number of nucleons is termed as mass number (A) of the atom.

Mass number (A) = number of protons (Z) + number of neutrons (n)

Isotopes: The atoms of the same element having the same atomic number but different mass number, e.g.,

${}_1{}^1\mathrm{H}\mathrm{ }\left(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{u}\mathrm{m}\right),\mathrm{ }{}_1{}^2\mathrm{D}\mathrm{ }\left(\mathrm{D}\mathrm{e}\mathrm{u}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{m}\right),\mathrm{ }{}_1{}^3\mathrm{T}\mathrm{ }\left(\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}\right)$.

Chemical properties of atoms are controlled by the number of electrons, which are determined by the number of protons in the nucleus. Therefore, all the isotopes of a given element show same chemical behaviour.

Isobars: Isobars are the atoms of different elements having the same mass number but different atomic number, e.g.,

${}_{18}{}^{40}\mathrm{A}\mathrm{r}\mathrm{ }\left(\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\right),\mathrm{ }{}_{\mathrm{ }20}{}^{40}\mathrm{C}\mathrm{a}\mathrm{ }\left(\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{i}\mathrm{u}\mathrm{m}\right)$.

Isotones: Isotones are the atoms of different elements having the same number of neutrons, e.g.,

${}_6{}^{14}\mathrm{C}\mathrm{ },\mathrm{ }{}_{\mathrm{ }7}{}^{15}\mathrm{N}$, etc.

Isoelectronic species: The atoms or ions, which have the same number of electrons. For example, Ne, Na⁺, F^–, Al³⁺, Mg²⁺, etc.

Atomic Models

Atomic models are structures showing arrangement of different sub atomic particles within the atom.

Thomson’s Model or Water Melon Model or Plum Pudding Model

This model was proposed by Joseph James Thomson in 1897. This model is also known as apple pie model.

• An atom is a positively charged sphere and the electrons are embedded in it. The mass of the atom is assumed to be uniformly distributed over the atom.


• The negative and positive charges are equal in magnitude. So, the atom as a whole is electrically neutral.

Merits of Thomson’s Model -Thomson’s model explained that atoms are electrically neutral.

Shortcomings - It could not explain the results of α (alpha) particle scattering experiment carried out by Rutherford.

Rutherford’s Model or Planetary Model

This model was given by Ernest Rutherford in 1911, based on his α particle scattering experiment

Alpha(α) particle scattering experiment

In this experiment, fast moving alpha (α)-particles were made to strike a thin gold foil.

• He selected a gold foil because he wanted as thin a layer as possible. This gold foil was about 1000 atoms thick.


• α-particles are doubly-charged helium ions. Since they have a mass of 4 u, the fast-moving α-particles have a considerable amount of energy.


• It was expected that α-particles would be deflected by the sub-atomic particles in the gold atoms. Since the α-particles were much heavier than the protons, he did not expect to see large deflections.

Observations

• Most of the fast moving α-particles passed straight through the gold foil.


• Some of the α-particles were deflected by the foil by small angles and a very small number by large angles.


• One out of every 12000 particles appeared to rebound

Conclusions

1. Most of the space inside the atom is empty because most of the α-particles passed through the gold foil without getting deflected.


2. Very few particles were deflected from their path, indicating that the positive charge of the atom occupies very little space.


3. A very small fraction of α-particles were deflected by 180⁰, indicating that all the positive charge and mass of the gold atom were concentrated in a very small volume within the atom.

Rutherford’s Model

1. All the positively charged particles are present in a small space in the centre of the atom. This small space is called nucleus.


2. Electrons (negatively charged) revolve around the nucleus in circular orbits with a high speed.


3. The size of the nucleus is very small compared to the size of atom.

Drawbacks of Rutherford’s Model

1. It is possible to have infinite number of orbits. In practice it is not the case.


2. The electron moving with acceleration must continuously lose energy and eventually will fall into the nucleus, which is contrary to the fact.


3. Rutherford model says nothing about the electronic structure of atoms i.e., how the electrons are distributed around the nucleus and what are the energies of these electrons.

• Why is the lead plate used in Rutherford’s alpha particle scattering experiment?

  Lead has a very high density and alpha particles cannot penetrate through it (lead).This helps to concentrate the beam of alpha particles towards the gold foil. Since the alpha particles coming out are harmful to human health (they are carcinogenic), the source of alpha particles is kept inside the lead box.

