Chemca Formula Sheet - Structure of Atom

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Structure of Atom

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h (Planck's) \(6.626 \times 10^{-34} \text{ J s}\)

c (Light speed) \(3 \times 10^{8} \text{ m/s}\)

Mass of \(e^-\) \(9.1 \times 10^{-31} \text{ kg}\)

1 eV \(1.602 \times 10^{-19} \text{ J}\)

1. Fundamental Particles & Rutherford's Model

Specific Charge (\(e/m\)) Ratio Order:

Electron > Proton > \(\alpha\)-particle > Neutron

Neutron's specific charge is zero.

Distance of Closest Approach (\(r_0\)): \[ r_0 = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{K.E._{initial}} \]

At \(r_0\), entire Kinetic Energy is converted to Electrostatic Potential Energy.

2. EM Radiation & Photoelectric Effect

Planck's Quantum Theory:

\[ E = nh\nu = \frac{nhc}{\lambda} \]

Shortcut: \(E (\text{eV}) \approx \frac{12400}{\lambda (\text{\AA})} \approx \frac{1240}{\lambda (\text{nm})}\)

Photoelectric Effect Equation:

\[ h\nu = \Phi + K.E._{max} \] \[ \frac{hc}{\lambda} = \frac{hc}{\lambda_0} + eV_0 \]

\(\Phi\) = Work Function, \(\lambda_0\) = Threshold Wavelength, \(V_0\) = Stopping Potential

Key Dependencies:

• Number of photoelectrons ejected \(\propto\) Intensity of incident light.
• Kinetic Energy (and \(V_0\)) \(\propto\) Frequency of incident light (Intensity has no effect on K.E.).

3. Bohr's Atomic Model (H and H-like species)

Applicable for 1-electron systems: \(H, He^+, Li^{2+}, Be^{3+}, \dots\)

Angular Momentum (Quantization):

\[ mvr = \frac{nh}{2\pi} \]

Radius of \(n^{th}\) orbit:

\[ r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA} \]

Velocity of \(e^-\):

\[ v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s} \]

Energy of \(n^{th}\) orbit:

\[ E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV/atom} \]

\(P.E. = 2 \times E_n\)

\(K.E. = -E_n\)

\(P.E. = -2 \times K.E.\)

\(T.E. = E_n\)

Frequency of Revolution (\(f\)):

\[ f = \frac{v}{2\pi r} \implies f \propto \frac{Z^2}{n^3} \]

Time Period (\(T\)):

\[ T = \frac{2\pi r}{v} \implies T \propto \frac{n^3}{Z^2} \]

4. Hydrogen Spectrum

Rydberg Equation (Wave Number): \[ \bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

\(R_H = 1.09677 \times 10^7 \text{ m}^{-1} \approx 109677 \text{ cm}^{-1}\)

Total Spectral Lines: Electron dropping from \(n_2\) to \(n_1\)

\[ \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]

Series	\(n_1\)	\(n_2\)	Region
Lyman	1	2, 3, 4...	Ultraviolet (UV)
Balmer	2	3, 4, 5...	Visible
Paschen	3	4, 5, 6...	Infrared (IR)
Brackett	4	5, 6, 7...	Infrared (IR)
Pfund	5	6, 7, 8...	Far Infrared

Max/Min Wavelength Trick:

• Longest Wavelength (\(\lambda_{max}\)) / Minimum Energy: First line of the series (\(n_2 = n_1 + 1\)). E.g., \(H_\alpha\) line.
• Shortest Wavelength (\(\lambda_{min}\)) / Maximum Energy: Series limit line (\(n_2 = \infty\)).

5. Quantum Mechanics: Dual Nature & Uncertainty

de-Broglie Wavelength:

\[ \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m(K.E.)}} \]

For an electron accelerated by \(V\) volts: \(\lambda \approx \frac{12.27}{\sqrt{V}} \text{ \AA}\)

Heisenberg Uncertainty Principle:

\[ \Delta x \cdot \Delta p \ge \frac{h}{4\pi} \] \[ \Delta x \cdot \Delta v \ge \frac{h}{4\pi m} \]

\(\Delta x\) = Uncertainty in position, \(\Delta p\) = Uncertainty in momentum

6. Quantum Numbers & Electronic Configuration

Quantum Number	Symbol	Possible Values	Physical Significance
Principal	\(n\)	\(1, 2, 3 \dots \infty\)	Size and Energy of the main shell.
Azimuthal (Angular)	\(l\)	\(0\) to \((n-1)\) (0=s, 1=p, 2=d, 3=f)	Shape of the subshell. Orbital angular momentum \(L = \sqrt{l(l+1)}\hbar\)
Magnetic	\(m_l\)	\(-l\) to \(+l\) (including 0) Total \((2l+1)\) values	Orientation of orbitals in space.
Spin	\(m_s\)	\(+1/2\) (\(\uparrow\)), \(-1/2\) (\(\downarrow\))	Spin direction. Spin angular momentum \(S = \sqrt{s(s+1)}\hbar\)

Nodes in Orbitals:

Radial (Spherical)

\(n - l - 1\)

Angular (Planes)

\(l\)

Total Nodes

\(n - 1\)

Magnetic Moment (\(\mu\)):

\[ \mu = \sqrt{n(n+2)} \text{ B.M.} \]

\(n\) = number of unpaired electrons.

Rules for Filling Electrons:

1. Aufbau Principle: Electrons fill orbitals in order of increasing energy (\((n+l)\) rule). If \((n+l)\) is same, lower \(n\) fills first.

Exceptions: \(Cr (3d^5 4s^1)\) and \(Cu (3d^{10} 4s^1)\) due to extra stability of half-filled/fully-filled orbitals (Symmetry & Exchange Energy).

2. Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. (An orbital holds max 2 electrons with opposite spins).

3. Hund's Rule of Maximum Multiplicity: Electron pairing in degenerate orbitals (like p, d, f) does not occur until each orbital is singly occupied with parallel spins.

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