The origin of spectral lines in the hydrogen atom (Hydrogen Spectrum) can be explained on the basis of Bohr’s theory. The hydrogen atom is said to be stable when the electron present in it revolves around the nucleus in the first orbit having the principal quantum number n = 1. This orbit is called the ground state.

The electron gains energy from the surrounding and jumps into a higher orbit with principal quantum number n = 2, 3, 4, 5, ….. These higher orbits are called excited states.  When electrons start revolving in the excited state the atom becomes unstable. To acquire stability the electron jumps from the higher orbit to lower orbit by the emission of the energy of value hν. Where ν is the frequency of radiation energy or radiation photon. This radiation is emitted in the form of spectral lines.

The Expression for the Wavelength of a line in the Hydrogen Spectrum:

Let E_n and E_p be the energies of an electron in the n^th and p^th orbits respectively (n > p) So when an electron takes a jump from the n^th orbit to the p^th orbit energy will be radiated in the form of a photon or quantum such that

E_n –  E_p = hν  ………… (1)

where ν is the frequency of radiation, h = Planck’s constant

This formula is called Bohr’s formula of spectral lines.

The wavelength λ obtained is characteristic wavelength due to jumping of the electron from n^th orbit to p^th orbit. We get different series of spectral lines due to the transition of the electron from different outer orbits to fixed inner orbit.

Energy Level Diagram for Hydrogen Atom:

Energy level diagrams indicate us the different series of lines observed in a spectrum of the hydrogen atom. The horizontal lines of the diagram indicate different energy levels. The vertical lines indicate the transition of an electron from a higher energy level to a lower energy level.

It is very important that as indicated in the diagram each transition corresponds to a definite characteristic wavelength. Thus different transitions give different series of lines. Different Series obtained are a) Lyman series, b)  Balmer series, c)  Paschen series, d)  Brackett series,  e)  Pfund series and f) Henry series

Different Series in Hydrogen Spectrum:

Lyman Series:

If the transition of electron takes place from any higher orbit (principal quantum number =  2, 3, 4,…….) to the first orbit (principal quantum number  = 1). We get a Lyman series of the hydrogen atom. It is obtained in the ultraviolet region.

This formula gives a wavelength of lines in the Lyman series of the hydrogen spectrum. Different lines of Lyman series are

• α line of Lyman series  p = 1 and n = 2
• α line of Lyman series  p = 1 and n = 3
• γ line of Lyman series  p = 1 and n = 4
• the longest line of Lyman series  p = 1 and n = 2
• the shortest line of Lyman series p = 1 and n = ∞

Balmer Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 3, 4, 5, …) to the second orbit (principal quantum number = 2). We get Balmer series of the hydrogen atom. It is obtained in the visible region.

This formula gives a wavelength of lines in the Balmer series of the hydrogen spectrum. Different lines of Balmer series area l

• α line of Balmer series  p = 2 and n = 3
• β line of Balmer series  p = 2 and n = 4
• γ line of Balmer series  p = 2 and n = 5
• the longest line of Balmer series  p = 2 and n = 3
• the shortest line of Balmer series p = 2 and n = ∞

Paschen Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 4, 5, 6, …) to the third orbit (principal quantum number = 3). We get Paschen series of the hydrogen atom. It is obtained in the infrared region.

This formula gives a wavelength of lines in the Paschen series of the hydrogen spectrum.

Brackett Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 5, 6, 7, …) to the fourth orbit (principal quantum number = 4). We get the Brackett series of the hydrogen atom. It is obtained in the far-infrared region.

This formula gives a wavelength of lines in Brackett series of the hydrogen spectrum

Pfund Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 6,7, 8, …….) to the fifth orbit (principal quantum number = 5). We get Pfund series of the hydrogen atom. It is obtained in the far-infrared region.

