Overview: Atom, Origin of Spectra : Bohr's Theory of Hydrogen Atom

• In the α-particle scattering experiment, most α-particles passed straight through the thin gold foil, but a very small number were deflected through large angles, even up to 180°.
• The undeflected particles indicate that the atom is mostly empty space rather than a solid sphere as previously assumed.
• The large-angle deflections proved that the positive charge and most of the mass of the atom are concentrated in a very small central region called the nucleus.
• The nucleus is extremely small (≈10⁻¹⁴ m) compared to the atom (≈10⁻¹⁰ m), so only a few α-particles come close enough to be strongly repelled.
• The scattering follows Coulomb’s law, and the positive charge on the nucleus is equal to Ze, where Z is the atomic number.

K = \[\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{r_0}\]

or

\[r_0=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{K}\]

The nucleus has protons and neutrons. The nuclear particles (protons and neutrons) are also called ‘nucleons'.

The total number of protons and neutrons is equal to the integral value of the atomic mass and is called the 'atomic mass number'.

The number of protons is called the 'atomic number'.

• In Bohr’s model, the positive charge is concentrated at the centre, and electrons revolve around the nucleus in certain permitted circular orbits called stationary orbits.
• The angular momentum of an electron in a stationary orbit is quantised and given by
  mvr = \[\frac {nh}{2π}\]​
  where n = 1,2,3,... is the principal quantum number.
• Electrons in stationary orbits do not radiate energy despite being accelerated; hence, the atom remains stable.
• When an atom absorbs energy, an electron moves to a higher orbit (excited state) and stays there for a very short time (~10⁻⁸ s).
• When the electron returns to a lower orbit, it emits radiation of frequency
  hν = E₂ − E₁​
  which is called Bohr’s frequency condition.

r = \[n^2\frac{h^2\varepsilon_0}{\pi mZe^2}\]

Z = 1

r = \[=n^2\frac{h^2\varepsilon_0}{\pi me^2}\]

\[r_n=(0.53)\frac{n^2}{Z}\mathrm{{Å}}\]

for hydrogen atom radius of nth orbit

\[r_n=(0.53)n^2\mathrm{{Å}}\]

v = \[\frac{Ze^2}{2h\varepsilon_0}\frac{1}{n}\]

Fine structure constant

\[v=\frac{1}{137}\left(\frac{cZ}{n}\right)\]

\[E=-\frac{mZ^{2}e^{4}}{8\varepsilon_{0}^{2}h^{2}}\left(\frac{1}{n^{2}}\right)\]

\[\frac{1}{\lambda}=\frac{v}{c}=\frac{mZ^{2}e^{4}}{8\varepsilon_{0}^{2}ch^{3}}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\]

\[\frac{me^4}{8\varepsilon_0^2ch^3}=R\] = Rydberg's constant

\[\frac{1}{\lambda}=Z^{2}R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\]

Z = 1

\[\frac{1}{\lambda}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\]

The minimum accelerating potential required to energise an electron which, on collision, can excite an atom is called the 'excitation potential' of that atom.

The minimum accelerating potential required to energise an electron which can ionise an atom is called the 'ionisation potential' of that atom.

The shifting of the atom from one energy state to another is called 'transition'.

\[\lambda=\frac{12375^*}{\Delta E\mathrm{~in~eV}}\mathrm{\r{A}}=\left[\frac{1237.5}{\Delta E\mathrm{~(in~eV)}}\right]\mathrm{nm}\]

• The hydrogen spectrum consists of bright discrete lines arranged in series such as the Lyman, Balmer, Paschen, Brackett, and Pfund series.
• Balmer gave the formula for visible lines:
  \[\frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{n^2}\right)\]
  where R is the Rydberg constant.
• According to Bohr’s theory, electrons emit radiation when they move from a higher energy level to a lower level.
• The general formula for the hydrogen spectrum is
  \[\frac{1}{\lambda}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\]
  with n₂ > n₁​.
• The Lyman series lies in the ultraviolet, the Balmer series in the visible, and the Paschen, Brackett, and Pfund series in the infrared.
• The shortest wavelength of a series (when n₂ = ∞) is called the series limit.
• In the emission spectrum, all series are observed, but in absorption at ordinary temperatures, the Lyman series is predominant because most atoms are in the ground state.

Energy required to move an electron from ground state of the atom to any other excited state of the atom is called excitation energy of that state.

Minimum energy required to move an electron from ground state to n = o or to knock a ground state electron completely out of the atom.

\[\frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{n^2}\right)\]