What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

Bohr atom

Unlike planets orbiting the Sun, electrons cannot be at any arbitrary distance from the nucleus; they can exist only in certain specific locations called allowed orbits. This property, first explained by Danish physicist Niels Bohr in 1913, is another result of quantum mechanics—specifically, the requirement that the angular momentum of an electron in orbit, like everything else in the quantum world, come in discrete bundles called quanta.

In the Bohr atom electrons can be found only in allowed orbits, and these allowed orbits are at different energies. The orbits are analogous to a set of stairs in which the gravitational potential energy is different for each step and in which a ball can be found on any step but never in between.

The laws of quantum mechanics describe the process by which electrons can move from one allowed orbit, or energy level, to another. As with many processes in the quantum world, this process is impossible to visualize. An electron disappears from the orbit in which it is located and reappears in its new location without ever appearing any place in between. This process is called a quantum leap or quantum jump, and it has no analog in the macroscopic world.

Because different orbits have different energies, whenever a quantum leap occurs, the energy possessed by the electron will be different after the jump. For example, if an electron jumps from a higher to a lower energy level, the lost energy will have to go somewhere and in fact will be emitted by the atom in a bundle of electromagnetic radiation. This bundle is known as a photon, and this emission of photons with a change of energy levels is the process by which atoms emit light. See also laser.

In the same way, if energy is added to an atom, an electron can use that energy to make a quantum leap from a lower to a higher orbit. This energy can be supplied in many ways. One common way is for the atom to absorb a photon of just the right frequency. For example, when white light is shone on an atom, it selectively absorbs those frequencies corresponding to the energy differences between allowed orbits.

Each element has a unique set of energy levels, and so the frequencies at which it absorbs and emits light act as a kind of fingerprint, identifying the particular element. This property of atoms has given rise to spectroscopy, a science devoted to identifying atoms and molecules by the kind of radiation they emit or absorb.

This picture of the atom, with electrons moving up and down between allowed orbits, accompanied by the absorption or emission of energy, contains the essential features of the Bohr atomic model, for which Bohr received the Nobel Prize for Physics in 1922. His basic model does not work well in explaining the details of the structure of atoms more complicated than hydrogen, however. This requires the introduction of quantum mechanics. In quantum mechanics each orbiting electron is represented by a mathematical expression known as a wave function—something like a vibrating guitar string laid out along the path of the electron’s orbit. These waveforms are called orbitals. See also quantum mechanics: Bohr’s theory of the atom.

In the quantum mechanical version of the Bohr atomic model, each of the allowed electron orbits is assigned a quantum number n that runs from 1 (for the orbit closest to the nucleus) to infinity (for orbits very far from the nucleus). All of the orbitals that have the same value of n make up a shell. Inside each shell there may be subshells corresponding to different rates of rotation and orientation of orbitals and the spin directions of the electrons. In general, the farther away from the nucleus a shell is, the more subshells it will have. See the table.

This arrangement of possible orbitals explains a great deal about the chemical properties of different atoms. The easiest way to see this is to imagine building up complex atoms by starting with hydrogen and adding one proton and one electron (along with the appropriate number of neutrons) at a time. In hydrogen the lowest-energy orbit—called the ground state—corresponds to the electron located in the shell closest to the nucleus. There are two possible states for an electron in this shell, corresponding to a clockwise spin and a counterclockwise spin (or, in the jargon of physicists, spin up and spin down).

The next most-complex atom is helium, which has two protons in its nucleus and two orbiting electrons. These electrons fill the two available states in the lowest shell, producing what is called a filled shell. The next atom is lithium, with three electrons. Because the closest shell is filled, the third electron goes into the next higher shell. This shell has spaces for eight electrons, so that it takes an atom with 10 electrons (neon) to fill the first two levels. The next atom after neon, sodium, has 11 electrons, so that one electron goes into the next highest shell.

periodic table showing the valence shells

In the progression thus far, three atoms—hydrogen, lithium, and sodium—have one electron in the outermost shell. As stated above, it is these outermost electrons that determine the chemical properties of an atom. Therefore, these three elements should have similar properties, as indeed they do. For this reason, they appear in the same column of the periodic table of the elements (see periodic law), and the same principle determines the position of every element in that table. The outermost shell of electrons—called the valence shell—determines the chemical behaviour of an atom, and the number of electrons in this shell depends on how many are left over after all the interior shells are filled.

