Definitions [4]
Completely removing an electron from atom is called ionisation.
The atom emits a photon with energy equal to the difference between the two energy levels, a phenomenon known as luminescence.
- The collection of different spectral lines obtained due to transition of an electron in hydrogen atom from upper energy levels to lower energy levels is called the Hydrogen Spectrum.
- The hydrogen spectrum consists of specific wavelengths of light emitted by hydrogen atoms. When transition of an electron in a hydrogen atom occurs between energy levels, it emits or absorbs photons of certain wavelengths, creating a series of lines known as the hydrogen spectrum.
The spectrum consisting of bright lines on a dark background, emitted when an atomic gas is excited at low pressure by passing an electric current through it, is called the Emission Line Spectrum.
Formulae [13]
\[L=mvr=\frac{nh}{2\pi},\quad n=1,2,3\ldots\]
\[h\nu=E_2-E_1=\frac{hc}{\lambda}\]
\[\bar{v}=\frac{1}{\lambda}=RZ^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\mathrm{m}^{-1}\]
where \[R=1.097\times10^7\mathrm{m}^{-1}\] (Rydberg constant)
\[E_\text{ionisation}=13.6Z^2\mathrm{~eV}\]
\[E_n=-Rhc\left(\frac{1}{n^2}\right)\]
Radius of the n-th Bohr Orbit (General):
\[r_n=\frac{\varepsilon_0n^2h^2}{\pi mZe^2}\]
\[\mathrm{i.e.,}r_n\propto n^2\mathrm{and}r_n\propto\frac{1}{Z}\]
Radius of n-th orbit for Hydrogen-like atom:
\[r_n=0.53\left(\frac{n^2}{Z}\right)\mathrm{\r{A}}\]
\[v_n=\frac{nh}{2\pi mr_n}\]
\[r=\frac{n^2h^2}{4\pi^2mkZe^2}\]
\[v=\frac{2\pi kZe^2}{nh}\]
For hydrogen atom (Z = 1):
\[v=\frac{2\pi ke^2}{nh}=\alpha\frac{c}{n}\]
where α is the fine structure constant and \[\alpha=\frac{1}{137}.\]
Total Energy of Electron in n-th Orbit (General):
- \[E_n=\frac{-Z^2me^4}{8\varepsilon_0^2n^2h^2}\]
Total Energy (Alternate form):
- \[E_n=-\frac{2\pi^2mk^2Z^2e^4}{n^2h^2}\]
Total Energy for Hydrogen-like Atom (Simplified):
- \[E_n=-13.6\frac{Z^2}{n^2}\mathrm{eV},\quad n=1,2,3\ldots\]
\[\Delta E=h\nu=E_i-E_f\]
\[\frac{1}{\lambda_{\mathrm{vac}}}=R_H\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]
where \[R_{H}=1.097\times10^{7}\mathrm{m}^{-1}\] (Rydberg constant)
General Formula for all Spectral Series:
\[\frac{1}{\lambda}=R\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]
where \[R=1.097\times10^7\mathrm{m}^{-1},n_1=\text{final state},n_2=\text{initial state},n_2>n_1\]
| Series | Final State (n₁) | Formula |
|---|---|---|
| Lyman | n₁ = 1 | \[\frac{1}{\lambda}=R\left[\frac{1}{1^2}-\frac{1}{n_2^2}\right]\] |
| Balmer | n₁ = 2 | \[\frac{1}{\lambda}=R\left[\frac{1}{2^2}-\frac{1}{n_2^2}\right]\] |
| Paschen | n₁ = 3 | \[\frac{1}{\lambda}=R\left[\frac{1}{3^2}-\frac{1}{n_2^2}\right]\] |
| Brackett | n₁ = 4 | \[\frac{1}{\lambda}=R\left[\frac{1}{4^2}-\frac{1}{n_2^2}\right]\] |
| Pfund | n₁ = 5 | \[\frac{1}{\lambda}=R\left[\frac{1}{5^2}-\frac{1}{n_2^2}\right]\] |
Theorems and Laws [1]
Bohr's First Postulate:
An atom consists of a small, massive central core called the nucleus, around which planetary electrons revolve. The centripetal force required for their rotation is provided by the electrostatic attraction between the electrons and the nucleus.
Bohr's Second Postulate (Quantum Condition):
The electrons are permitted to circulate only in those orbits in which the angular momentum of an electron is an integral multiple of \[\frac{h}{2\pi}\]; h being Planck's constant.
Bohr's Third Postulate:
While revolving in the permissible orbits, an electron does not radiate energy. These non-radiating orbits are called stationary orbits.
