chemistry-class-11-unit-02-chapter-06-structure-of-the-atom-l-6-8-j0zdenlfp5y

Structure Of The Atom L-6

Quantum number

Magnetic quantum number (m)
The magnetic quantum number of an electron is one of the four quantum numbers that state the position of the electron with respect to the nucleus

Structure Of The Atom L-6

Quantum number

Orientation of orbital (in space)
Value for m depends of 'l'
Number of orientation
l = 0 $\qquad\quad\quad$ (1)
l = 1 $\qquad\quad\quad$ (3)
l = 2 $\qquad\quad\quad$ (5)
l = 3 $\qquad\quad\quad$ (7)

Structure Of The Atom L-6

Quantum Number

s $\rightarrow$ 1
p $\rightarrow 3(p_x ~ p_y ~ p_z)$
d $\rightarrow 5(d_{x y} ~ d_{y z} ~ d_{z x} ~ d_{x^2-y^2} ~ d_{z^2})$
f $\rightarrow$ 7

Structure Of The Atom L-6

Quantum number

$\text{Total number of orientation} (2l+1)$
$-l,-l+1,..0,..,l-1,+l$

l	$(2l+1)$	$m$	Orbital
0	1	0	s
1	3	$-1,0,+1$	$p_x ~ p_y ~ p_z$
2	5	$-2,-1,0,+1,+2$	$d_{x y} ~ d_{y z} ~ d_{z x} ~ d_{x^2-y^2} ~ d_{z^2}$
3	7	$-3,........,+3$

Structure Of The Atom L-6

Examples

For n=3
$n=3 \quad l=0 , 1 , 2$
3s $\rightarrow$ 3s (1)
3p $\rightarrow$ $3p_x ~ 3p_y ~ 3p_z$ (3)
3d $\rightarrow$ $3d_{xy} ~ 3d_{yz} ~ 3d_{zx} ~ 3d_{x^2-y^2} ~ 3d_{z^2}$ (5)

Structure Of The Atom L-6

Examples

For $n$ $\rightarrow$ $n^2$ orbitals
- 1 orbital $\quad$ 1 $s$
- 4 orbitals $\quad$ 2 $s$ $\quad$ 2 $px$ $\quad$ 2 $py$ $\quad$ 2 $pz$
- 9 orbitals

Structure Of The Atom L-6

Spin quantum number

Spin orientation of electron
Two spin orientations $m_s=+\frac{1}{2} \uparrow \alpha , -\frac{1}{2} \downarrow \beta$

Structure Of The Atom L-6

Summary

n	l	m	$m_s$
size	shape	orientation	spin orientation

$n$ = $\quad$ 1, 2, 3, ..........
$l$ = $\quad$ 0,1,2, .... n-1
$m$ = $\quad -l,-l+1, \cdots 0, \cdots, l-1, l$
$m_s$ = $\pm \frac{1}{2},-\frac{1}{2}$

Structure Of The Atom L-6

Schrodinger equation

$H \psi=E \psi$
$\psi \equiv \psi_{n l m}=\psi(n, l, m)$
$\left|\psi_{n l m}\right|^2=$ Probability density of the electron at a given point
${r(x, y, z)}$ - $\text { "spherical polar coordinates" } (r, \theta, \phi)$
$\psi_{nlm} (r, \theta, \phi)$ = $R_{nl}$ (r) $\cdot Y_{lm} (\theta ,\phi)$
${R_{nl^{(r)}}}$ -radial component (nl) $\rightarrow$ radial node
${Y_{nlm}(\theta ,\phi)}$ - angular component (lm) $\rightarrow$ angular node
Node - region where probability of finding the electron is zero
$\psi \rightarrow 0 (\psi)^2$ = 0

Structure Of The Atom L-6

Angular nodes

${Y_{\ell m}(\theta, \phi)=0}$
Number of angular nodes $=l$

l	number of angular nodes
0	0
1	1
2	2

Structure Of The Atom L-6

Radical nodes

$\quad$ $R_{n l}(r)=0$

Structure Of The Atom L-6

Radical nodes

Radial nodes $R_{n l}(r)=0$
Number of radial nodes
$1s \rightarrow 0$
$2 s \rightarrow 1$
$3 s \rightarrow 2$
$2 p \rightarrow 0$
$3 p \rightarrow 1$

Structure Of The Atom L-6

Radical nodes

$4 p \rightarrow 4-1-1=2$
$3 d \rightarrow 3-2-1=0$
$4 d \rightarrow 4-2-1=1$
$4 f \rightarrow 4-3-1=0$
Number of radial nodes = $n-l-1$
Number of angular nodes = l
Total number of nodes
$n-l-1+l=n-1$

Structure Of The Atom L-6

Boundary surface diagrams

Structure Of The Atom L-6

Boundary surface diagrams

90% of the probability density should be accounted for

Structure Of The Atom L-6

Energy of the orbitals

$H \psi = {E} \psi$
For hydrogen atom( hydrogen like $\mathrm{He}^{+}, \mathrm{Li}^{2+}, \ldots$ )
$\psi_{nlm}$
$E_n$ (depends only on $\ddot{n}$ )
$\psi_{1 s} \rightarrow E_{1 s}$
$\psi_{2 s} \rightarrow E_{2 s}$
$\underbrace{(\psi_{2 p x} , \psi_{2 p y} , \psi_{2 p_z})}_ {E_{2p}}$

Structure Of The Atom L-6

Energy of the orbitals

$\psi_{3 s}\rightarrow E_{3 s}$
$\underbrace{(\psi_{3 p x} , \psi_{3 p y} , \psi_{3 p_z})}_ {E_{3p}}$
$\underbrace{(E_{1S})}_ {1}$ < $\underbrace{( E_{2S} = E_{2px} = E_{2py} = E_{2pz})}_ {4}$ < $\underbrace{(E{3S} = E{3p} = E{3d})}_ {9}$
1,4,9 showing degenerated states

Structure Of The Atom L-6

Energy of the orbitals

For multielectronic systems
$\psi_{nlm} \quad E_{n l} \equiv$ depends on n & l
$1s \quad 2s \quad 2p \quad 3s \quad 3p \quad 3d \quad 4s \quad 4p \quad 4 d \quad 4f$
$n + l$ = $1 \quad 2 \quad 3 \quad 3 \quad 4 \quad 5 \quad 4 \quad 5 \quad 6 \quad 7$
Energy increases with $n+l$
$E_{1s} < E_{2s} < E_{2p} < E_{3s} < E_{3p} < E_{4s} < E_{3d} < E_{4p}$

Structure Of The Atom L-6

Structure of atom

Structure Of The Atom L-6

Quantum number

Structure Of The Atom L-6

Quantum number

Structure Of The Atom L-6

Quantum Number

Structure Of The Atom L-6

Quantum number

Structure Of The Atom L-6

Examples

Structure Of The Atom L-6

Examples

Structure Of The Atom L-6

Spin quantum number

Structure Of The Atom L-6

Summary

Structure Of The Atom L-6

Schrodinger equation

Structure Of The Atom L-6

Angular nodes

Structure Of The Atom L-6

Radical nodes

Structure Of The Atom L-6

Radical nodes

Structure Of The Atom L-6

Radical nodes

Structure Of The Atom L-6

Boundary surface diagrams

Structure Of The Atom L-6

Boundary surface diagrams

Structure Of The Atom L-6

Energy of the orbitals

Structure Of The Atom L-6

Energy of the orbitals

Structure Of The Atom L-6

Energy of the orbitals

Structure Of The Atom L-6

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