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1. Discovery of subatomic particles:

Discovery & their discoverers:

Name of Particles Scientist
Electron J.J. Thomson
Proton Goldstein
Neutron Chadwick
Cathode Ray William Crooke's
Anode Ray Goldstein
Nucleus Rutherford
Atomic No. Moseley
Nomenclature of e- Stoney
Charge of e- Millikan
Specific charge on e-(e/m) J.J. Thomson

Properties of the three main subatomic particles:

Name Electron Proton Neutron
Symbol e p n
Absolute charge -1.6022×10-19C +1.6022×10-19C 0
Relative Electric charge -1 +1 0
Mass kg 9.10939×10-31 1.67262×10-27 1.67493×10-27
Molar Mass (g) 0.00054 1.00727 1.00867
Approximate mass(u) 0 1 1

2. Atomic number, mass number and the atomic species:

(i) Atomic number Z= number of protons.

Mosely's relation ⇒v=az-b.

(ii) Mass number A= number of n+p.

(iii) Isotopes = Same Z but different A.

(iv) Average atomic weight


, where m1,m2,m3 are mass of the isotopes and

x1,x2,x3 are their percentage abundance or relative ratio.

(v) Isobars = Same A but different Z.

(vi) Isotones/Isoneutronic/Isotonic = same number of neutrons.

(vii)Isodiapheres = Same (n-p)

(viii)Isosteres = Molecules with the same number of atoms and electrons.

(ix) Isoelectronic = Same number of e-s.

3. Atomic Models:

(i) Thomson Model of an atom:

J. J. Thomson, in 1898, proposed that an atom possesses a spherical shape (radius approximately 10-10m ) in which the positive charge is uniformly distributed. The electrons are embedded into it in such a manner as to give the most stable electrostatic arrangement. Many different names are given to this model, for example, plum pudding, raisin pudding or watermelon.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

Limitation: This model was able to explain the overall neutrality of the atom but was not consistent with the results of later experiments.

(ii) Rutherford’s alpha particle scattering experiment and model:

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?


(i) most of the α– particles passed through the gold foil were undeflected.

(ii) a small fraction of the α –particles was deflected by small angles.

(iii) a very few α – particles ( ∼1 in 20,000) bounced back, that is, were deflected by nearly 180°.


(i) Most of the space in the atom is empty as most of the α –particles passed through the foil undeflected.

(ii) A few positively charged α– particles were deflected. The deflection must be due to the enormous repulsive force showing that the positive charge of the atom is not spread throughout the atom as Thomson had presumed. The positive charge has to be concentrated in a very small volume that repelled and deflected the positively charged α – particles.

(iii) Calculations by Rutherford showed that the volume occupied by the nucleus is negligibly small as compared to the total volume of the atom. The radius of the atom is about 10-10m, while that of the nucleus is 10-15m.

Rutherford proposed the nuclear model of atoms (after the discovery of protons):

(i) The positive charge and most of the mass of the atom was densely concentrated in an extremely small region. This very small portion of the atom was called the nucleus by Rutherford.

(ii) The nucleus is surrounded by electrons that move around the nucleus with a very high speed in circular paths called orbits. Thus, Rutherford’s model of atom resembles the solar system in which the nucleus plays the role of the sun and the electrons of the revolving planets.

(iii) Electrons and the nucleus are held together by electrostatic forces of attraction.

Estimation of the closest distance of approach (derivation).