This formula gives a wavelength of lines in the Pfund series of the hydrogen spectrum

Notes:  Shortest wavelength is called series limit

Continuous or Characteristic X-rays:

Characteristic x-rays are emitted from heavy elements when their electrons make transitions between the lower atomic energy levels. The characteristic x-ray emission which is shown as two sharp peaks in the illustration at left occurs when vacancies are produced in the n=1 or K-shell of the atom and electrons drop down from above to fill the gap.

The x-rays produced by transitions from the n=2 to n=1 levels are called K-alpha x-rays, and those for the n=3 to n = 1 transition are called K-beta x-rays. For a particular material, the wavelength has definite value. Hence these x rays are called continuous or characteristic X-rays. The values of energy are different for different materials.

The frequencies of the characteristic x-rays can be predicted from the Bohr model. Moseley measured the frequencies of the characteristic x-rays from a large fraction of the elements of the periodic table and produced a plot of them which is now called a “Moseley plot”.

Characteristic x-rays are used for the investigation of crystal structure by x-ray diffraction. Crystal lattice dimensions may be determined with the use of Bragg’s law in a Bragg spectrometer.

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In the last article, we have studied Rutherford’s model of an atom, its merits, and demerits. In this article, we shall study Bohr’s Model of an atom, its merits, and demerits.

To overcome the limitations of Rutherford’s model of an atom, Neil Bohr put forward his theory of atom using Planck’s quantum theory. Bohr’s theory is applicable to the hydrogen atom.

Bohr’s Atomic Theory of Hydrogen Atom:

Postulate I (Postulate of Circular Orbit):

In a hydrogen atom, the electron revolves around a circular orbit around the nucleus.  The electrostatic force of attraction between the positively charged nucleus and the negativity charged electron provide necessary centripetal force for circular motion.

The Expression For the First Postulate:

Let m be the mass of an electron revolving around the nucleus in a circular orbit of radius r with a constant speed v round the nucleus.  Let – e and + e be the charges on the electron and the nucleus respectively.

By the first postulate,

Centripetal force =  Electrostatic force

Where ε_o is the electrical permittivity of free space

Postulate – II (Postulate of Selected Orbit):

The electron can revolve only in a certain selected orbit in which the angular momentum of the electron is equal to an integral multiple of nh/2π, where h is the Planck’s constant.  These orbits are called stationary or permissible orbits.  The electron does not radiate energy while revolving in these orbits.

The Expression for the Second Postulate:

Let m be the mass of an electron revolving around the nucleus in a circular orbit of radius r with a constant speed v round the nucleus.

Where, n   =    1, 2, 3………..

n = Principal quantum number

h = Planck’s constant

The integer n is called the principal quantum number and it denotes the number of the orbit.

Postulate –III (Postulate of The Origin of Spectral Lines):

When an electron takes a jump from a higher energy orbit to a lower energy orbit, energy is radiated in the form of a quantum or photon of energy hν, which is equal to the difference of energies of the electron in the two orbits.

Expression for the Third Postulate:

Let E_n and E_p be the energies of an electron in the n^th and p^th orbits respectively (n > p) So when an electron takes a jump from the n^th orbit to the p^th orbit energy will be radiated in the form of a photon or quantum such that

E_n –  E_p = hν

Where ν is the frequency of radiation.

Expression for Radius of Bohr’s Orbit:

Let m be the mass of an electron revolving in a circular orbit of radius r with a constant speed  v around the nucleus.  Let – e and + e be the charges on the electron and the nucleus, respectively.

By the first postulate,

Centripetal force =  Electrostatic force

Where ε_o is the electrical permittivity of free space

From the second postulate of Bohr’s theory

From equation (1) and (2)

This is the required expression for the radius of Bohr’s orbit. Since ε_o, h, π, m, e are constant

∴  r    ∝   n²

Thus the radius of the Bohr’s orbit of an atom is directly proportional to the square of the principal quantum number.

The Expression for Velocity of Electron in Bohr’s orbit:

From the second postulate of Bohr’s theory

Where, n = 1, 2, 3………..

n = Principal quantum number

h = Planck’s constant

This is the required expression for the velocity of the electron in Bohr’s orbit of an atom. Since ε_o, h, π, e are constant

∴   v  ∝ 1 / n

Thus the velocity of the electron in Bohr’s orbit of an atom is inversely proportional to the principal quantum number.