Developments Leading to the Bohr’s Model of Atom

1. Dual character of the electromagnetic radiation which means that radiations possess both wave like and particle like properties,


2. Experimental results regarding atomic spectra which can be explained only by assuming quantized electronic energy levels in atoms.

1. Wave Nature of Electromagnetic Radiation

   When electrically charged particle moves under acceleration, alternating electrical and magnetic fields are produced and transmitted. These fields are transmitted in the forms of waves called electromagnetic waves or electromagnetic radiation.

   

Properties of electromagnetic radiations

1. The oscillating electric and magnetic fields produced by oscillating charged particles are perpendicular to each other and both are perpendicular to the direction of propagation of the wave.


2. They do not require a medium to travel. That means they can even travel in vacuum.


3. Different types of electromagnetic radiations differ from one another in wavelength (or frequency). Collection of electromagnetic waves over a range of wavelengths (frequencies), is called electromagnetic spectrum. Example, Radio waves, light, xrays, gamma rays etc.


4. Wavelength - The distance between two neighbouring crests or troughs of wave. It is denoted by λ.


5. Frequency (ν) - The number of waves which pass through a particular point in one second. Or the number of oscillations per second.


6. Velocity (c) - The distance travelled by a wave in one second. In vacuum all types of electromagnetic radiations travel with the same velocity. Its value is 3 ×10⁸ m sec^-1. This is also the velocity of light, c.

   c = νλ



7. Wave number- Wave number,$\mathrm{ }\bar{\mathrm{\nu }}\mathrm{ }$is defined as the number of wavelengths per unit length. The SI unit is m^–1. However commonly used unit is cm^–1.

$$\bar{\mathrm{\nu }}=\frac{1}{\mathrm{\lambda }}$$

Particle Nature of Electromagnetic Radiation: Planck’s Quantum Theory

1. Some of the observations which could not be explained with the help of even the electromagnetic theory (wave theory)


2. The nature of emission of radiation from hot bodies (black body radiation)


3. Ejection of electrons from metal surface when radiation strikes it (photoelectric effect)


4. Variation of heat capacity of solids as a function of temperature


5. Line spectra of atoms with special reference to hydrogen

Quantum Theory

• The radiant energy is emitted or absorbed not continuously but discretely in the form of small discrete packets of energy called ‘quanta’ (plural for quantum). In case of light , the quantum of energy is called a ‘photon’


• The energy of each quantum is directly proportional to the frequency of the radiation, i.e.

  E ∝ ν     or E = hν,

  where h = Planck’s constant = 6.626 x 10^-27 Js



• Energy is always emitted or absorbed as integral multiple of this quantum.

  E = n hν, where n=1, 2, 3, 4, .....

  1. Black body radiation

     An ideal body, which emits and absorbs all frequencies, is called a black body. The radiation emitted by such a body is called black body radiation.

     • The blackbody spectrum depends only on the temperature of the object, and not on what it is made of. An iron horseshoe, a ceramic vase, and a piece of charcoal - all emit the same blackbody spectrum if their temperatures are the same. As the temperature of an object increases, it emits more energy at all wavelengths.
     
     
     • As the temperature of an object increases, the peak wavelength of the blackbody spectrum becomes shorter (bluer). For example, blue stars are hotter than red stars.
     
     
     • The black body spectrum always becomes small at the left-hand side (the short wavelength, high frequency side).

     

• According to classical physics each frequency of vibration should have the same energy. Since there is no limit to how great the frequency can be, there is no limit to the energy of the vibrating electrons at high frequencies. This means that there should be no limit to the energy of the light produced by the electrons vibrating at high frequencies. However experimentally, the blackbody spectrum always becomes small at the left-hand side (short wavelength, high frequency).


• Planck said that an electron vibrating with a frequency ν could only have  an energy of 1hν, 2hν, 3hν, 4hν, ...; that is, energy of vibrating electron = (any integer) x hν. But the electron has to have at least one quantum of  energy if it is going to vibrate. If it doesn't have at least energy of 1hν, it will not vibrate at all and can't produce any light. Or at high frequencies the amount of energy in a quantum, hν, is so large that the high-frequency vibrations can never vibrate. This is why the blackbody spectrum always becomes small at the left-hand (high frequency) side.

Photoelectric effect

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

Observations

• When beam of light falls on a metal surface electrons are ejected immediately.


• Number of electrons ejected is proportional to intensity or brightness of light


• Threshold frequency (ν_o) - For each metal there is a characteristic minimum frequency below which photoelectric effect is not observed. This is called threshold frequency.