An α - particle is projected from infinity with a velocity V0 towards the nucleus of an atom having the atomic  number equal to Z, then find out (i) the closest distance of approach R, (ii) what is the velocity of the α -particle at the distance R1R1>R from the nucleus.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

From energy conservation, P.E1+KE1=P.E2+KE2


R=4KZe2mαVα2 (the closest distance of approach)

Let velocity at R1 is V1

From energy conservation, P.E1+KE1=P.E3+KE3


4. Size of the nucleus:

The volume of the nucleus is very small and is only a minute fraction of the total volume of the atom. Nucleus has a diameter of the order of 10-12 to 10-13cm and the atom has a diameter of the order of 10-8cm. Thus, diameter (size) of the atom is 100,000 times the diameter of the nucleus.

The radius of a nucleus is proportional to the cube root of the number of nucleons within it.


Where R0 can be 1.1×10-13 to 1.44×10-13cm; A= mass number; R= Radius of the nucleus.

Nucleus contains protons and neutrons except hydrogen atoms, which does not contain neutrons in the nucleus.

5. Wave Nature of Electromagnetic Radiation:

Electromagnetic wave radiation:

The oscillating electrical/magnetic field are electromagnetic radiations. Experimentally, the direction of the oscillations of the electrical and the magnetic field are perpendicular to each other.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?
E→=Electric field, B→=Magnetic field

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?
Direction of the wave propagation.

Some important characteristics of a wave:

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

Wavelength of a wave is defined as the distance between any two consecutive crests or troughs. It is

represented by λ (lambda) and is expressed in A˙ or m or cm or nm (nanometer) or  pm (picometer).



Frequency of a wave is defined as the number of waves passing through a point in one second. It is

represented by υ (nu) and is expressed in Hertz Hz or  cycles/sec or simply sec-1⁡or s-1.

1Hz=1 cycle /sec

Velocity of a wave is defined as the linear distance travelled by the wave in one second. It is represented by v and is expressed in cm/sec or m/sec ms-1.

Amplitude of a wave is the height of the crest or the depth of the trough. It is represented by 'a' and is

expressed in the units of length.

Wave number is defined as the number of waves present in 1 cm length. Evidently, it will be equal to the reciprocal of the wavelength. It is represented by υ_ (read as nu bar).


If λ is expressed in cm, υ_ will have the units cm-1.

The relationship between the velocity, the wavelength, and the frequency of a wave. As the frequency is the number of waves passing through a point per second and λ is the length of each wave, hence, their product will give the velocity of the wave. Thus, v=υ×λ.

6. Spectrum:

(i) Electromagnetic Spectrum:

The arrangement of various types of electromagnetic radiations in the order of their increasing or decreasing wavelength or frequency is known as electromagnetic spectrum.

Radiation Wavelength (A) Frequency (Hz)
Gamma rays 0.01 to  0.1 3×1019 to  3×1020
X-rays 0.1 to 150 2×1016 to  3×1019
UV radiations 150 to  3800 7.9×1014 to  2×1016
Visible rays 3800 to  7600 3.95×1014 to  7.9×1014
Microwaves 6×106 to  3×109 1×105 to  1×109

(ii) Order of wavelength in electromagnetic spectrum:

cosmic rays  <γ- rays  < X-rays < Ultraviolet rays  < Visible  < Infrared  < Microwaves  < Radio waves.

(iii) Spectrum (Important points):

Continuous emission spectrum is given by incandescent sources.

Emission line spectrum is given by atoms.

Emission band spectrum is given by molecules.

Angle of deviation ∝ frequency of radiation:

More lines are observed in the emission spectrum than the absorption spectrum.

Hydrogen Spectrum  n2 →n1  
Lyman ⟶ Any higher orbit ⟶1 [Found in U.V. region]
Balmer ⟶ Any higher orbit ⟶2 [Found in Visible region]
 Paschen → Any higher orbit ⟶3 [Found in  I.R. region]
 Brackett → Any higher orbit ⟶4  [Found in  I.R. region]
 Pfund ⟶ Any higher orbit ⟶5  [Found in I.R. region]

7. Rydberg Equation:





Where C= the velocity of the electromagnetic waves.