The Expression for Angular Velocity of Electron in Bohr’s Orbit:

Now, ε_o, m, h, π, e are constant

∴ ω  ∝ 1 / n³

Thus the angular velocity of the electron in Bohr’s orbit of an atom is inversely proportional to the cube of the principal quantum number.

Notes:

• The frequency of electron in Bohr’s orbit of an atom is inversely proportional to the cube of the principal quantum number.
• The time period of the electron in Bohr’s orbit of an atom is directly proportional to the cube of the principal quantum number.
• The centripetal acceleration of electron in Bohr’s orbit of an atom is inversely proportional to the fourth power of the principal quantum number

The Expression for Energy of Electron in Bohr’s Orbit:

Let m be the mass of an electron revolving in a circular orbit of radius r with a constant speed v around the nucleus.  Let – e  and + e be the charges on the electron and the nucleus, respectively.

The  potential energy of electron having charge,  – e is given by

The total energy of the electron is given by

The total energy of electron  =  Kinetic energy of electron + Potential energy of the electron

This is the required expression for the energy of the electron in Bohr’s orbit of an atom. Since ε_o, m, h, π, e are constant

∴   E ∝  1 / n²

Thus the energy of an electron in Bohr’s orbit of an atom is inversely proportional to the square of the principal quantum number.

The negative sign indicates that the electron is bound to the nucleus by attractive force and to remove the electron from the atom energy must be supplied to the electron to overcome the attractive force. This energy is called the binding energy of the electron.

Merits of Bohr’s Model of Hydrogen Atom:

• This theory explains the spectrum of hydrogen atom completely.
• The concept of electronic configuration i.e. the distribution of electrons in different orbits was introduced.
• This theory is capable of explaining the line spectra of elements in general.
• This theory can be used to find the ionization potential of an electron in an atom.
• The value of Rydberg can be calculated using this theory.

Demerits of Bohr’s Model of Hydrogen Atom

• Though spectra of a simple atom like hydrogen is explained by Bohr’s Theory, it fails to account for elements containing
  more than one electron.
• A line in an emission spectrum splits up into a number of closely spaced lines when the atomic source of radiation is placed in
  the magnetic field. This is known as the Zeeman Effect. Bohr Theory could not explain this.
• A line in an emission spectrum splits up into a number of closely spaced lines when the atomic source of radiation is placed in an electric field, which is known as the Stark effect. Bohr Theory has no explanation for it.

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In this article, we shall study Dalton’s atomic theory, Rutherford’s model of an atom, its merits, and demerits.

Dalton’s Atomic Theory:

This theory was proposed by English chemist John Dalton in 1808. The main propositions of the theory are as follows:

• Every element is made up of extremely small particles called an atom.
• The atoms are indivisible and they can neither be created nor be destroyed.
• Atoms of the same element resemble each other in all respects but differ from the atoms of other elements.
• Atom is the smallest unit of matter which takes part in a chemical reaction.
• When chemical compounds are formed they do so by the combination of atoms of different elements in the simple proportion of whole numbers.
• Atoms of different elements may combine in more than one proportion to form different compounds.

Fundamental Particles of an Atom:

Electrons:

• Electrons were discovered by British scientist Sir J.J. Thomson.in 1897 (Cathode ray tube experiment).
• The term electron was coined by Stoney.
• The charge on the electron was found by Milikan
• Specific charge ratio or charge to mass ratio or e/m ratio is obtained by J.J. Thomson.

Protons:

• Proton was discovered by E.Goldstein.
• They are located in the nucleus.
• They are positively charged particles.

Neutrons:

• Neutron was discovered by Chadwick in 1932.
• Neutrons have no charge i.e. they are electrically neutral.
• They are located in the nucleus of an atom.

