It states that electron pairing in the degenerate orbitals does not take place until all the available orbitals contain one electron each.

As we know there are three different orientations in space that are possible for a p orbital. All the three p orbitals, namely, px, py and pz have same energies and are called degenerate orbitals. However, in the presence of magnetic or electric field the degeneracy is lost.

\(E_n={(-1312.8)Z^2\over n^2}KJ\ mol^{-1}\)
Where Z = atomic number, n = principal quantum number.

(i) \(\lambda =\frac{h}{mv}\). This equation implies that a moving particle can be considered as a wave and a wave can exhibit the properties of a particle.
(ii) For a particle with high linear momentum (mv) the wavelength will be so small and cannot be observed.
(iii) For a microscopic particle such as an electron, the mass is of the order of 10^-31 kg, hence the wavelength is much larger than the size of atom and it becomes significant.
(iv) For the electron, the de Broglie wavelength is significant and measurable while for the iron ball it is too small to measure, hence it becomes insignificant.

J. J. Thomson's cathode ray experiment revealed that atoms consist of negatively charged particles called electrons. He proposed that atom is a positively,charged sphere in which the electrons are embedded like the seeds in the watermelon.

The distribution of electrons into various orbitals of an atom is called its electronic configuration.
N (Z = 7)
 

(i) Cr (Z = 24) Is² 2s² 2p⁶ 3s² 3p⁶ 3d⁵ 4s¹
Cr³⁺ - Is² 2s² 2p⁶ 3s² 3p⁶ 3d⁴.
It contains 4 unpaired electrons.
(ii) Ne (Z = 10) 1s²2s² 2p⁶. No unpaired electrons in it.

The charge of an electron is 1.602 x 10^-19 coulomb
The mass of an electron is 9.11 x 10-³¹ kg

According to J. C. Maxwell, whenever an electron is subjected to acceleration, it emits radiation and loses energy. As a result of this, its orbit should become smaller and smaller and finally it should drop into the nucleus by following a spiral path. This means that atom would collapse and thus Rutherford's model failed to explain stability of atoms. Another drawback of the Rutherford's model is that it gives no information about the electronic structure of an atom.

a) Principal quantum number defines energy and size of an orbital.
b) Azimuthal quantum number defines shape of an orbital
c) Magnetic quantum number defines spatial orientation (direction) of an orbital.

Quantum numbers are a set of four numbers which are used to specify the position, energy, shape, size, angular momentum, magnetic properties spin and orientation of an electron in an atom.

The set of quantum numbers is not possible because, for n = 1,1 cannot be equal to 1. It can have 0 value.

(i) Boron (Z = 5) ; 1s² 2s² 2p¹
(ii) Neon (Z = 10) ; 1s² 2S²2p⁶
(iii) Aluminium (Z = 13) ;1s² 2S²2p⁶ 3s² 3p¹
(iv) Chlorine(Z = 17) ; 1s² 2s² 2p⁶ 3s² 3p⁵
(v) Calcium (Z = 20) ;1s² 2S²2p⁶ 3s² 3p⁶ 4s²
(vi) Rubidium (Z = 37) ; 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 5s¹

(i) The principal quantum number represents the energy level in which electron revolves around the nucleus and is denoted by the symbol 'n'.
(ii) The 'n' can have the values 1,2,3, ... n = 1 represents K shell; n = 2 represents L shell and n = 3, 4, 5 represent the M, N, O shells, respectively.
(iii) The maximum number of electrons that can be accommodated in a given shell is 2n².
(iv) 'n' gives the energy of the electron,
\(E_n=\frac{(-1312.8)Z^2}{n^2}\ kJ\ mol^{-1}\) and the distance of the electron from the nucleus is given by \(r_n=\frac{(0.529)n^2}{Z}A\)