Where RH= Rydberg constant =109678 cm-1

=10967800 m-1.


RHCh= Energy of the Ist orbit of hydrogen.

RHChZ2= Energy of the Ist orbit of any hydrogen-like species.

Important points:

(i) α line/First line/Starting line/Initial line (First line of any series).

(ii) Last line/Limiting line/Marginal line (Last line of any series).

(iii) Total number of lines in the emission spectrum for ( n2→n1)

(T.E.L) Total emission lines =n2-n1n2-n1+12

But for (n→1), T.E.L =n(n-1)2

8. Particle Nature of Electromagnetic Radiation:

Particle Nature of Electromagnetic Radiation: Planck's Quantum Theory

Some of the experimental phenomenon such as diffraction and interference can be explained by the wave nature of the electromagnetic radiation. However, following are some of the observations which could not be explained

(i) The nature of the emission of radiation from hot bodies (black - body radiation).

(ii) The ejection of electrons from metal surface when radiation strikes it (photoelectric effect).

Black Body Radiation:

When solids are heated, they emit radiation over a wide range of wavelengths. The ideal body, which emits and absorbs all frequencies, is called a black body and the radiation emitted by such a body is called the black body radiation. The exact frequency distribution of the emitted radiation (i.e., intensity versus frequency curve of the radiation) from a black body depends only on its temperature.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

The above experimental results cannot be explained satisfactorily on the basis of the wave theory of light.

Planck suggested that atoms and molecules could emit (or absorb) energy only in discrete quantities and

not in a continuous manner.

9. Quantum theory of light:

The smallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation is called as quantum of light.

According to Planck, the light energy coming out from any source is always an integral multiple of a smallest energy value called quantum of light.

Let quantum of light be =E0, then total energy coming out is =nE0(n= Integer).

Quantum of light  = Photon (Packet or bundle of energy).

Energy of one photon is given by

E0=hυ (υ- Frequency of light )

h=6.625×10-34J.secsec  (h-planck's constant )

E0=hcλ  (c- speed of light )

(λ.- wavelength)

Order of magnitude of E0=10-34×10810-10=10-16J

One electron volt (eV): The energy gained by an electron when it is accelerated from rest through a potential difference of 1 volt.

Note: Positive charge always moves from a high potential to a low potential and -ve charge always moves from a low potential to a high potential if set free.

If a charge 'q' is accelerated through a potential difference of "V' volt, its kinetic energy will be increased by qV.

1eV=1.6×10-19C×1 volt


10. Photoelectric effect:

When certain metals (for example Potassium, Rubidium, Cesium etc.) were exposed to a beam of light, the electrons were ejected as shown in the figure:

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

The phenomenon is called Photoelectric effect. The results observed in this experiment were:

(i) The electrons are ejected from the metal surface as soon as the beam of light strikes the surface, i.e., there is no time lag between the striking of light beam and the ejection of electrons from the metal surface.

(ii) The number of electrons ejected is proportional to the intensity or brightness of light.

(iii) For each metal, there is a characteristic minimum frequency υo (also known as threshold frequency) below which photoelectric effect is not observed. At a frequency υ>υo, the ejected electrons come out with a certain kinetic energy. The kinetic energies of these electrons increase with the increase of frequency of the light used.

The kinetic energy of the ejected electron is given by the equation:


, where me is the mass of the electron and v is the velocity associated with the ejected electron.

11. Bohr's Model for the Hydrogen Atom:

(a) Applicable for single e- species only, like H,He+,Li+2,Be+3,Na+10, etc.

(b) Related with the particle nature of electron.

(c) Based on Planck’s quantum theory.

Important Formulae:

(i) Angular momentum in an orbit is quantized.


