Atomic model

Bohr model

• electron orbit
• principal quantum number $n$
  • the radius of the orbit gets larger as $n$ increases
• $L=n\hslash$
• Bohr radius
• Balmer series (or Balmer lines)
• Rydberg formula for hydrogen
  • $\frac{1}{\lambda }=(R_{\mathrm{H}})\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

Atomic orbitals

(source: commons.wikimedia.org)

• electron shell

  • orbitals with the same $n$ are in the same shell
  • electrons with the same $n$ are in the same shell

• subshell

  • Electrons with the same $n$ and $\ell$ are said to be in the same subshell
  • Orbitals with the same $n$ and $\ell$ are said to be degenerate, and they are in the same subshell (or sublevel) and they have the same energy level
  • For every subshell (given by $n$ and $\ell$):
    • The number of orbitals is $2\ell +1$.
    • The maximum number of electrons is $2(2\ell +1)$.
  • for a given value of $n$, the energies of subshells folllow order: $ns<np<nd<nf$

• atomic orbital

  • every atomic orbital can hold a maximum of two electrons
  • electrons with the same $n$, $\ell$, and $m_l$ are in the same orbital. (they have the same energy level, shape, and orientation)
  • electron cloud

• principal quantum number ($n$) (מספר קוונטי ראשי, מספר קוונטי יסודי)

  • $n=1,2,3,4,\dots$ (in X-ray notation: $\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\dots$)
  • determines the electron shell
  • higher $n$ → larger orbital
  • higher $n$ → higher energy level (and larger distance from the nucleus)
  • The number of orbitals in a shell is $n^2$.
  • The maximum number of electrons in a shell is $2n^2$.

• azimuthal quantum number ($\ell$) (or orbital angular momentum quantum number)

  • determines the shape of the orbital
  • $\ell =0,1,2,3,\dots ,n-1$ (letters: $s,p,d,f$)

• magnetic quantum number ($m_l$)

  • determines the orientation of the orbital
  • $m_l=-\ell ,\dots ,0,\dots ,\ell$

• spin magnetic quantum number ($m_s$)

  • (sometimes called spin quantum number)
  • determines the orientation of the electron’s spin
  • spin-up ($\uparrow$) $m_s=+\frac{1}{2}$
  • spin-down ($\downarrow$) $m_s=-\frac{1}{2}$

• energy level

• Pauli exclusion principle

  • No two electrons in an atom can have the same set of four quantum numbers.

• Aufbau principle (or Aufbau rule)

  • “a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy subshells are filled before electrons are placed in higher-energy orbitals.” (Wikipedia)
  • Madelung energy ordering rule:
    • subshells with a lower value of $n+\ell$ are filled first
    • if two subshells have the same $n+\ell$ value, the subshell with the lower $n$ is filled first
    • exceptions

• When removing electrons to form a cation, electrons are always removed from the subshell with the highest $n$.

• Hund’s rule of maximum multiplicity

  • ”in a subshell of an atom, electrons are first singly filled with same spin before they are filled doubly” (Wikipedia)

• octet rule

• electron density (or electronic density)

$\ell$	subshell letter	electrons $2(2\ell +1)$	shells containing it	orbitals
$0$	$s$	$2$	Every shell	$s$
$1$	$p$	$6$	2nd shell onwards	$p_x$, $p_y$, $p_z$
$2$	$d$	$10$	3rd shell onwards	$d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$, $d_{z^2}$
$3$	$f$	$14$	4th shell onwards	…
$4$	$g$	$18$	5th shell onwards (theoretically)	…

shell name	$n$	$\ell =0$	$1$	$2$	$3$	shell max electrons $2n^2$
$K$	$1$	$1s$				$2$
$L$	$2$	$2s$	$2p$			$8=2+6$
$M$	$3$	$3s$	$3p$	$3d$		$18=2+6+10$
$N$	$4$	$4s$	$4p$	$4d$	$4f$	$32=2+6+10+14$
$O$	$5$	$5s$	$5p$	$5d$	$5f$	$50=2+6+10+14+18$

• $R(r)$ is the radial wavefunction
• $Y(\theta ,\varphi )$ is the angular wavefunction
• $\psi (r,\theta ,\varphi )=R(r)\cdot Y(\theta ,\varphi )$
• radial probability density $R(r{)}^2$
  • radial probability function $4\pi r^{2}R(r{)}^2$
  • nodes and peaks