Energy Levels In Atoms And Line Spectra (Copy)

• Discrete Energy Levels in Atoms

• Electrons in isolated atoms (e.g. hydrogen) can only occupy specific, quantised energy levels.
• Each level is associated with a particular energy value, typically expressed in electronvolts (eV).
• These energy levels are negative, indicating that energy must be supplied to free the electron.
• The ground state is the lowest energy level (most stable).
• Higher levels are called excited states.
• Energy levels are numbered:
  n = 1, 2, 3, … where n = 1 is the ground state.
• For hydrogen:
  Eₙ = –13.6 / n² eV
  (only for hydrogen-like atoms)

• Electron Transitions and Photons

• Electrons can move between energy levels by absorbing or emitting photons:
  • Absorption: electron gains energy, moves to higher level.
  • Emission: electron loses energy, drops to lower level, emitting a photon.
• The energy of the photon is equal to the energy difference between the two levels:

  hf = E₁ – E₂

  where:
  h = Planck’s constant = 6.63 × 10⁻³⁴ J·s
  f = frequency of emitted/absorbed photon
  E₁, E₂ = energy of higher and lower levels respectively
  (E₁ > E₂ for emission, E₁ < E₂ for absorption)

• Emission Line Spectra

• Produced when electrons in excited atoms return to lower energy levels, emitting photons.
• Each transition corresponds to a specific wavelength/frequency.
• Results in a line spectrum – bright lines at discrete wavelengths on a dark background.
• The set of lines forms a characteristic fingerprint of the element (e.g. hydrogen emission spectrum).

• Absorption Line Spectra

• Occurs when white light passes through a cool gas.
• Electrons in atoms absorb specific wavelengths to move to higher energy levels.
• These wavelengths are missing from the continuous spectrum – appears as dark lines.

• Spectral Series in Hydrogen Atom

• Energy–Frequency–Wavelength Relationships

1. Energy of a photon:
   E = hf
2. Frequency–wavelength relation:
   c = fλ
   ⇒ f = c / λ
3. Energy in terms of wavelength:
   E = hc / λ

where:
c = speed of light = 3.00 × 10⁸ m/s
λ = wavelength of photon

• eV and Joules Conversion

• 1 eV = 1.60 × 10⁻¹⁹ J
• To convert eV to J: multiply by 1.60 × 10⁻¹⁹
• To convert J to eV: divide by 1.60 × 10⁻¹⁹

• Practical Applications of Line Spectra

• Identifying elements in stars via spectroscopy.
• Flame tests in chemistry.
• Quantum energy level structure study.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course