• If frequency of light is less than the threshold frequency there is no ejection of electrons no matter how long it falls on surface or how high is its intensity.


• According to wave theory, small energies from the incident radiation should accumulate over time and we should be able to observe the photo electric effect as soon as the energy is sufficient for ejection of electron. However this is not the case as observed above.


• Work function (W_o) - The minimum energy required to eject electrons from the surface of a metal, is called photoelectric work function.

  W_o= hν_o.

  It is characteristic property of a metal.



• Kinetic energy of the ejected electrons = Energy of photon – Work function

  ½ m_ev² = hν - hν_o



• Total amount of energy absorbed or emitted, for n photons, will be,

  $$\mathrm{E}\mathrm{ }=\mathrm{ }\mathrm{n}\mathrm{ }\mathrm{h}\mathrm{ }\mathrm{\nu }\mathrm{ }=\mathrm{ }\mathrm{n}\mathrm{h}\frac{\mathrm{c}}{\mathrm{\lambda }}$$



• The amount of energy possessed by 1 mole of photons is called 1 Einstein of energy, and is given by,

  $$\mathrm{E}\mathrm{ }=\mathrm{ }{\mathrm{N}}_{\mathrm{A}}\mathrm{ }\mathrm{h}\mathrm{ }\mathrm{\nu }\mathrm{ }=\mathrm{ }{\mathrm{N}}_{\mathrm{A}}\mathrm{h}\frac{\mathrm{c}}{\mathrm{\lambda }}$$

Spectrum

When a white light is passed through a prism, it splits into a series of coloured bands known as spectrum.

• Continuous spectrum - The spectrum which consists of all the wavelengths is called continuous spectrum.


• Line spectrum - A spectrum in which only specific wavelengths are present is known as a line spectrum. It has bright lines with dark spaces between them.


• Electromagnetic spectrum is a continuous spectrum. It consists of a range of electromagnetic radiations arranged in the order of increasing wavelengths or decreasing frequencies. It extends from radio waves to gamma rays.


• 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 absorbed radiation are said to be “excited” or move to higher energy state. When they come to lower energy states from the excited state, they emit radiation.


• Absorption spectrum - The spectrum obtained when radiation is passed through a sample of material. The sample absorbs radiation of certain wavelengths. The wavelengths which are absorbed are missing and come as dark lines.


• The study of emission or absorption spectra is referred as spectroscopy.


• The emission spectra of atoms in the gas phase, on the other hand, do not show a continuous spread of wavelength, rather they emit light only at specific wavelengths with dark spaces between them. Such spectra are called line spectra or atomic spectra.


• Line spectrum of element is unique and there is regularity in the line spectrum of each element.

Rydberg’s Formula - Wave number of emitted radiation, when an electron falls from energy level n₂ to n₁, is given by

$$\mathit{}\bar{\mathrm{\nu }}=109,677\mathrm{ }\left(\frac{1}{{{\mathrm{n}}_1}^2}-\frac{1}{{{\mathrm{n}}_2}^2}\right)$$

Where n₁ = 1, 2........, n₂ = n₁ + 1, n₁ + 2......,

and R = Rydberg’s constant = 109677 cm^-1

Spectral Lines for Atomic Hydrogen

• Balmer had discovered the visible light spectrum (corresponding to n₁ = 2) and it was named after him.


• Rydberg later on generalized the concept. Different series of spectrum are named after their discoverers.

Bohr’s Atomic Model

This model was given by Neils Bohr in 1913.

According to this model,

1. The electron in the hydrogen atom can move around the nucleus in a circular path of fixed radius and energy, known as orbits, stationary states or allowed energy states. These orbits are arranged concentrically around the nucleus.


2. The energy of an electron in the orbit does not change with time. While revolving in discrete orbits the electrons do not radiate energy.
   • The electron will move from a lower stationary state to a higher stationary state when required amount of energy is absorbed by the electron.
   
   
   • Energy is emitted when electron moves from higher stationary state to lower stationary state.
   
   
   • The energy change does not take place in a continuous manner.