(ii) Radius of Bohr orbit =r=n2h24π2mKZe2

On solving r =0.529×n2Z A˙ 

, where 0.529 A˙=a0 is called the atomic unit of length (Bohr).

(iii) Velocity of an electron in Bohr orbit:


On solving v=2.18×106Zn m/s


(iv) Energy of an electron in Bohr orbit:

Potential Energy PE=-KZe2r  i.e., At r=∞,PE=0

Kinetic Energy (KE)=12KZe2r   i.e., At r=∞,KE=0

Total Energy TE=-2π2mK2Z2e4n2h2

On solving TE =-2.18×10-18×Z2n2 J/atom

=-13.6×Z2n2 eV/atom

=-313.6×Z2n2 Kcal/mol

=-1313.6×Z2n2 KJ/mol

"Relation between " TE, PE and KE




What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

Important Shortcuts:

T.E of any H- like species = TE of Hydrogen ×Z2

(For same orbit)

ΔE for H- like species =ΔE (For Hydrogen) ×Z2

(For same transition)

Energy in nth orbit-for H like species =E1n2 [For same atom]

(I.E.) "ionization Energy ⇒n=1⟶n=∞

(S.E.) Separation Energy

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

(E.E) Excitation Energy =

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

(v) Failures / limitations of Bohr's theory:

(a) He could not explain the line spectra of atoms containing more than one electron.

(b) He also could not explain the presence of multiple spectral lines.

(c) He was unable to explain the splitting of spectral lines in magnetic field (Zeeman effect) and in electric field (Stark effect).

(d) No conclusion was given for the principle of quantization of angular momentum.

(e) He was unable to explain the de-Broglie's concept of the dual nature of matter.

(f) He could not explain Heisenberg's uncertainty principle.

12. Towards quantum mechanical model of atom:

(i) Dual nature of electron (de-Broglie Hypothesis):

(a) Einstein had suggested that light can behave as a wave as well as like a particle, i.e., it has a dual character.

(b) In 1924, de-Broglie proposed that an electron behaves both as a material particle and as a wave.

(c) This proposed a new theory called the wave mechanical theory of matter. According to this theory, the electrons protons and even atoms, when in motion, possess wave properties.

(d) According to de-Broglie, the wavelength associated with a particle of mass m , moving with a velocity v, is given by the relation,


, where h is Planck's constant




KE= Kinetic Energy

V= Potential

λ=h2mqV  for e- if solved) then λ=150VA˙

(ii) de-Broglie’s explanation for Bohr’s model:



mv=hλ putting this in mvr=nh2π


⇒λ=2πrn de Broglie wavelength

(iii) Important Points:

(a) When an e- revolves in orbit then no. of waves made by e-= orbit number (n).

(b) Frequency of matter waves.

υ=vλ=vPh=mv2h=2KEh [υ= frequency ]

13. Heisenberg's uncertainty principle:

The exact position and momentum of a fast-moving particle cannot be calculated precisely at the same

moment of time. If Δx is the error in the measurement of the position of the particle and if Δp is the error in the measurement of momentum of the particle, then:

Δx,Δp≥h4π or Δx⋅(mΔv)≥h4π

, where Δx= uncertainty in the position

Δp= uncertainty in the momentum

h= Plank's constant

m= mass of the particle

Δv= uncertainty in the velocity

h4π=5.27×10-35 J.sec. (In SI unit)

14. Quantum mechanical model of an atom:

Orbital Wave Functions, Probability Distribution and Shape of Orbitals:

(i) Wave function (ψ) provides useful information about an electron. These are:

(a) Energy of the electron in an orbital.

(b) Position of the electron in space in a particular allowed energy state.

(ii) On the basis of values of |ψ|2, i.e., the probability of finding electron, electron clouds may be drawn and the region where they are dense shows the region of high probability.

