Unit 2: Structure of Atom: Exam-Ready Study Resource

Aston Mass Spectrograph

The Aston Mass Spectrograph is an instrument used to measure the masses of ions and separate different isotopes of an element. It works on the principle that ions of different masses are deflected differently when subjected to electric and magnetic fields.

Principle and Working

Positive ions, typically produced by an electric discharge, are accelerated through an electric field. These accelerated ions then pass through a slit into a region containing parallel electric and magnetic fields. The electric field deflects ions based on their charge, while the magnetic field, perpendicular to the electric field, deflects them based on their mass-to-charge ratio (m/q). By carefully adjusting these fields, ions with different m/q values are separated and focused onto different points on a photographic plate or detector. This allows for the precise determination of the mass of each isotope.

Bohr’s Theory of Hydrogen Atom

Niels Bohr proposed a model for the hydrogen atom in 1913, addressing the shortcomings of Rutherford’s model regarding atomic stability and discrete spectra.

Postulates of Bohr’s Theory

1. Stationary Orbits: Electrons revolve around the nucleus in certain stable, non-radiating orbits, called stationary orbits. While in these orbits, electrons do not emit or absorb energy.
2. Quantization of Angular Momentum: An electron can only revolve in those orbits for which its angular momentum is an integral multiple of h/2π (where h is Planck’s constant).

   L = m_evr = n(h/2π) = nħ
   Where n = 1, 2, 3, ... (principal quantum number)

3. Energy Transitions: An atom radiates or absorbs energy only when an electron jumps from one stationary orbit to another. The frequency (ν) of the emitted or absorbed radiation is given by:

   hν = E_i - E_f
   Where E_i is the energy of the initial higher energy state and E_f is the energy of the final lower energy state.

Expression of Radii of Electrons

Based on Bohr's postulates, the radius (r_n) of the n^th stationary orbit for a hydrogen-like atom with atomic number Z is given by:

r_n = n²h²ε₀ / (πm_ee²Z)
For hydrogen (Z=1), the radius of the first orbit (n=1) is called the Bohr radius (a₀):
a₀ = h²ε₀ / (πm_ee²) ≈ 0.529 × 10^-10 m
Thus, r_n = n²a₀/Z

This shows that the radii of allowed orbits are quantized and proportional to n².

Expression of Energies

The total energy (E_n) of an electron in the n^th orbit of a hydrogen-like atom with atomic number Z is:

E_n = - m_ee⁴Z² / (8ε₀²n²h²)
Upon substituting constants, this simplifies to:
E_n = -13.6 Z²/n² eV

The negative sign indicates that the electron is bound to the nucleus. For hydrogen (Z=1), the energy levels are E_n = -13.6/n² eV.

Hydrogen Atom Spectrum

When an electron transitions from a higher energy state (n_i) to a lower energy state (n_f), it emits a photon with energy equal to the difference in energy levels. This gives rise to the discrete line spectrum of hydrogen.

1/λ = ν̄ = (E_i - E_f) / hc = RZ² (1/n_f² - 1/n_i²)
Where R is the Rydberg constant (R ≈ 1.097 × 10⁷ m^-1).

Different series of spectral lines are observed depending on the final energy level (n_f):

• Lyman Series: n_f = 1, n_i = 2, 3, 4,... (Ultraviolet region)
• Balmer Series: n_f = 2, n_i = 3, 4, 5,... (Visible region)
• Paschen Series: n_f = 3, n_i = 4, 5, 6,... (Infrared region)
• Brackett Series: n_f = 4, n_i = 5, 6, 7,... (Infrared region)
• Pfund Series: n_f = 5, n_i = 6, 7, 8,... (Infrared region)

Effect of Nuclear Motion on Atomic Spectra

Bohr's original theory assumed an infinitely massive, stationary nucleus. However, the nucleus also has a finite mass and moves, albeit much less, around the common center of mass with the electron. This motion slightly affects the energy levels and spectral lines.

Reduced Mass

To account for nuclear motion, the electron's mass (m_e) is replaced by the reduced mass (μ) of the electron-nucleus system. The reduced mass is given by:

μ = m_em_N / (m_e + m_N)
Where m_e is the mass of the electron and m_N is the mass of the nucleus.

Since m_N is much greater than m_e, μ is slightly less than m_e (μ ≈ m_e).

Modified Rydberg Constant and Wave Number

When the reduced mass (μ) is used instead of m_e in the formulas for energy and wavelength, the Rydberg constant also gets modified.

Modified Rydberg Constant (R_μ) = R_∞ (μ/m_e)
Where R_∞ is the Rydberg constant assuming an infinitely massive nucleus.

The wave number (ν̄) for transitions is then given by:

ν̄ = R_μZ² (1/n_f² - 1/n_i²)

This explains the slight differences in spectral lines observed for isotopes of hydrogen, such as hydrogen (¹H), deuterium (²H), and tritium (³H), as their nuclear masses (m_N) differ.

Evidences in Favour of Bohr’s Theory

• Explanation of Hydrogen Spectrum: Successfully predicted the discrete spectral lines and their wavelengths for hydrogen and hydrogen-like ions.
• Stability of Atoms: Explained why electrons do not continuously radiate energy and spiral into the nucleus.
• Calculated Energy Levels and Radii: The theoretically calculated values for energy levels and orbital radii matched experimental observations for hydrogen.
• Frank-Hertz Experiment: Provided direct experimental evidence for the existence of discrete energy levels in atoms.

Correspondence Principle

Bohr’s Correspondence Principle: For very large quantum numbers (i.e., when n → ∞), the results of quantum mechanics must agree with those of classical physics.

