Revision Notes: Atoms

Table of Contents
1. Dalton's Atomic Theory
2. Thomson's Atomic Model
3. Rutherford's Atomic Model
4. Bohr's Atomic Model
5. Hydrogen Spectrum Series
6. Wave Model
7. Quantum Numbers
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Dalton's Atomic Theory

According to Dalton's atomic theory, all matter is composed of tiny, indivisible particles called atoms. Atoms of the same element are identical in mass and chemical properties, while atoms of different elements differ in these aspects. 
Dalton proposed that atoms cannot be created, destroyed, or converted into atoms of another element in ordinary chemical reactions; instead, they combine in simple whole-number ratios to form compounds.

Thomson's Atomic Model

Thomson proposed that an atom is a uniformly positively charged sphere of radius of the order of 10^-10 m, with the electrons embedded in it like "plums in a pudding". The atom as a whole is electrically neutral because the positive charge of the sphere balances the negative charge of the electrons.

Limitations of Thomson's Model

• Thomson's model could not explain the origin of the spectral lines (line spectrum) of hydrogen and other elements.
• It failed to account for the large-angle scattering of α-particles observed in scattering experiments, which required a concentrated positive centre in the atom.

Rutherford's Atomic Model

Rutherford, based on α-particle scattering experiments, proposed a nuclear model of the atom. His key observations and conclusions were:

• The entire positive charge and almost all the mass of the atom is concentrated in a very small central region called the nucleus, of the order of 10^-15 m.
• The electrons are negatively charged and revolve around the nucleus in the space outside it.
• The total positive charge of the nucleus equals the total negative charge of the electrons, so the atom as a whole is electrically neutral.
• The centripetal force required for the electrons to revolve is provided by the electrostatic attraction between the electrons and the nucleus.

Distance of Closest Approach

The distance of closest approach is the minimum separation between an α-particle and the nucleus when the α-particle is deflected and comes momentarily to rest before being repelled. At this point, the kinetic energy of the α-particle is converted into electrostatic potential energy. The relation used to calculate this distance is given in the accompanying formula image.

The symbols in the formula represent:

• E_k - kinetic energy of the α-particle
• Z - atomic number of the target nucleus
• e - magnitude of electronic charge
• ε₀ - permittivity of free space

Impact Parameter

The impact parameter is the perpendicular distance between the initial velocity vector of the α-particle (when far away) and the centre of the nucleus. It determines how close the particle will approach the nucleus and therefore the scattering angle.

In the diagram, Z = atomic number of the nucleus, E_k = kinetic energy of the α-particle, and θ = scattering angle.

Rutherford's Scattering Formula

The angular distribution of scattered α-particles is given by Rutherford's scattering formula. The relation is shown in the image below and expresses the number of particles scattered into a given solid angle as a function of θ, Z, incident energy and target thickness.

The quantities in the formula are:

• N(θ) - number of α-particles scattered per unit time into angle θ
• N_i - number of incident α-particles on the foil
• n - number of atoms per unit volume of the foil
• Z - atomic number of target atoms
• E - kinetic energy of the incident α-particles
• t - thickness of the foil

Limitations of Rutherford's Atomic Model

• About the stability of the atom: According to classical electromagnetism (Maxwell's theory) an accelerating charge (an electron in circular orbit) should continuously radiate energy as electromagnetic waves. Consequently, the electron should lose energy, spiral into the nucleus and the atom would be unstable, but atoms are observed to be stable.
• About the line spectrum: Rutherford's model could not explain why atoms emit light at specific discrete wavelengths (line spectra).

Bohr's Atomic Model

• Electrons revolve in certain discrete non-radiating orbits called stationary orbits.
  For these orbits the angular momentum of the electron is quantised:
   mvr = n h / 2π, where n = 1, 2, 3, ... (principal quantum number).
• Energy is emitted or absorbed only when an electron jumps from one permitted orbit to another. The energy of the emitted or absorbed photon is 
  hν = E₂ - E₁, where E₁ and E₂ are energies of the initial and final orbits respectively.
• The radius of the n-th permitted orbit is given by:
  r = n²h² / (4π² m K Z e²)  ⇒  r ∝ n² / Z,
  where K = 1 / (4π ε₀), 
  h = Planck's constant, 
  m = mass of electron, 
  Z = atomic number
  e = electronic charge.
• The velocity of the electron in the n-th orbit is:
  v = 2π K Z e² / (n h)  ⇒  v ∝ Z / n.
• The frequency of revolution of the electron in the n-th orbit is:
  ν = K Z e² / (n h r) = (4π² Z² e⁴ m K²) / (n³ h³)  ⇒  ν ∝ Z³ / n³.
• The kinetic energy of the electron in the n-th orbit is:
  E_k = (2π² m e⁴ Z² K²) / (n² h²) = 13.6 Z² / n² eV.
• The potential energy of the electron in the n-th orbit is:
  E_p = - (4π² m e⁴ Z² K²) / (n² h²) = -27.2 Z² / n² eV  ⇒  E_p ∝ - Z² / n².
• The total energy of the electron in the n-th orbit is:
  E = E_k + E_p = - (2π² m e⁴ Z² K²) / (n² h²) = -13.6 Z² / n² eV.