(iii) The plane and the point where |ψ|2 is zero (i.e., no probability) are called nodal plane or nodal point, respectively

(iv) Orbital wave function ψ is the product of two functions; radial function and angular function, i.e.,


ψ(r) is the function of the distance (r) from the nucleus while ψ(ω) is the function of the two spherical co-ordinates θ and ϕ as shown in figure.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

(v) ψ(r), i.e., the radial part of the wave function depends upon the quantum number n and ∣ and decides the size of an orbital.

(vi) The angular part of the wave function ψ(ω), depends upon the quantum numbers ∣ and m and describes the shape of orbital.

For the sake of convenience, the ψ(r) vs. r and ψ(ω)vs. angle are plotted separately.

15. Orbital:

An orbital may be defined as the region of space around the nucleus where the probability of finding an electron is maximum (90% to 95% ).

Orbitals do not define a definite path for the electron, rather they define only the probability of the electron being in various regions of space around the nucleus.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

The s-orbitals:

(i) All the s-orbitals are spherical and show spherical symmetry. Being symmetrical, they do not have directional dependence.

(ii) The size (or radii) of s-orbital depends upon the value of n and increases in proportionality of n2.

(iii) The area covered under orbital corresponds to high probability (>90%) of finding an electron.

The p-orbitals:

(i) P-orbitals are not spherically symmetrical and have dumb-bell shape.

(ii) p- subshell is constituted by three p -orbitals px,py and pz named with respect to the axis.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

Figure: Electron-density distribution in a 2p- orbital

(iii) Each p- orbital consists of two lobes joined together by a node (nucleus).

(iv) For a p -orbital to exist, minimum values of n=2, because for p- subshell l=1, and the minimum value of n which gives l=1 is 2. Thus, 1 p- orbital does not exist and they start from 2p.

(v) For l=1 value of m=-1, 0,+1 and each other of m indicates one p -orbital. e.g., 2p- subshell contains 2px' 2py' and 2pz orbitals.

All the three have the same shape and energy (i.e., degenerate) and differ in orientation only.

The wave function for pz has maximum amplitude along z- axis. It is zero along xy plane. Similarly, px and py has the maximum probability of finding electron along x and y -axis.

The d-orbitals:

(i) For d- subshell, value of l=2 and minimum value of n, which gives l=2, is 3. Thus,  1 d or 2 d- orbitals have no existence, and they start from 3d.

(ii) For l=2 , the values of m=-2, -1, 0,+1,+2, (five values), thus, each d-subshell is made up of five d-orbitals which are dxy' dyz,dzx,dx2-y2,dz2.

(iii) Except dz2, they have four lobes separated by a node (nucleus) while dz2 has only two lobes. Their shapes have been shown below:

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

(iv) Lobes of dxy'dyz and dzx lie between the axis, while dx2-y2 and dz2 lie on the axis.

(v) dz2 has two lobes on z- axis and doughnut in xy - plane. It looks like a baby soother.

(vi) All the d-orbitals belonging to a subshell have same energy and so are degenerate (fivefold degeneracy).

The f-orbitals:

(i) For  f-subshell l=3 and thus  m will have seven values  m=-3,-2,-1, 0,+1,+2,+3. For l=3, minimum value of n must be four thus 1f, 2f of 3f have no existence and f- orbitals start from n=4 (i.e., 4f).

(ii) In a f- subshell, there are seven f- orbitals. Their contour diagram which are 3-D, are difficult to draw on a paper.

(iii) The seven f-orbitals are: fxx2-y2,fyx2-y2,fzx2-y2,fxyz,fz3,fyz2 and fxz2.


(a) Sometimes, an orbital is represented in terms of ψn,l,m also. Usually for m=0,z - axis is chosen.

e.g.,  ψ420 reveals that n=4,l=2 and m=0, thus it is 4dz2.

(b) For an orbital (or subshell), as the value of n increases, the size of the orbital increases, i.e., 1s<2s<3s, etc.

(c) As r approaches to 0,ψ also approaches zero except s- orbital. It concludes that only s- electron can

penetrate through nucleus and has the probability to be found at the nucleus.

