Atoms Class 12 Physics Chapter 12 Notes

19th Century Atomic Hypothesis:

By the 19th century, accumulating evidence supported the atomic hypothesis of matter.
In 1897, J.J. Thomson’s experiments on electric discharge through gases revealed the presence of negatively charged constituents (electrons) in all atoms.
Atoms as a whole are electrically neutral, indicating the existence of positive charges to balance electrons.
The arrangement of positive charges and electrons inside the atom, or the atom’s structure, was unknown.

Thomson’s Plum Pudding Model (1898):

J.J. Thomson proposed the first model of the atom.
According to this model, the positive charge is uniformly distributed throughout the atom, and electrons are embedded within it, like seeds in a watermelon.
Subsequent studies showed that the actual distribution of electrons and positive charges differs from this model.

Electromagnetic Radiation in Condensed Matter and Gases:

Condensed matter (solids, liquids) and dense gases emit electromagnetic radiation with a continuous distribution of wavelengths.
This radiation results from atom and molecule oscillations and interactions with neighbors.
In contrast, rarefied gases in flame or electrically excited emit light with discrete wavelengths, appearing as bright lines.
This is due to the emission by individual atoms in these rarefied gases.

Characteristic Spectra and Hydrogen:

Each element has a characteristic spectrum of radiation.
For example, hydrogen consistently emits a set of lines with fixed relative positions.
There is a connection between the internal structure of an atom and the emitted spectrum.

Rutherford’s Planetary Model (Nuclear Model):

Ernst Rutherford, a former student of J.J. Thomson, proposed an experiment using alpha particles to investigate atomic structure.
Hans Geiger and Ernst Marsden conducted the experiment in 1911.
The results led to Rutherford’s planetary model of the atom, where most of the mass and positive charge are concentrated in the nucleus, and electrons revolve around it.

Challenges in Rutherford’s Model:

Rutherford’s model couldn’t explain why atoms emit light with only specific, discrete wavelengths.
Particularly, it couldn’t explain the complex spectrum of hydrogen, which consists of a single electron and a single proton.
The classical atom model, where electrons orbit the nucleus like planets around the sun, had serious difficulties.

Alpha-particle scattering experiment and Rutherford’s nuclear model of the atom

Alpha-Particle Scattering Experiment (1911):

Ernst Rutherford suggested the experiment.
Conducted by H. Geiger and E. Marsden.
A beam of 5.5 MeV alpha particles from a radioactive source was directed at a thin gold foil.
The apparatus included a rotatable detector to observe scattered alpha particles.
The experiment’s goal was to study the scattering of alpha particles as a function of scattering angle.

Experimental Results:

A typical graph of the number of alpha particles scattered at different angles was obtained.
Only about 0.14% of alpha particles scattered by more than 1°, and around 1 in 8000 deflected by more than 90°.
Rutherford concluded that for an alpha particle to deflect significantly, it must experience a large repulsive force.
This suggested the presence of a small, dense, positively charged nucleus within the atom.

Rutherford’s Nuclear Model of the Atom:

In Rutherford’s model, the atom has a small, dense, positively charged nucleus at its center.
Most of the atom’s positive charge and mass are concentrated in the nucleus.
Electrons are located at a distance from the nucleus and move in orbits, similar to planets orbiting the sun.

Atomic Spectra

Characteristics of Atomic Spectra:

Every element has a unique and characteristic spectrum of radiation.
When an atomic gas or vapor is excited at low pressure (typically by passing an electric current through it), the resulting emitted radiation contains specific, discrete wavelengths.
This type of spectrum is known as an emission line spectrum.
In an emission line spectrum, bright lines appear against a dark background, representing the specific wavelengths of light emitted.

Use as a Fingerprint:

The emission line spectrum of a material serves as a distinctive “fingerprint” for the identification of the gas or element.
Each element’s emission line spectrum is unique, helping in its identification.

Absorption Spectrum:

When white light passes through a gas and the transmitted light is analyzed using a spectrometer, dark lines appear in the spectrum.
These dark lines correspond to the wavelengths found in the emission line spectrum of the gas.
This is known as the absorption spectrum of the material or gas.
The absorption spectrum represents the wavelengths of light that are absorbed by the material.

Bohr model of the hydrogen atom

Rutherford’s Atom Model and its Issues:

Rutherford’s atom model, with a central nucleus and revolving electrons, resembles a sun-planet system.
In this model, charged objects interact through Coulomb’s Law of force, unlike gravitational force that holds planets in orbit.
Accelerating charged particles, according to classical electromagnetic theory, emit electromagnetic waves.
This radiation should lead to a continuous decrease in the energy of accelerating electrons, causing them to spiral inward and eventually collide with the nucleus, rendering the atom unstable.
The classical theory suggests that the frequency of emitted electromagnetic waves would continuously change as the electrons spiral inwards, leading to a continuous spectrum, which contradicts the observed line spectrum.

Niels Bohr’s Contribution:

Niels Bohr, who studied in Rutherford’s laboratory, recognized the limitations of classical electromagnetism in explaining atomic-scale phenomena.
In 1913, Bohr introduced a modified atomic model that combined classical and early quantum concepts, addressing the shortcomings of classical mechanics and electromagnetism.

Bohr’s Three Postulates:

Stable Orbits: Electrons in an atom can revolve in certain stable orbits without emitting radiant energy. These stable states have definite total energy and are called stationary states.
Quantized Angular Momentum: Electrons only revolve in orbits where the angular momentum is quantized, specifically as an integral multiple of Planck’s constant (h/2π).
Quantum Transitions: Electrons can transition between non-radiating orbits, and during this transition, they emit a photon with an energy equal to the difference between the initial and final states.

