\(\triangle\)x = 0.6\(\mathring{A}\) = 0.6 x 10^-10m
\(\triangle p\) = ?
\(\triangle x.\triangle p\ge\frac{h}{4\pi}\)
\(\triangle x.\triangle p\ge5.28\times10^{-35}kgm^{2}s^{-1}\)
\((0.6 \times10^{-10}) \triangle p \ge5.28\times10^{-35}\)
\(\Rightarrow \triangle \ge \frac{5.28\times10^{-35}kgm^{2}s^{-1}}{0.6510^{-1}m}\)
\(\triangle p \ge8.8\times10^{-25}kgms^{-1}\)

Electronic configuration of oxygen

ஃ 8^th electron present in 2p_x orbital and the quantum numbers are
n = 2,l = 1,m₁ = either + 1 or -1 and s = -1/2
Electronic configuration of chlorine 

15^th electron present in 3P_z orbital and the quantum numbers are n = 3, l = 1, m₁ = either +1 or -1 and ms = +1/2

i) ₂₅Mn - 1s², 2S^2, 2p⁶, 3s², 3p⁶, 4s², 3d⁵
₂₃Mn²⁺ - 1s², 2S^2, 2p⁶, 3s², 3p⁶, 3d⁵
ii) ₂₄Cr - 1s^2, 2S²,2p⁶, 3s², 3p⁶, 4s¹, 3d³
₂₁Cr³⁺ - 1s^2, 2S²,2p⁶, 3s², 3p⁶, 3d³

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.

d sub-shell has five orientations. Each orientation can accommodate one electron with spin quantum number +1/2 and another electron with -1/2 Hence number of electrons with s+1/2 in 'd' sub shell is 5.

Magnetic quantum number accounts for the orientation of the electron in orbital.

The electronic configuration of sulphur is 1s² 2s² 2p⁶ 3s² 3p⁴
For 1s², n,+ l = 1+ 0 = 1 ; For 2p⁶, n + 1 = 2 + 1 = 3
2s², n + I = 2 + 0 = 2;      For 3s², n + I = 3 + 0 = 3
3p⁴, n + 1 = 3 + 1 = 4
Thus n + 1 = 3 for 2p⁶ electron and 3s² electron = 8 electrons.

(i) The last electron is the 'one' electron present in 'p' sub level.
i.e., n = 2; l = 1; m = -1,0 or + 1,s = + 1/2 or - 1/2
(ii) If the first electron present in 'p' subshell has the-following quantum numbers.
n = 3, l = 1, m = -1,0 (or) + 1, s = + 1/2,
the last electron has n = 3, l = 1, m = -1, 0 (or) + 1, s = - 1/2.

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.

Energy electrons symmetry
This is due to the symmetrical distribution and exchange energy of given d- electrons. Symmetry leads to stability.
Exchange energy:
If two or more electrons with the same spin are present in degenerate orbitals, there is a possibility for exchanging their positions. During exchange process, the energy is released and the released energy is called exchange energy. If more number of exchanges are possible, more exchange energy in released. More number of exchanges are possible only in case of half filled and fully filled configurations.
For example, in chromium the electronic configuration is [Ar]3d⁵ 4s¹. The 3d orbital is half filled and there are ten possible exchanges as shown in figure. On the other hand only six exchanges are possible for [Ar]3d⁴ 4s² configuration. Hence, exchange energy for the half filled configuration is more. This increases the stability of half filled 3d orbitals.

The exchange energy is the basis for Hund's rule, which allows maximum multiplicity, that is electron pairing is possible only when all the degenerate orbitals contain one electron each.

The variation of the probability of locating the electron on a sphere with nucleus at its centre depends on the azimuthal quantum number of the orbital in which the electron is present.

For 1s orbital, l = 0, m = 0,\(f(\theta)=\frac{1}{\sqrt 2}\)and \(g(\varphi)=\frac{1}{\sqrt 2\pi}\) Therefore, the angular distribution function is equal to \(\frac{1}{2\sqrt \pi }\) 
i.e. it is independent of the angle \(\theta\) and \(\varphi\). Hence, the probability of finding the electron is independent of the direction from the nucleus. So, the shape of the s orbital is spherical.

For p orbitals, l = 1 and the corresponding m values are -1, 0 and +1. The angular distribution functions are quite complex and are not discussed here, The shape of the p orbital is shown in Figure (b). The three different m values indicates that there are three different orientations possible for p orbitals. These orbitals are designated as P_x, P_y and P_z and the angular distribution for these orbitals shows that the lobes are along the x, y arid z axis respectively. As seen in the Figure the 2p orbitals have one nodal plane

For 'd' orbital l = 2 and the corresponding m values are -2, -1, 0, +1, +2. The shape of the d orbital looks like a 'clover leaf'.
The five m values give rise to five d orbitals namely d_xy, d_yz, d_zx, d_x²-y² and d_z². The 3d orbitals contain two nodal planes.

For 'f' orbital, l = 3 and the m values are -3, -2,-1,0, +1, +2, +3 corresponding to seven f orbitals. f_z³, f_xz²,f_yz²,f_xyz,f_{z(x² - y²)},f_{x(x² - 3y²)},f_{y(3x² - y²)} which are shown in Figure. There are 3 nodal planes in the f-orbitals.

(i) Magnetic quantum number
1. It is denoted by the letter 'm_l'. It takes integral values ranging from -I to +1 through 0. i.e. if l = 1; m = -1, 0 and +1.
2. The Zeeman Effect (the splitting of spectral lines in a magnetic field) provides the experimental justification for this quantum number.
3. The magnitude of the angular momentum is determined by the quantum number l while its direction is given by magnetic quantum number.
(ii) Spin quantum number
1. The spin quantum number represents the spin of the electron and is denoted by the letter 'm_s'.
2. The electron in an atom revolves not only around the nucleus but also spins. It is usual to write this as electron spins about its own axis either in a clockwise direction or in anti-clockwise direction.
3. Corresponding to the clockwise and anti-clockwise spinning of the electron, maximum two values are possible for this quantum number.
4. The values of 'm_s' is equal to \(-\frac{1}{2}\) and  \(+\frac{1}{2}.\)

m 160 g = 160 x 10^-3 kg
\(v=140 km hr^{-1}=\frac{104\times10^{3}}{60\times60}ms^{-1}\)
\(v=38.88ms^{-1}\)
\(\lambda=\frac{h}{mv}\)
\(=\frac{6.626\times^{-34}Kgm^{2}s^{-1}}{160\times10^{-3}Kg\times38.88ms^{-1}}\)
\(\lambda=1.065\times10^{-34}m\)

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.