(d) For one electron system (e.g., H+,Li2+,Be3+ etc.), the energy of the orbital depends only on the value of  n and not on l. Thus, all the subshells belonging to same shell will have the same energy.

(e) There is no unique way of representing the angular dependence functions of all seven f-orbitals. An

alternative way may be as fz3,fxz2,fyx2,fzy2,fxx2-3y2fyy2-3z2 and fzz2-3x2*.

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

The radial probability density for some orbitals of the hydrogen atom. Ordinate is proportional to

4πr2R2, and all distributions are to the same scale.

16. Node and Nodal Plane:

(i) Node represents the region where probability of finding an electron is zero  (i.e.,  ψ and ψ2=0 ). Similarly, nodal plane represents the plane having zero probability of finding an electron.

(ii) Nodes are of two types:

(a) Radial node

(b) Angular node

A radial node is the spherical region around nucleus having ψ and ψ2 equal to zero. An orbital having a higher number of nodes has more energy.

(iii) Calculation of the number of nodes:

Radial nodes =n-l-1

Angular nodes =l

Total nodes =n-1

n and l are principal and azimuthal quantum numbers. e.g., in 3p- orbital.

Radial nodes =3-1-1=1 (=n-l-1)

Angular nodes =1 (=l)

Total nodes  =2 (one radial, one angular)

(iv) For one electron system, the energy depends only upon the number of nodes (n-1) and not upon the types of nodes.

17. Degenerate and Non-degenerate Orbitals:

Orbitals belonging to the same subshell and having the same energy are known as degenerate orbitals. Degeneracy is maintained in the absence of a magnetic field but destroyed if the degenerate orbitals are kept in the magnetic field. Thus,

What will be the number of waves formed by a Bohr electron in one complete revolution in its second orbit?

18. Quantum numbers:

The set of four integers required to define the state of an electron in an atom are called quantum numbers. The quantum numbers are:

(i) Principal quantum number (n).

(ii) Azimuthal quantum number l.

(iii) Magnetic quantum number m.

(iv) Spin quantum number (s).

(i) Principal quantum number, (n), relates to the amplitude (i.e., size) of an electron wave and also the total energy of the electron. It has integral values of 1, 2, 3, 4,… etc., also denoted as K,L,M,N…. etc.

(ii) Azimuthal quantum number (l), tells us about the sub-energy shell of an electron. For each main energy shell, there can be 'n' number of sub-energy shells. These sub-energy shells are designated by different values of l. For each value of n, l can have the values from 0, 1, 2, 3…n-1.

(iii) Magnetic quantum number, (m), explains the behavior of an electron in the external magnetic field or in other words, it tells us about orbitals of the electrons. The values of m gives the number of orbitals associated with a particular sub-shell in the shell. For each value of l, m can have values from –l to +l including zero.

e.g., when l=1,m=-1, 0,+1;  l=2, m=-2,-1, 0, +1, +2

(iv) Spin quantum number, (s), Each spinning electron can have two values of s = +12 or -12

19. Filling of Electrons in Orbitals:

Rules for filling of orbitals:

(i) Aufbau principle: The electrons are filled up in the increasing order of the energy in sub-shells.

 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10.

(ii) (n+l) rule: The sub-shell with the lowest (n+l) value is filled up first, but when two or more sub-shells have the same (n+l) value, the sub-shell with the lowest value of n is filled up first.

(iii) Pauli exclusion principle: Pauli stated that no two electrons in an atom can have the same values of all four quantum numbers.

(iv) Hund's rule of maximum multiplicity: Electrons are distributed among the orbitals of a sub-shell in such a way as to give maximum number of unpaired electrons with parallel spin.


(i)  24Cr=[Ar]4s2,3d4 (Not correct)

 [Ar]4s1,3d5 (correct: as d5 structure is more stable than d4 structure)

(ii)  29Cu=[Ar]4s1,3d10correct: as d10 structure is more stable than d9 structure