3. The frequency of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ΔE, is given by ,

   $$\mathrm{\nu }=\frac{\mathrm{\Delta }\mathrm{E}}{\mathrm{h}}=\frac{{\mathrm{E}}_2-\mathrm{ }{\mathrm{E}}_1}{\mathrm{h}}\mathit{}\left(\mathrm{B}\mathrm{o}\mathrm{h}\mathrm{r}\mathrm{’}\mathrm{s}\mathrm{ }\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{ }\mathrm{r}\mathrm{u}\mathrm{l}\mathrm{e}\right)\mathit{}$$

   Where E₁ and E₂ are the energies of the lower and higher allowed energy states respectively..



4. The angular momentum of an electron in a given stationary state can be expressed as

   $${\mathrm{m}}_{\mathrm{e}}\mathrm{v}\mathrm{ }\mathrm{r}=\frac{\mathrm{n}\mathrm{h}}{2\mathrm{\pi }},\mathrm{ }\mathrm{ }\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{n}=\mathrm{1,2},3\dots \mathrm{ }$$

• The stationary states for electron are numbered n = 1, 2, 3... These integral numbers are called Principal quantum numbers.


• The radii of the nth stationary state is expressed as r_n = n² a₀, where a₀ = radius of the first stationary state = 52.9 pm. This is called the Bohr radius.


• The energy of stationary state is given by

  $${\mathrm{E}}_{\mathrm{n}}\mathrm{ }=\mathrm{ }-\frac{{\mathrm{R}}_{\mathrm{H}}}{{\mathrm{n}}^2},\mathrm{ }\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{n}\mathrm{ }=\mathrm{ }1,\mathrm{ }2,\mathrm{ }3....\mathrm{ }$$

  And, R_H = h c R

  The value of R_H is 2.18×10^-18 J.

  The energy of the lowest state for n = 1, called the ground state, is

  E₁ = –2.18×10^–18 J

  $$\mathrm{A}\mathrm{n}\mathrm{d}\mathrm{ }{\mathrm{E}}_2\mathrm{ }=\mathrm{ }–\frac{2.18\times {10}^{-18}}{4}\mathrm{ }\mathrm{J}\mathrm{ }=\mathrm{ }–0.545\times {10}^{-18}\mathrm{ }\mathrm{J}$$



• What does the negative electronic energy (E_n) for hydrogen atom mean?

  The negative sign of hydrogen atom means that the energy of the electron in the atom is lower than the energy of a free electron at rest. A free electron at rest is an electron that is infinitely far away from the nucleus and is assigned the energy value of zero. E_∞=0. As the electron gets closer to the nucleus (as n decreases), E_n becomes larger in absolute value and more and more negative. The most negative energy value is given by n=1 which corresponds to the most stable orbit, called the ground state.



• Energies of the stationary states associated with hydrogen like species (having one electron only), for example, He⁺ Li²⁺, Be³⁺ and so on, are given by

  $${\mathrm{E}}_{\mathrm{n}}\mathrm{ }=\mathrm{ }\frac{-{\mathrm{R}}_{\mathrm{H}}{\mathrm{Z}}^2\mathrm{ }}{{\mathrm{n}}^2}\mathrm{ }\mathrm{J}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{r}}_{\mathrm{n}}\mathrm{ }=\mathrm{ }\frac{{\mathrm{a}}_0\times {\mathrm{Z}}^2}{\mathrm{n}}$$



• Radius of orbit of electron is given by

  $$\mathrm{r}=\frac{4\mathrm{\pi }{\mathrm{\epsilon }}_0{\mathrm{n}}^2{\mathrm{h}}^2}{4{\mathrm{\pi }}^2{\mathrm{m}}_{\mathrm{e}}\mathrm{Z}{\mathrm{e}}^2}=\frac{{\mathrm{\epsilon }}_0{\mathrm{n}}^2{\mathrm{h}}^2}{\mathrm{\pi }{\mathrm{m}}_{\mathrm{e}}\mathrm{Z}{\mathrm{e}}^2}$$

  where,

  n = principle quantum number,

  h = Planck’s constant,

  m_e = mass of an electron,

  Z = atomic number and

  e = electronic charge.