This principle asserts that quantum theory should reproduce classical physics in the limit where quantum effects become negligible. It serves as a crucial bridge between classical and quantum mechanics, ensuring consistency across different scales of physical phenomena.

Fine Structure of Spectral Lines and Sommerfeld’s Relativistic Atom Model

Fine Structure of Spectral Lines

High-resolution spectroscopes reveal that many spectral lines, which appear as single lines in lower resolution, are actually composed of several very closely spaced lines. This splitting is known as fine structure. It cannot be explained by Bohr's original theory and is primarily due to relativistic effects and spin-orbit coupling.

Sommerfeld’s Relativistic Atom Model (Qualitative Idea)

Arnold Sommerfeld extended Bohr’s model by introducing two key ideas qualitatively:

1. Elliptical Orbits: He proposed that electrons could move in elliptical orbits in addition to circular ones. Each energy level (n) could have several elliptical sub-orbits with different eccentricities, characterized by an azimuthal quantum number (k, later ℓ).
2. Relativistic Mass Variation: He incorporated the relativistic variation of electron mass with velocity, particularly for electrons in elliptical orbits where velocity changes. This relativistic correction caused the elliptical orbits to precess (perihelion shift), leading to slightly different energy values for orbits within the same principal quantum number.

These modifications partially explained the fine structure of spectral lines by lifting the degeneracy of energy levels, meaning states with the same principal quantum number (n) but different orbital angular momentum quantum numbers (k or ℓ) would have slightly different energies. While a significant step, Sommerfeld's model was still incomplete and did not fully explain all aspects of fine structure (e.g., electron spin).

Excitation and Ionisation Potentials

Excitation Potential: The minimum accelerating potential (voltage) required to provide an electron with enough energy to excite an atom from its ground state to a higher energy level. This corresponds to the energy difference between the ground state and the excited state (ΔE = E_excited - E_ground).

Ionisation Potential: The minimum accelerating potential (voltage) required to provide an electron with enough energy to completely remove an electron from an atom in its ground state, leading to the formation of an ion. This energy corresponds to the binding energy of the electron in the ground state (E_ionisation = |E_ground|).

For hydrogen, the first excitation potential (from n=1 to n=2) is approximately 10.2 eV, and the ionization potential (from n=1 to n=∞) is 13.6 eV.

Frank and Hertz Experiment

The Frank-Hertz experiment (1914) provided direct experimental proof for the existence of discrete energy levels in atoms, thus validating Bohr's quantization condition.

Experimental Setup and Observations

Electrons emitted from a hot cathode are accelerated towards a grid by a variable potential difference (V_acc), passing through a gas (e.g., mercury vapor) before reaching an anode. A small opposing potential (V_retard) is applied between the grid and the anode to repel electrons that have lost energy.

As V_acc increases, the anode current initially increases. However, at specific discrete values of V_acc (e.g., 4.9 V for mercury), the anode current suddenly drops. This happens because electrons gain enough kinetic energy to make inelastic collisions with mercury atoms, exciting them to their first excited state. In these collisions, electrons lose a significant portion of their kinetic energy (equal to the excitation energy of mercury, ~4.9 eV) and are unable to overcome the retarding potential to reach the anode, causing the current to drop. As V_acc increases further, electrons gain enough energy to excite more atoms or excite atoms to higher levels, leading to subsequent drops in current at multiples of the excitation potential (e.g., 9.8 V, 14.7 V).

Significance

The experiment demonstrated that atoms absorb energy only in specific, quantized amounts, confirming the existence of discrete energy levels predicted by Bohr's theory.

Characteristic X-ray Spectra

When high-energy electrons strike a heavy metal target, two types of X-rays are produced: continuous X-rays (bremsstrahlung) and characteristic X-rays.

Characteristic X-rays: These X-rays are produced when incident electrons eject inner-shell electrons (e.g., from the K or L shell) of the target atom. The vacancy created is then filled by an electron from a higher energy shell (e.g., L or M shell) transitioning downwards. The energy difference is emitted as an X-ray photon with a specific, characteristic wavelength unique to the target material.

The different series of characteristic X-rays are named after the shell where the vacancy is created:

• K-series: Vacancy in the K-shell. K_α (L → K), K_β (M → K), etc.
• L-series: Vacancy in the L-shell. L_α (M → L), L_β (N → L), etc.

These spectra consist of sharp, distinct lines, providing a fingerprint for the elements.

Moseley’s Law

Moseley’s Law: The square root of the frequency (ν) of a characteristic X-ray spectral line is directly proportional to the atomic number (Z) of the element.

√ν = a(Z - b)

Where 'a' and 'b' are constants that depend on the specific spectral series (e.g., K_α, L_α) and are independent of the element. 'a' is related to the Rydberg constant and 'b' is a screening constant.

Moseley's law, discovered in 1913, was a pivotal finding because it established the atomic number (Z) as the fundamental property defining an element, rather than atomic weight. This led to the rearrangement of elements in the periodic table based on increasing atomic number.

Bragg’s Law

Bragg’s Law: It describes the condition for constructive interference when X-rays are scattered by a crystalline lattice.

nλ = 2d sinθ

Where:
n is an integer representing the order of diffraction (n = 1, 2, 3, ...).
λ is the wavelength of the incident X-ray.
d is the spacing between the atomic planes (interplanar spacing) in the crystal lattice.
θ is the glancing angle (Bragg angle) between the incident X-ray beam and the crystal planes.

Bragg's Law is fundamental to X-ray crystallography, a technique used to determine the atomic and molecular structure of crystals. When the condition for constructive interference is met, a strong diffracted beam (Bragg peak) is observed, providing information about the crystal's atomic arrangement and lattice parameters.