(i) The Bohr's atom model is applicable only to species having one electron such as hydrogen, Li²⁺ etc and not applicable to multi-electron atoms.
(ii) It was unable to explain the splitting of spectral lines in the presence of magnetic field (Zeeman effect) or an electric-field (Stark effect).
(iii) Bohr's theory was unable to explain why the electron is restricted to revolve around the nucleus in a fixed orbit in which the angular momentum of the electron is equal to \(\frac{nh}{2\pi}\) 

(i) Rutherford bombarded a thin gold foil with a stream of fast moving a-particles.
(ii) It was observed that most of the \(\alpha\)-particles passed through the foil.
(iii) Some of them were deflected through a small angle.
(iv) Very few α-particles were reflected back by 180°.
(v) Based on these observations, he proposed that in an atom, there is a tiny positively charged nucleus and the electrons are moving around the nucleus with high speed.

In an atom the wave function \(\Psi \) for an electron has no physical significance as such. However, its square i.e.\(\Psi ^{ 2 }\) at any point gives the intensity of the electron wave at that point.
In view of Heisenberg's uncertainty principle, it shows the probability of finding the electron at that point and therefore, termed as probability density

It is impossible to accurately determine both the position as well as the momentum of a microscopic particle simultaneously.
\(\Delta x.\Delta p\ge h/4\pi \)
where, Δx and Δp are uncertainties in determining the position and momentum, respectively

Schrodinger Wave Equation:
\(\frac { \partial ^{ 2 }\Psi }{ \partial x^{ 2 } } +\frac { \partial ^{ 2 }\Psi }{ \partial y^{ 2 } } +\frac { \partial ^{ 2 }\Psi }{ \partial z^{ 2 } } +\frac { 8\pi ^{ 2 }m }{ { h }^{ 2 } } (E-V)\Psi =0\)
\(\Psi \) = amplitude of wave; E = total energy of electron
V = potential energy; m = mass of electron

Isoelectronic - Species having same number of electrons
₁₁Na⁺ = 11 - 1 = 10e^-; 19K⁺ = 19 - 1 = 18e^-
₁₂Mg²⁺= 12 - 2 = 10e^-; .₂₀Ca²⁺ = 20 - 2 = 18e^-
₁₆S^2- 16 + 2 = 18e^-; _I8Ar = 18e^-
Hence the isoelectronic species are
(i) Na⁺ and Mg2+
(ii) K⁺, Ca²⁺, S^2- and Ar

If a substance which gives a line emission spectrum is placed in an external electric field, its lines get split into a number of closely spaced lines. This phenomenon is known as Stark effect.

If a substance which gives a line emission spectrum, is placed in a magnetic field, the lines of the spectrum get split up into a number of closely spaced lines. This phenomenon is known as Zeeman effect.

K.E = 2.5 \(\times\)10^-24 J
Kinetic energy = \(\frac { 1 }{ 2 } \)mv² = 2.5 \(\times\)10^-24
m = 9.1\(\times\) 10-31 kg
v = \(\left( \frac { 2K.E }{ m } \right) ^{ \frac { 1 }{ 2 } }=\left[ \frac { 2\times 2.5\times { 10 }^{ -24 } }{ 9.1\times { 10 }^{ -31 } } \right] ^{ \frac { 1 }{ 2 } }\)
= 2.34 \(\times\)10³ ms^-1
\(\boxed{v = 2.34 103\ {ms}^{-1}}\)
Wavelength , \(\lambda =\frac { h }{ mv } \)
= \(\frac { 6.626\times { 10 }^{ -34 } }{ 9.1\times { 10 }^{ -31 }\times 2.34\times { 10 }^{ 3 } } =311.1\times { 10 }^{ -9 }m\)
\(\boxed{\lambda =311.1\times 10^{ -9 }m(or)311.1\ nm}\)

To obtain atomic number of an element fill the orbitals in order of their increasing energies up to the given outer orbital configuration
(a) Is²2s², 2p^6, 3s² (Z = 12)
(b) Is² 2S², 2p³ (Z = 7)
(c) Is²2s², 2p⁶, 3s², 3p⁵ (Z = 17)

It states that electron pairing in the degenerate orbitals does not take place until all the available orbitals contains one electron each.