Bohr's Fourth Postulate:
An atom can emit or absorb radiation in the form of discrete energy photons only when an electron jumps from a higher to a lower orbit or from a lower to a higher orbit, respectively.
Key Points
- Most alpha particles passed through the gold foil without any deflection, proving the atom is mostly empty space.
- Around 0.14% of incident alpha particles are scattered by more than 1°.
- Around 1 in 8000 alpha particles are deflected by more than 90°.
- Large-angle deflections indicated a small, dense, positively charged nucleus at the centre of the atom.
- The gold foil used had a thickness of 2.1 × 10⁻⁷ m; the alpha particles had an energy of 5.5 MeV.
- Electrons have negligible mass and do not affect the trajectory of incident alpha particles.
- This experiment disproved Thomson's plum-pudding model and established the nuclear structure of the atom.
- Rutherford's model could not explain the stability of an atom despite the revolving electrons around the nucleus.
- Electrons revolving around the nucleus emit radiation and subsequently lose energy.
- Loss of energy causes electrons to spiral inward and eventually fall into the nucleus, leading to the collapse of the atom - but this is untrue.
- If continuous energy loss occurred, electrons would fall into the nucleus, making the atom unstable - which contradicts actual observations.
- If electrons emit continuous energy, they should form a continuous spectrum, but actually, a line spectrum is obtained, which Rutherford's model fails to explain.
- Bohr modified Rutherford's model - electrons move in fixed orbital shells, each with fixed energy levels.
- The centripetal force for electron revolution is provided by electrostatic attraction between the electron and the nucleus.
- An electron does not radiate energy while revolving in a stationary orbit.
- Energy is emitted or absorbed only during electron transitions between orbits.
- Limitations of Bohr's Model:
- Fails to explain the Zeeman Effect (effect of high magnetic fields on atomic spectra).
- Contradicts the Heisenberg Uncertainty Principle.
- Unable to explain the spectra of larger/multi-electron atoms.
- For hydrogen (Z = 1): ground state energy = −13.6 eV; at n = ∞, energy = 0 eV.
- Energy levels for hydrogen: n=1: −13.6 eV, n=2: −3.4 eV, n=3: −1.511 eV, n=4: −0.850 eV, n=5: −0.544 eV.
- In normal conditions, electrons are in the ground state, occupying orbitals closest to the nucleus.
- Beyond ionisation potential, the electron is no longer bound — energy levels form a continuum (starts at 13.6 eV above ground in hydrogen).
- Electrons in orbitals close to the nucleus are stable (need more energy to remove); electrons farther away are less stable.
- If energy supplied ≥ ionisation energy, ionisation occurs.
- Spectral Series (from energy level transitions): Lyman, Balmer, Paschen, Bracket, Pfund series.
- The radius of Bohr's orbit is proportional to \[n^{2}\] and inversely proportional to Z.
- For hydrogen (Z = 1), the ground state (n = 1) radius is 0.53 Å, known as Bohr's radius.
- The velocity of an electron decreases as the orbital number (n) increases.
- For hydrogen, orbital speed of electron equals \[\alpha\frac{c}{n}\], where \[\alpha=\frac{1}{137}\].
- The total energy of an electron in any orbit is negative, indicating a bound state.
- For hydrogen-like atoms, the energy of an electron in the n-th orbit is \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .
- Lyman series — transitions to n = 1; region: ultraviolet
- Balmer series — transitions to n = 2; region: visible
- Paschen series — transitions to n = 3; region: infrared
- Brackett series — transitions to n = 4; region: infrared
- Pfund series — transitions to n = 5; region: infrared
- The spectrum of hydrogen is important as most of the universe is made of hydrogen.
- Balmer series involves transitions starting/ending with the first excited state (n = 2) of hydrogen.
- Lyman series — n₁=1, n₂: 2→∞; converges toward 91–122 nm; UV region
- Balmer series — n₁=2, n₂: 3→∞; converges toward 365–657 nm; visible region (Hα = Red, Hβ = Blue-green, Hγ = Blue)
- Paschen series — n₁=3, n₂: 4→∞; converges toward 821–1876 nm; infrared (IR)
- Brackett series — n₁=4, n₂: 5→∞; converges toward 1459–4053 nm; IR region
- Pfund series — n₁=5, n₂: 6→∞; converges toward 2280–7462 nm; IR region
- Humphreys series — n₁=6, n₂: 7→∞; converges toward 3283 nm–∞; IR region
- Setting n₁=1 and n₂ from 2 to ∞ gives the Lyman series converging to 91 nm