Thomson’s Model of an Atom:

Thomson proposed this model in 1903. According to him, an atom consists of protons and electrons. The total number of protons is equal to the total number of electrons. Thus the net charge of the atom is zero. An atom consists of a positively charged sphere with negatively charged electrons embedded in it. This model of an atom is called the watermelon model or plum (dot) pudding model of an atom.

Geiger Marsden Experiment:

A narrow beam of alpha particles from the radioactive source was incident on a thin gold foil. The scattering of alpha particles takes place. The scattered alpha particles were detected by a detector fixed on a stand. The deviation of alpha particles from their original path is called the scattering angle. They observed that most of the alpha particles just passed through without any deviation as if there is empty space. A few alpha particles were deflected through smaller angles. A few alpha particles deviated through larger angles. This larger deflection is possible only if alpha-particles collide with heavy and positively charged particles inside the atom because like charges only repel each other.  This massive +ve charge is at the centre of the atom and called the nucleus. Very few alpha particles were rebounded i.e. they deviated through 180°. This concludes that the nucleus is very small as compared to the volume of the atom.

Rutherford’s Model of an Aom:

From the observations of the above experiment, Rutherford put forward the concept of his atomic model. The atom consists of a centrally located positively charged nucleus. The whole mass of an atom is concentrated in the nucleus. Around the nucleus, there is empty space in which the negatively charged electrons revolve in different orbits. The total positive charge of the nucleus is equal to the total negative charge on orbiting electrons. Hence atom is electrically neutral. Rutherford’s model of an atom is also called as a planetary model of an atom.

Limitations of Rutherford’s Model of an Atom:

• The negatively charged electrons revolve around the nucleus in a circular orbit, hence they possess centripetal acceleration. According to the classical theory of electromagnetism, accelerated charge radiates energy continuously. Therefore, the electron should radiate energy while going around the nucleus losing its energy continuously. therefore, it should approach nearer the nucleus while going round emitting radiations of increasing frequency and finally falling in the nucleus. Thus it should move in a spiral path and should emit a continuous spectrum and thus structure atom is not stable. Actually, the spectrum observed is the line spectrum of definite frequency, and hence a modification to Rutherford’s atom model was necessary.
• This model of an atom fails to explain the distribution of electrons in different orbit around the nucleus. According to Rutherford’s model of an atom, the atomic spectrum should be continuous. But the atomic spectrum is found to be discontinuous. Rutherford’s model fails to explain the discontinuity of the atomic spectrum.
• This model also fails to explain the line spectra of atoms, which show discrete lines, each line corresponds to a fixed frequency.

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1.1.1.1 What is Physics?

Science word is derived from the Latin word ‘Scientia’ which means ‘to know’. Science has many disciplines, Physics being one of them. The word Physics is derived from the Greek word ‘Fusis’ meaning ‘nature’. Physics is that branch of Science which deals with the study of matter and energy or matter or motion i.e. Physics is a study of matter and energy in its different forms. In other words, physics is the study of nature and its laws. We expect that all the different events taking place in nature always take place according to some basic rules and revealing these rules of nature from the observed events in physics.

As physics is a study of nature and its behaviour it is real science. No one has been given authority to frame the rules. Sir Issac Newton, Einstein are the great physicist because using the observations available at that time, they could guess and frame the laws of physics, which explain these events and the observations in a convincing way. If a new phenomenon is observed which can not be explained using existing laws or rules we are always free to change the rules.

Knowledge of Physics overlaps with other sciences considerably, hence such overlapping gives rise to subjects like Biophysics, Biochemistry, Astrophysics, Geophysics, etc.

Physics can be conveniently divided into two parts, classical Physics (Pre-1900) and modern Physics (Post – 1900). Classical physics includes the study of mechanics, gravitation, heat, sound, light, electricity and magnetism. Modern Physics includes the study of quantum mechanics, relativity, atoms, molecules, nuclei, elementary particles, and condensed matter.

