Structure of Atom

Electron ($e^-$)

Discovered by J.J. Thomson.
Charge: $-1.6 \times 10^{-19}\text{ C}$
Mass: $9.1 \times 10^{-31}\text{ kg}$

Proton ($p^+$)

Discovered by Goldstein.
Charge: $+1.6 \times 10^{-19}\text{ C}$
Mass: $1.672 \times 10^{-27}\text{ kg}$

Neutron ($n^0$)

Discovered by Chadwick.
Charge: $0$
Mass: $1.675 \times 10^{-27}\text{ kg}$

Based on $\alpha$-particle scattering on a gold foil.

Conclusions:

• Most space in an atom is empty.
• Positive charge is concentrated in a tiny volume.
• The nucleus is incredibly dense.

Ejection of $e^-$ when light of suitable frequency hits a metal surface.

Einstein's Equation:

$$ h\nu = h\nu_0 + \frac{1}{2} m_e v^2 $$

• $\nu$ = Frequency of incident light
• $\nu_0$ = Threshold frequency
• $K.E.$ = $\frac{1}{2} m_e v^2$

Electrons revolve in well-defined stationary orbits.

Angular Momentum: $$ mvr = \frac{nh}{2\pi} $$ Radius of $n^{\text{th}}$ orbit: $$ r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA} $$ Energy of electron: $$ E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV} $$

1. de Broglie Equation

Matter exhibits dual behavior (particle and wave).

$$ \lambda = \frac{h}{mv} = \frac{h}{p} $$

2. Heisenberg Uncertainty

Impossible to determine exactly both position and momentum simultaneously.

$$ \Delta x \times \Delta p \ge \frac{h}{4\pi} $$

Result: Bohr's fixed orbits are ruled out. We use probability.

• Principal ($n$):
  Main shell ($K, L, M...$), size & energy.

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• Azimuthal ($l$):
  Subshell shape ($s=0, p=1, d=2, f=3$).

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• Magnetic ($m_l$):
  Spatial orientation in magnetic field. Values: $-l$ to $+l$.

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• Spin ($m_s$):
  Electron spin ($+1/2, -1/2$).

• $s$-orbital: Spherical
• $p$-orbital: Dumbbell
• $d$-orbital: Double dumbbell

Nodes Hack:

Angular nodes = $l$
Radial nodes = $n - l - 1$
Total nodes = $n - 1$

• Aufbau Principle: Fill lowest $(n+l)$ energy orbital first.
• Pauli Exclusion: Max 2 electrons per orbital, opposite spins.
• Hund's Rule: Singly occupy degenerate orbitals before pairing.

Exceptions (High Stability):

Cr (24): $[\text{Ar}] 4s^1 3d^5$ (Half-filled)
Cu (29): $[\text{Ar}] 4s^1 3d^{10}$ (Fully-filled)