(a) ₁H = 1s¹, H- = 1s2
(b) 11Na = 1s² 2s²,2p⁶,3s¹,Na+ = 1s²2s²2p⁶
(c) 8O = 1s² 2s²,2p⁴,O^2- = 1s²2s²²p⁶
(d) 9F = 1s² 2s²,2p^5,F- =1s²2s2²p⁶

(i) It states that, the lower the value of (n + 1) for an orbital, the lower is its energy. If two orbitals have the same value of (n + 1), the orbital with lower value of n will have the lower energy.
(ii) (a) Principal quantum number : 1s < 2s = 2p < 3s = 3p = 3d <4s = 4p = 4d = 4f < 5s = 5p = 5d = 5f < 6s = 6p = 6d = 6f < 7s
(b) Based on the (n+1) rule: 1s < 2s < 2p < 3s < 3p < 4s <3d <4p <5s <4d < 5p <6s <4f < 5d < 6p < 7s < 5f < 6d.

Erwin Schrodinger expressed the wave nature of electron in terms of a differential equation. This equation determines the change of wave function in space depending on the field of force in which the electron moves. The time independent Schrodinger equation can be expressed as,
\(\overset { \wedge }{ H } \psi =E\psi \)          .........(1)
Where \(\overset { \wedge }{ H } \) is called Hamiltonian operator, \(\psi \) is the wave function and is a function of position coordinates of the particle and is denoted as \(\psi \) (x, y, z) E is the energy of the system
\(\overset { \wedge }{ H } =\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 } } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V \right] \)
can be written as
\(\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 }m } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V\Psi \right] =E\Psi \)
Multiply by \(\frac { 8{ \pi }^{ 2 }m}{ { -h }^{ 2 } } \)and rearranging
\(\frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } +\frac { 8{ \pi }^{ 2 }m }{ { -h }^{ 2 } } (E-V)\Psi =0\)          ........(2)
The above Schrodinger wave equation does not contain time as a variable and is referred to as time independent Schrodinger wave equation. This equation can be solved only for certain values of E, the total energy. i.e. the energy of the system is quantised. The permitted total energy values are called eigen values and corresponding wave functions represent the atomic orbitals.

Ferrous and ferric salts: In ferrous salts Fe²⁺,  the configuration is 1s²2s²,2p⁶,3s², 3p⁶, 3d⁶.In ferric salts Fe3⁺ ,the configuration is 1s²2s²,2p⁶,3s², 3p⁶, 3d⁵.
As half-filled 3d⁵ configuration is more stable therefore ferric salts are more stable than ferrous salts.

The word Aufbau in German means 'building up'. In the ground state of the atoms, the orbitals are filled in the order of their increasing energies. That is the electrons first occupy the lowest energy orbital available to them.
Once the lower energy orbitals are completely filled, then the electrons enter the next higher energy orbitals. The order of filling of various orbitals as per the Aufbau principle which is in accordance with (n + l) rule.

Statement : "No two electrons in an atom can have the same set of values of all four quantum numbers"
Explanation : It means that, each electron must have unique values for the four quantum numbers (n, l, m and s).
For the lone electron present in hydrogen atom, the four quantum numbers are: n = 1; l = 0; m = 0 and s = +1/2. For the two electrons present in helium, one electron has the quantum numbers same as the electron of hydrogen atom, n = 1.
l = 0, m = 0 and s = +1/2. For other electron, the fourth quantum number is different i.e., n = 1, l = 0, m = 0 and s = -1/2.
As we know that the spin quantum number can have only two values +1/2 and - 1/2, only two electrons can be accommodated in a given orbital in accordance with pauli exclusion principle.