The complex physical phenomena involving wide range of length, mass. and time can be easily understood due to following reasons:

• A quantitative study of various natural phenomena shows that there is some regularity and symmetry even in the most complex phenomenon which helps us to understand it.
• All these phenomena can be explained in terms of only a few basic laws.
• Complex phenomena can be separated into simpler phenomena and by understanding these simple phenomena, the complex phenomena can be understood.

Scientific Methods:

The study of science and particularly in Physics is based on systematic observation, logical reasoning, model making, and theoretical prediction and necessary modifications. All the four steps taken together constitute what we call the ‘scientific method’. The scientific method helps us to describe the given physical phenomenon or behavior of a physical system in terms of a limited number of laws. This gives us what we call ‘theory’. The theory should be self-consistent and consistent with known experimental data. The discrepancy between the theory and experimental data has to lead to new theories in Physics.

Relation Between Physics and Mathematics:

Physics is directly related to maths because the description of nature becomes easy if we have the freedom to use mathematics. In physics, we use mathematical techniques like algebra, trigonometry, and calculus. Thus mathematics is a language of physics. Without knowledge of mathematics, it would be much more difficult to discover, understand and explain the laws of nature. But we should note that mathematics itself is not physics. To understand nature is a journey of physics, mathematics is the mean of the journey.

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1.1.1.2 Scope of Physics:

The scope of physics is broad and encompasses the study of the fundamental principles governing the natural world. Physics not only explores the properties and behaviour of matter and energy but also plays a crucial role in advancing technology, contributing to other scientific disciplines, and addressing fundamental questions about the nature of the universe. Here are key aspects of the scope of physics:

Mechanics:

Mechanics is a branch of physics, which deals with the motion of material bodies. In this branch, the forces responsible for producing or changing the motion of the body are studied. The energy involved is also studied. Newton’s laws of motion, the law of conservation of momentum and energy, Newton’s gravitation law forms the base of this branch of Physics.

Heat:

Heat is the energy that a body possesses by virtue of the motion of the molecules of which it is composed and the potential energy due to interatomic forces. The term heat is also used to indicate the energy in the process of transfer between an object and its surroundings because a difference exists between their temperatures. Thermodynamics is the name given to the branch of physics which studies the relationship between heat and mechanics.

Acoustics:

Acoustic is a branch which studies sound. Wave motion is studied in this branch.  An object in a state of vibration can set medium particles in the vibration and this disturbance in the medium can travel from one point to another. Thus sound is wave motion itself.

Optics:

Optics is a branch of science which studies electromagnetic waves to which the eye responds (light). Propagation of light means the propagation of electromagnetic waves with varying electric and magnetic fields through a vacuum or a transparent medium. It has two broad branches: geometric optics and physical optics.

Electricity and Magnetism:

These topics are interrelated with each other. We have to take the help of another topic when we are studying one of them individually. Electricity deals with the forces on charged particles, the effect of such forces. It also studies the phenomenon caused by the motion of charged particles. Magnetism can have an effect on the electric current. magnetic materials can be used in producing electric currents. Electronics is the branch of electricity.

Modern Physics:

Modern physics is the branch of physics which deals with the recent developments in the science-related to physics such as Radioactivity, X-Rays, Cathode Rays, Atomic and Molecular Structure, Quantum Theory and wave mechanics, etc.

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1.1.1.3 Pioneers of Physics

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1.1.1.4 Click Here to find the List of Noble Prize Winners in Physics

Conclusion:

Physics is a branch of science that seeks to understand the fundamental principles governing the natural world, encompassing everything from the smallest particles to the largest galaxies. It involves the study of matter, energy, space, and time, as well as the interactions between these elements. Physicists explore the fundamental laws and forces that govern the behaviour of the universe and seek to explain and predict the observed phenomena. Physics relies heavily on mathematical models and experimental observations. The scientific method is a fundamental aspect of physics, involving the formulation of hypotheses, experimentation, and the development of theories that can be tested and refined through further observations and experiments.

Related Topics:

• 1.1.2 Scientific Methods
• 1.1.3 Scientific View
• 1.1.4 Physics and Other Sciences

For More Topics in Physics Click Here

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