Atomic Structure

Planck’s Quantum Theory:

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Energy of one photon $=\mathrm{h} \nu=\frac{\mathrm{hc}}{\lambda}$

Photoelectric Effect:

$$ h v=h v_{0}+\frac{1}{2} m_{e} v^{2} $$

Bohr’s Model for Hydrogen like atoms:

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(1) $m v r=n \frac{h}{2 \pi} $ (Quantization of angular momentum)

(2) $ E_n = - \frac {E_1}{n^2} z^2 $= $-2.178 \times 10^{-18}$

$ \frac{z^2}{n^2} J/atom = -13.6 \frac{z^2}{n^2}eV$

$ E_{1}=\frac{-2 \pi^{2} m e^{4}}{n^{2}} $

(3) $r_{n}=\frac{n^{2}}{z} \times \frac{h^{2}}{4 \pi^{2} e^{2} m}=\frac{0.529 \times n^{2}}{z}A^o$

(4) $\quad \mathrm{v}=\frac{2 \pi z \mathrm{e}^{2}}{\mathrm{nh}}=\frac{2.18 \times 10^{6} \times \mathrm{z}}{\mathrm{n}} \mathrm{m} / \mathrm{s}$

De-Broglie wavelength:

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$$ \lambda=\frac{\mathrm{h}}{\mathrm{mc}}=\frac{\mathrm{h}}{\mathrm{p}} \text { (for photon) } $$

$$ \lambda= \frac{h}{\sqrt2mK.E} $$

Wavelength of emitted photon:

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$$ \frac{1}{\lambda}=\bar{v}=R Z^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right) $$

Line Spectrum of Hydrogen

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The hydrogen spectrum is the line spectrum emitted by a hydrogen atom when an excited hydrogen atom returns to its ground state.

The various levels of excited state from which a hydrogen atom can return to its lower energy state give rise to various series.

These series are the Lyman series, Balmer series, Paschen series, Brackett series, and Pfund series.

Rydberg’s formula gives the wavelength of the hydrogen spectrum as,

$$ \frac{1}{\lambda}=\bar{v}=R Z^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right) $$

Degeneracy

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Degeneracy of orbitals means that the orbitals are of equal energy. Such orbitals are called degenerate orbitals. In hydrogen the level of energy degeneracy is as follows: 1s, 2s = 2p, 3s = 3p = 3d, 4s = 4p = 4d = 4f,…

No. of photons emitted by a sample of $\mathrm{H}$ atom:

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$$ \frac{\Delta \mathrm{n}(\Delta \mathrm{n}+1)}{2} $$

Heisenberg’s uncertainty principle:

$\Delta \mathrm{x} . \Delta \mathrm{p}>\frac{\mathrm{h}}{4 \pi}$ or

$\mathrm{m} \Delta \mathrm{x} \cdot \Delta \mathrm{v} \geq \frac{\mathrm{h}}{4 \pi}$ or

$\Delta \mathrm{x} \cdot \Delta \mathrm{v} \geq \frac{\mathrm{h}}{4 \pi \mathrm{m}}$

Quantum Numbers:

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  • Principal quantum number $(n)=1,2,3,4 \ldots$. to $\infty$.

  • Orbital angular momentum of electron in any orbit $=\frac{n h}{2 \pi}$.

  • Azimuthal quantum number $(\ell)=0,1, \ldots .$. to $(n-1)$.

  • Number of orbitals in a subshell $=2 \ell+1$

  • Maximum number of electrons in $n^{th}$ orbit $=2 n^{2}$

  • Maximum number of electrons in particular subshell $=2 \times(2 \ell+1)$

  • Hund’s rule : (n + l)

  • Orbital angular momentum

$\mathrm{L}=\frac{\mathrm{h}}{2 \pi} \sqrt{\ell(\ell+1)}=\hbar \sqrt{\ell(\ell+1)}$

$ [\hbar=\frac{\mathrm{h}}{2 \pi}] $

Electron Configurations

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The electron configuration of an element describes how electrons are distributed in its atomic orbitals. Electron configurations of atoms follow a standard notation in which all electron-containing atomic subshells (with the number of electrons they hold written in superscript) are placed in a sequence. For example, the electron configuration of sodium is $1 s^2 2 s^2 2 p^6 3 s^1$.

Writing Electron Configurations

Shells The maximum number of electrons that can be accommodated in a shell is based on the principal quantum number ( $n$ ). It is represented by the formula $2 n^2$, where ’ $n$ ’ is the shell number. The shells, values of $n$, and the total number of electrons that can be accommodated are tabulated below.

Shell and ’ $n$ ’ value Maximum electrons present in the shell
K shell, $n=1$ $2^* 1^2=2$
L shell, $n=2$ $2^* 2^2=8$
M shell, $n=3$ $2^* 3^2=18$
N shell, $n=4$ $2^* 4^2=32$

$ \newline$

Subshells

  • The subshells into which electrons are distributed are based on the azimuthal quantum number (denoted by ‘T).
  • This quantum number is dependent on the value of the principal quantum number, $\mathrm{n}$. Therefore, when $n$ has a value of 4 , four different subshells are possible.
  • When $\mathrm{n}=4$. The subshells correspond to $\mathrm{l}=0, \mathrm{l}=\mathrm{l}, \mathrm{l}=2$, and $\mathrm{l}=3$ and are named the $\mathrm{s}, \mathrm{p}, \mathrm{d}$, and f subshells, respectively.
  • The maximum number of electrons that can be accommodated by a subshell is given by the formula $2^*(21+1)$.
  • Therefore, the $s, p, d$, and $f$ subshells can accommodate a maximum of $2,6,10$, and 14 electrons, respectively.

Notation

  • The electron configuration of an atom is written with the help of subshell labels.
  • These labels contain the shell number (given by the principal quantum number), the subshell name (given by the azimuthal quantum number) and the total number of electrons in the subshell in superscript.
  • For example, if two electrons are filled in the ’ $s$ ’ subshell of the first shell, the resulting notation is $1 \mathrm{~s}^2$.
  • With the help of these subshell labels, the electron configuration of magnesium (atomic number 12) can be written as $1 s^2 2 s^2 2 p^6 3 s^2$.

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Henry Moseley Equation

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Henry Moseley observed regularities in the characteristic $X$-ray spectra of the elements.

A plot of $\sqrt{v}$ (where $V$ is frequency of $X$-rays emitted) against atomic number $(Z)$ gave a straight line and not the plot of $\sqrt{v}$ vs atomic mass.

It showed that the atomic number is a more fundamental property of an element than its atomic mass.

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Radial nodes

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The radial nodes (i.e., probability density function is zero), the probability density functions for the $\mathrm{n} p$ and $\mathrm{nd}$ orbitals are zero at the plane (s), passing through the nucleus (origin). For example, in case of $p_z$ orbital, xy-plane is a nodal plane, in case of $d_{x y}$ orbital, there are two nodal planes passing through the origin and bisecting the xy plane containing $\mathrm{z}$-axis. These are called angular nodes and number of angular nodes are given by ’ $l$, i.e., one angular node for $p$ orbitals, two angular nodes for ’ $d$ ’ orbitals and so on. The total number of nodes are given by ( $n-1$ ), i.e., sum of $l$ angular nodes and $(n-l-1)$ radial nodes.



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