Revision: Atoms and Nuclei >> Atoms Physics (Theory) ISC (Science) ISC Class 12 CISCE

When an atomic gas or vapour at low pressure is excited, it emits radiation of certain wavelengths. The emitted radiation, when analysed with a spectroscope, shows a series of bright lines on a dark background. This type of spectrum is called an emission line spectrum.

When white light from a source passes through an atomic gas or vapour, the gas absorbs radiation of certain wavelengths. As a result, dark lines appear in the otherwise continuous spectrum. This type of spectrum is called an absorption spectrum.

• 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.

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

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.

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

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.

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

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.

If the distance between the alpha-particle and the nucleus is rr, then the electrostatic force between them is given by:

F = \[\frac {1}{4πε_0}\] ⋅ \[\frac{2e\cdot Ze}{r^2}\]

\[\Delta E=h\nu=E_i-E_f\]

\[\frac{1}{\lambda_{\mathrm{vac}}}=R_H\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]

where \[R_{H}=1.097\times10^{7}\mathrm{m}^{-1}\] (Rydberg constant)

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

Fine structure constant

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

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

Z = 1

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

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

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

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

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

or

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

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

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

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

\[r_n=(0.53)n^2\mathrm{{Å}}\]

• Proposed by Ernest Rutherford in 1911 based on the gold foil (α-particle scattering) experiment.
• Most α-particles passed straight through, showing that the atom is mostly empty space.
• Some α-particles were deflected, indicating the presence of a positively charged centre.
• Very few α-particles were deflected at large angles or bounced back, proving a dense nucleus.
• All the positive charge and most of the mass are concentrated in a tiny nucleus (~10⁻¹⁵ m).
• Electrons revolve around the nucleus in circular orbits.
• The electrostatic force of attraction between nucleus and electrons keeps them in orbit.
• Limitation: Could not explain stability of atom and line spectra of hydrogen.

• Bohr accepted Rutherford’s nuclear model but modified it using quantum ideas.
• Classical mechanics and electromagnetism could not explain atomic-scale behaviour fully.
• Only certain orbits are allowed for the electron in the hydrogen atom.
• These orbits have definite total energy.
• Electron transitions between energy levels lead to photon emission.
• The energy of the hydrogen atom is negative because the electron is bound to the nucleus.

Two key developments provided the foundation for Bohr's model:

(i) Wave-Particle Duality of Electromagnetic Radiation

Electromagnetic radiation has a dual nature — it behaves both as a wave and as a stream of particles called photons. Each photon carries energy:

where h = Planck's constant = 6.626 × 10⁻³⁴ J·s and ν = frequency.

Key wave properties:

• Wavelength (λ): Distance between two consecutive crests or troughs

• Frequency (ν): Number of waves passing a given point per second (unit: Hz or s⁻¹)

• Wave number \[(\bar{\nu})\]: Number of wavelengths per unit length = 1/λ (unit: cm⁻¹ or m⁻¹)

Relation between speed, frequency, and wavelength:

\[c=\nu\lambda\quad\Rightarrow\quad\nu=\frac{c}{\lambda}\]

Longer wavelength → smaller frequency → lower energy of radiation.

(ii) Quantisation of Energy

Results of atomic spectra showed that atoms absorb or emit energy only in discrete amounts. This gave evidence that energy is quantised — it comes in fixed packets (quanta).

Evolution of Quantum Theory (Timeline):

Classical Theory (matter = particles, radiation = waves)

↓

Einstein & Planck Energy is Quantised

↓

Bohr Line Spectra (Bohr's H atom model)

↓

de Broglie Matter has Wave Nature

↓

Heisenberg Uncertainty Principle

↓

Schrödinger Quantum Theory (matter & radiation both have wave-particle duality)

• 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.

• 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.

• 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.

• 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.