Potential Energy of Electron (Hydrogen)

• For the hydrogen atom the potential energy may be written as 
  E = - m e⁴ / (8 n² h² ε₀²), which is equivalent to the expressions given above when constants are combined appropriately.
• The wavelength of the radiation emitted when an electron falls from orbit n₂ to n₁ is shown in the figure below.

In quantum mechanics, energies of bound systems are discrete (quantised). For example, the energy levels of a particle of mass m confined in a one-dimensional box of length L are given by
E_n = n² h² / (8 m L²), where n = 1, 2, 3, ...

Hydrogen Spectrum Series

Each element emits a characteristic line spectrum, consisting of distinct isolated lines corresponding to transitions between quantised energy levels. The hydrogen atom shows several well-known series of spectral lines:

• Lyman series: transitions to n = 1 from n = 2, 3, 4, ...; lies in the ultraviolet region.
• Balmer series: transitions to n = 2 from n = 3, 4, 5, ...; lies in the visible region.
• Paschen series: transitions to n = 3 from n = 4, 5, 6, ...; lies in the infrared region.
• Brackett series: transitions to n = 4 from n = 5, 6, 7, ...; lies in the infrared region.
• Pfund series: transitions to n = 5 from n = 6, 7, 8, ...; lies in the infrared region.

Spectral Series

Wave Model

• The wave model treats particles such as electrons using wave concepts. In this model an electron is described by a wave function that gives the probability amplitude of finding the electron at different positions and times.
• In the wave model, particles are considered to be associated with wave functions, which provide information about the probability of finding a particle in a particular location at a given time. The wave function is a mathematical representation of the particle's wave-like behaviour and is central to understanding phenomena such as interference and diffraction.
• Wave-particle duality states that particles exhibit both wave-like and particle-like behaviour depending on the experiment (for example, the double-slit experiment).
• The wave model also incorporates the idea of quantization, where certain physical properties, such as energy, are restricted to discrete values. This is in contrast to classical physics, where these properties can take on a continuous range of values. The quantization of energy levels in atoms, for example, leads to the discrete spectral lines observed in atomic emission and absorption spectra.

Quantum Numbers

• Principal quantum number (n): determines the energy level and size of the orbital; 
  n = 1, 2, 3, ... . Larger n corresponds to higher energy and larger average distance from the nucleus.
• Orbital angular momentum quantum number (l): (also azimuthal quantum number) determines the shape of the orbital. For a given 
  n, l = 0, 1, 2, ..., n-1. Values l = 0, 1, 2, 3 correspond to s, p, d, f orbitals respectively.
  The values of l correspond to different orbital shapes: s (l = 0), p (l = 1), d (l = 2), and f (l = 3).
• Magnetic quantum number (m_l): specifies the orientation of the orbital in space.This quantum number is particularly important in determining how orbitals are oriented in a magnetic field.
  For a given l, m_l = -l, -l+1, ..., 0, ..., +l.
• Spin quantum number (m_s): describes the intrinsic spin of the electron; which can be thought of as a type of intrinsic angular momentum
  m_s = +1/2 or -1/2. The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers.

Some Solved Examples:

Example 1: In H-atom, a transition takes place from n = 3 to n = 2 orbit. Calculate the wavelength of the emitted photon, will the photon be visible? To which spectral series will this photon belong? (Take, R = 1.097 × 10⁷ m^-1.)

Sol. 

The wavelength of the emitted photon is given by the Rydberg formula:

For a transition from n = 3 to n = 2:

The calculated wavelength lies in the visible (red) part of the spectrum; therefore the photon is visible. This transition corresponds to the first member of the Balmer series.

Example 2: In a head-on collision between an α-particle and a gold nucleus, the closest distance of approach is 4 × 10^-14 m. Calculate the initial kinetic energy of the α-particle.

Sol.

Given: closest distance of approach r₀ = 4 × 10^-14 m, atomic number Z = 79.

The electrostatic potential energy at distance r₀ equals the initial kinetic energy of the α-particle; use the relation shown below.

Substitute the values to obtain the initial kinetic energy (calculation carried out using the constants for e and ε₀).

Example 3: It is found experimentally that 13.6 eV energy is required to separate a H-atom into a proton and an electron. Compute the orbital radius and velocity of the electron in a H-atom.

Sol.

Total energy of the electron in ground state of hydrogen,
TE = -13.6 eV = -13.6 × 1.6 × 10^-19 J.

Therefore TE = -2.2 × 10^-18 J.

The total energy is given by the Bohr expression (shown in the image):

From energy relations and quantisation condition, the orbital radius (Bohr radius for n = 1) and the velocity of the revolving electron are obtained as shown:

Summary

This chapter summarises the historical development of atomic models from Dalton through Thomson, Rutherford and Bohr, introduces the wave description of electrons and the concept of quantised energy levels, and explains hydrogen's spectral series and quantum numbers. The solved examples illustrate applications of the Rydberg formula, electrostatic energy for scattering problems, and Bohr model relations for energy, radius and velocity of electrons.