Atomic Structure Formulas

1. ThomsonΓÇÖs model of an atom

Entire mass and positive charge of an atom is uniformly distributed in a sphere of radius about 10^-10 m and negatively charged electrons are embedded in the sphere.

2. RutherfordΓÇÖs model of an atom

Entire mass, positive charge of an atom are concentrated in a very small j region of radius about 10^-15m, called nucleus. Electrons revolve round this nucleus in circular orbits.

3. Rutherford scattering, impact parameter and distance of closest approach

(i) The relation between the number of ╬▒-particles scattered N (╬╕) and the angle of scattering ╬╕ is
N(╬╕) Γê¥ cosec⁴ (╬╕/2)

(ii) Impact parameter
b = \(\frac{2 \mathrm{Ze}^{2} \cot (\theta / 2)}{4 \pi \varepsilon_{0} \mathrm{mv}_{0}^{2}}=\frac{\mathrm{kZe}^{2} \cot (\theta / 2)}{\mathrm{E}}\)

(iii) Distance of closest approach, If b = 0
r₀ = \(\frac{2 \mathrm{Zeq}}{4 \pi \varepsilon_{0} \mathrm{mv}_{0}^{2}}=\frac{2(\mathrm{Ze})(2 \mathrm{e})}{4 \pi \varepsilon_{0} \mathrm{mv}_{0}^{2}}=\frac{\mathrm{k} 2 \mathrm{Ze}^{2}}{\mathrm{E}}\)
E = Kinetic energy of a particle

4. BohrΓÇÖs model of an atom

(i) Electrons revolve round the nucleus in circular orbits of constant energy.
Centripetal force = Coulomb force
\(\frac{m v^{2}}{r}=k \frac{Z e^{2}}{r^{2}}\)

(ii) The orbits in which the angular momentum of an electron is integral multiple h/2╧Ç, are stable orbits. (Bohr’s quantum condition)
L = mvr = \(\frac{n h}{2 \pi}\)

(iii) A photon is emitted or absorbed due to transition of an electron from one energy level to another energy level. Thus
E₂ – E₁ = h╬╜ (in emission)
or E₁ – E₂ = h╬╜ (in absorption)
(Bohr’s frequency condition)

5. State of an electron in an atom

(i) The radius of n-th orbit:
r_n = \(\frac{n^{2} h^{2}}{4 \pi^{2} m K e^{2} Z}\)
= \(a_{0} \frac{n^{2}}{Z}\)

r_n ∝ \(\frac{n^{2}}{Z}\)

(ii) The velocity of electron in n-th orbit:
v_n = \(\frac{2 \pi \mathrm{Ke}^{2} \mathrm{Z}}{\mathrm{nh}}\)
= 2.188 x 10⁶ \(\frac{\mathrm{Z}}{\mathrm{n}}\) m/s
= \(\frac{c}{137}\left(\frac{Z}{n}\right)\)
v_n ∝ \(\frac{\mathrm{Z}}{\mathrm{n}}\)
For hydrogen

(iii) The period of an electron in n-th orbit
T_n = \(\frac{2 \pi r_{n}}{v_{n}}=\frac{n^{3} h^{3}}{4 \pi^{2} K^{2} m c^{4} Z^{2}}\) sec, T_n ∝ \(\frac{\mathrm{n}^{3}}{\mathrm{Z}^{2}}\)

(iv) Energy of an electron in n-th orbit (in hydrogen atom) For a BohrΓÇÖs atom of atomic number Z:
(a) Kinetic energy KE = \(\frac{\mathrm{KZe}^{2}}{2 \mathrm{r}_{\mathrm{n}}}=\frac{\mathrm{R} \mathrm{ch} \mathrm{Z}^{2}}{\mathrm{n}^{2}}, \mathrm{KE} \propto \frac{\mathrm{Z}^{2}}{\mathrm{n}^{2}}\)

(b) Potential energy U = \(-\frac{\mathrm{KZe}^{2}}{\mathrm{r}_{\mathrm{n}}}=-\frac{2 \mathrm{R} \mathrm{ch} \mathrm{Z}^{2}}{\mathrm{n}^{2}}, \mathrm{U} \propto \frac{\mathrm{Z}^{2}}{\mathrm{n}^{2}}\)

(c) Total energy

E ∝ \(\frac{Z^{2}}{n^{2}}\)

(d) Binding Energy B.E. = \(\frac{13.6}{n^{2}}\) Z² eV
BE ∝\(\frac{Z^{2}}{n^{2}}\)

6. Energy, frequency and wave number of emitted or abosrbed photons by an atom

(a) Energy of photon
h╬╜ = (E_n2 – E_n1)
= \(\frac{2 \pi^{2} \mathrm{K}^{2} \mathrm{me}^{4}}{\mathrm{h}^{2}}\left[\frac{1}{\mathrm{n}_{1}^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right]\)

(b) Frequency of photon
\(\bar{v}=\frac{2 \pi^{2} K^{2} m e^{4}}{h^{3}}\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]\)

(c) Wave number

Z = 1 for H-atom
Where R is Rydberg constant
R = \(\frac{2 \pi^{2} K^{2} m e^{4}}{c h^{3}}\) = 1.1 ├ù 10⁷ m^-1
For a Bohr atom of atomic number

7. Various series of hydrogen spectrum

(i) Lyman series (in ultra violet region)
\(\bar{v}=\frac{1}{\lambda}=R\left[\frac{1}{1^{2}}-\frac{1}{n_{2}^{2}}\right]\), n₂ = 2, 3, 4……….
Minimum wavelength λ_min = 912 Å

(ii) Balmar Series (In visible region)
\(\vec{v}=\frac{1}{\lambda}=R\left[\frac{1}{2^{2}}-\frac{1}{n_{2}^{2}}\right]\), n₂ = 3, 4, 5 ………….
Minimum wavelength λ_min = 3645 Å

(iii) Paschen series (Infra red region)
\(\bar{v}=\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{3^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right]\), n₂ = 4, 5, 6…….
Minimum wavelength λ_min = 8210 Å

(iv) Brackett series
n₁ = 4 and n₂ = 5, 6, 7……….

(v) Pfund series
n₁ = 5 and n₂ = 6, 7, 8………

(vi) If an electron transition takes place from n-th orbit, then maximum number of lines m the spectrum = \(\frac{n(n-1)}{2}\)

8. Ioniztion and excitation energies, ionization and excitation potentials

Ionization energy = (E_Γê₧ – E_n) eV
Excitation energy = (E_n+1 – E_n) eV
Ionization Potential = (E_Γê₧ – E_n) V
Excitation Potential = (E_n+1 – E_n) V

9. Quantum numbers

• Principal quantum number (n): possible values of n = 1, 2, 3…….
• Orbital quantum number (l): possible values from 0 to (n – 1).
• Orbital magnetic quantum number (m₁): possible values are (2l + 1) integral from -l to +l.
• Spin quantum number (s) – possible values are (+\(\frac{1}{2}\)) and (-\(\frac{1}{2}\))

Orbital angular momentum = \(\frac{h}{2 \pi} \sqrt{l(l+1)}\)
Spin angular momentum = \(\frac{h}{2 \pi} \sqrt{s(s+1)}\)

10. Numbers of electrons

• Maximum number of electrons in an orbit 2n².
• Maximum number of electrons in a sub-orbit = 2(2l + 1)

11. PauliΓÇÖs exclusion principle
No two electrons in an atom can have the same set of four quantum numbers.

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