Determining Energy Levels:

Bohr’s model provides an expression to determine the energies of different energy states for a hydrogen atom.
It requires the radius of the electron orbit, which is obtained using the second postulate and quantization of angular momentum.

Expression for Radius (r) and Total Energy (E):

The radius of the nth possible orbit is determined using the quantization condition: r = n²h²ε₀/(4π²me²)
The total energy of the electron in stationary states of the hydrogen atom is calculated by substituting the radius into the energy expression: E = -me⁴Z²/(8ε₀²n²h²)

Energy Units:

Atomic energies are often expressed in electron volts (eV).
1 eV = 1.6 x 10⁻¹⁹ J.
The negative sign of total energy indicates that the electron is bound to the nucleus.

Hydrogen atom

Energy Levels in the Hydrogen Atom:

The energy of an atom is most negative (least) when its electron is in the orbit closest to the nucleus (n = 1).
As the value of n increases (n = 2, 3, …), the absolute value of the energy (E) decreases, making the energy larger in the outer orbits.
The lowest state of the atom, known as the ground state, has the lowest energy, with the electron revolving in the orbit of the smallest radius (Bohr radius, a₀).
The energy of the ground state (n = 1), denoted as E₁, is -13.6 eV.
The minimum energy required to free the electron from the ground state of the hydrogen atom is 13.6 eV, which is called the ionization energy. This prediction by Bohr’s model matches experimental values well.

Excited States:

At room temperature, most hydrogen atoms are in the ground state.
When a hydrogen atom gains energy, for example, through electron collisions, it can move to higher energy states, becoming an excited state.
Energy levels are labeled by the principal quantum number (n) in ascending order of energy.

Energy Differences between States:

The energy difference between energy levels (Eₙ) can be calculated using Bohr’s model.
For example, to excite an electron from the ground state (n = 1) to the first excited state (n = 2), it requires an energy of 10.2 eV (E₂ – E₁ = 10.2 eV).
As n increases, the minimum energy required to free the electron from an excited atom decreases.
Electrons in excited states can fall back to lower-energy states, emitting photons in the process.

Energy Level Diagram:

An energy level diagram for stationary states of a hydrogen atom, calculated using Bohr’s model (Eq. 12.10), shows the energy levels labeled by the principal quantum number (n).
The highest energy state in the diagram corresponds to an electron completely removed from the nucleus and at rest, with an energy of 0 eV.

Line spectra of the hydrogen atom

Bohr’s Third Postulate:

According to the third postulate of Bohr’s model, when an atom transitions from a higher energy state with quantum number ni to a lower energy state with quantum number nf (where nf < ni), the energy difference is carried away by a photon.

Frequency of Emitted Photon:

The frequency of the emitted photon (nif) is related to the energy difference between the initial (Eni) and final (Enf) energy states: hvif = Eni – Enf

Emission Lines in Atomic Spectra:

When electrons jump from higher energy states to lower energy states, they emit photons.
These emitted spectral lines are called emission lines.
Emission lines represent discrete frequencies corresponding to transitions between different atomic energy levels.
These lines appear in the atomic spectrum when observed through a spectrometer.

Absorption Process:

Absorption occurs when an atom absorbs a photon with precisely the same energy required for an electron in a lower energy state to transition to a higher energy state.
In this process, the atom absorbs specific frequencies of photons, resulting in dark spectral absorption lines in the otherwise continuous spectrum.

Significance of Bohr’s Model:

Bohr’s model provided a brilliant explanation for the hydrogen atom spectrum.
It laid the foundation for the modern quantum theory and significantly contributed to the understanding of atomic structure and spectral lines.
Niels Bohr was awarded the Nobel Prize in Physics in 1922 for his work in this field.

Louis de Broglie’s explanation of Bohr’s second postulate of quantization

Bohr’s Second Postulate:

Bohr’s second postulate states that the angular momentum (L) of an electron orbiting the nucleus is quantized, meaning it can only have values that are integral multiples of h/2π, where h is Planck’s constant (L = nh/2π; n = 1, 2, 3 …).

De Broglie’s Explanation:

Louis de Broglie explained Bohr’s second postulate by considering the wave-particle duality of matter.
De Broglie’s hypothesis, introduced in Chapter 11, states that material particles, such as electrons, also have wave properties.
The wave nature of electrons was experimentally verified by C. J. Davisson and L. H. Germer in 1927.
De Broglie argued that the electron in its circular orbit, as proposed by Bohr, should be viewed as a particle wave.
Particle waves can lead to standing waves under resonant conditions, similar to waves on a string. Only certain wavelengths form standing waves with nodes at the ends.
For an electron in the nth circular orbit of radius rₙ, the total distance traveled by the wave is 2πrₙ, which should be an integral number of wavelengths (nλ), where λ is the de Broglie wavelength.
This leads to the equation: 2πrₙ = nλ.
Using the de Broglie wavelength (λ = h/p) and the magnitude of the electron’s momentum (p = mvₙ), the equation becomes: m * vₙ * rₙ = nh/2π.
This equation corresponds to Bohr’s quantum condition for the angular momentum of the electron (L = nh/2π), where n, the principal quantum number, quantizes the angular momentum.
De Broglie’s explanation shows that the quantization of electron orbits and energy levels in the hydrogen atom is due to the wave nature of electrons, and only resonant standing waves are allowed.

Limitations of Bohr’s Model:

While Bohr’s model correctly predicts the frequencies of radiation emitted or absorbed by hydrogenic atoms, it has several limitations:
- It is applicable only to hydrogenic atoms and cannot be extended to more complex atoms.
- The model does not account for the interactions between electrons in multi-electron atoms.
- It cannot explain the relative intensities of spectral lines, which vary in emission spectra, as some transitions are favored over others.
For complex atoms, a new theory based on Quantum Mechanics is necessary to provide a more complete understanding of atomic structure.