• Velocity of electron in any orbit is given by

  $$\mathrm{v}=\frac{{\mathrm{e}}^2\mathrm{Z}}{2{\mathrm{\epsilon }}_0\mathrm{n}\mathrm{h}}=\frac{\mathrm{c}}{137}.\frac{\mathrm{Z}}{\mathrm{n}}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\Rightarrow \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{v}\mathrm{ }\propto \mathrm{ }\frac{\mathrm{Z}}{\mathrm{n}}$$

  • Frequency of electron in any orbit is given by
    $$\mathrm{f}=\frac{\mathrm{v}}{2\mathrm{\pi }\mathrm{r}}=\frac{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}{4{{\mathrm{\epsilon }}_0}^2{\mathrm{n}}^3{\mathrm{h}}^3}$$
  
  
  • Time Period

    $$\mathrm{T}=\frac{1}{\mathrm{f}}=\frac{4{{\mathrm{\epsilon }}_0}^2{\mathrm{n}}^3{\mathrm{h}}^3}{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}=\mathrm{ }\frac{{\left(4\mathrm{\pi }{\mathrm{\epsilon }}_0\right)}^2{\mathrm{n}}^3{\mathrm{h}}^3}{4{\mathrm{\pi }}^2{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}$$

  
  
  • Kinetic Energy

    $$\mathrm{K}\mathrm{ }=\frac{1}{2}{\mathrm{m}}_{\mathrm{e}}{\mathrm{v}}^2=\frac{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}{8{{\mathrm{\epsilon }}_0}^2{\mathrm{n}}^2{\mathrm{h}}^2}$$

  
  
  • Potential Energy

    $$\mathrm{U}\mathrm{ }=-\frac{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}{4{{\mathrm{\epsilon }}_0}^2{\mathrm{n}}^2{\mathrm{h}}^2}$$

  
  
  • Total Energy

    $${\mathrm{E}}_{\mathrm{n}}=\mathrm{K}+\mathrm{U}\mathrm{ }=-\frac{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4{\mathrm{Z}}^2}{8{{\mathrm{\epsilon }}_0}^2{\mathrm{n}}^2{\mathrm{h}}^2}$$

    $${\mathrm{E}}_{\mathrm{n}}=-13.6\frac{{\mathrm{Z}}^2}{{\mathrm{n}}^2}\mathrm{ }\mathrm{e}\mathrm{V}$$

    E₁ = -13.6 eV, E₂ = -3.4 eV, E₃ = -1.5 eV, E_∞ = 0

  
  
  • Wave number

    $$\frac{1}{\mathrm{\lambda }}=\mathrm{R}{\mathrm{Z}}^2\left(\frac{1}{{{\mathrm{n}}_2}^2}-\frac{1}{{{\mathrm{n}}_1}^2}\right)\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{R}=\frac{{\mathrm{m}}_{\mathrm{e}}{\mathrm{e}}^4}{8{{\mathrm{\epsilon }}_0}^2{\mathrm{h}}^3\mathrm{c}}=1.09678\mathrm{ }\mathrm{X}{\mathrm{ }10}^7\mathrm{ }{\mathrm{m}}^{-1}$$

Explanation of Line Spectrum for Hydrogen

• The absorption or emission of radiation energy occurs only when an electron jumps from one permitted orbit to another permitted orbit. Energy of emitted photon is,

  $$\mathrm{\Delta }\mathrm{E}=\mathrm{h}\mathrm{\nu }={\mathrm{E}}_2\mathrm{ }–\mathrm{ }{\mathrm{E}}_1=-{\mathrm{R}}_{\mathrm{H}}\left(\frac{1}{{{\mathrm{n}}_2}^2}-\frac{1}{{{\mathrm{n}}_1}^2}\right)$$

  Where E₁ and E₂ are energies of electron in orbits n₁ and n₂.



• In case of absorption spectrum, n₂ > n₁ and the term in the parenthesis is positive and energy is absorbed. On the other hand in case of emission spectrum n₁ > n₂, Δ E is negative and energy is released.


• Further, each spectral line, whether in absorption or emission spectrum, can be associated to the particular transition in hydrogen atom. In case of large number of hydrogen atoms, different possible transitions can be observed and thus leading to large number of spectral lines. The brightness or intensity of spectral lines depends upon the number of photons of same wavelength or frequency absorbed or emitted.

Limitations of Bohr’s Model of Atom

1. It fails to account for the finer details (like doublet, that is two closely spaced lines) of the hydrogen atom spectrum observed by using sophisticated spectroscopic techniques.

   [Doublets occur due to presence of two electrons with different spin orientation in each orbital]



2. This model is also unable to explain the spectrum of atoms other than hydrogen, for example, helium atom which possesses only two electrons.


3. Bohr’s theory was also unable to explain the splitting of spectral lines in the presence of magnetic field (Zeeman effect) or an electric field (Stark effect).


4. It could not explain the ability of atoms to form molecules by chemical bonds.