Definition and explanation of principal quantum numbers

Principal Quantum Number (n)

According to Bohr’s atomic model, electrons in an atom revolve around the nucleus in certain designated circular paths of fixed energy. These paths are known as orbits, shells, or principal energy levels. The integers used to represent or designate these orbits or energy levels are called Principal Quantum Numbers.

Key Characteristics:

• The principal quantum number is denoted by the lowercase letter n.
• The value of n is always a positive integer, i.e., n = 1, 2, 3, 4, … etc.
• According to Bohr’s theory:
  • When n = 1, it indicates the 1st energy level or K shell.
  • When n = 2, it indicates the 2nd energy level or L shell.
  • When n = 3, it indicates the 3rd energy level or M shell.
  • When n = 4, it indicates the 4th energy level or N shell.
• The sequential order of their energy level is: K < L < M < N (where energy increases as n increases).
• The principal quantum number primarily indicates the orbit number. The maximum electron-holding capacity of any principal energy level is given by the formula 2n².

Maximum Electron Capacity of Shells:

Applications of Principal Quantum Number

With the help of the principal quantum number (n), several crucial atomic properties and parameters can be precisely determined:

• The total number of electrons within any principal energy level can be calculated using 2n².
• The angular momentum (L) of a revolving electron in a specific energy level can be determined using the formula:
  mvr =

  nh 2π
• The radius (r_n) of a specific principal energy level can be determined using the formula:
  r_n =

  n²h² 4π²me²Z
• The velocity (v_n) of an electron revolving within a specific orbit can be determined using the formula:
  v_n =

  2πZe² nh
• The total energy (E_n) of an electron in a specific principal shell can be determined using the formula:
  E_n = −

  2π²mZ²e⁴ n²h²
• When an electron transitions between energy levels, the wavelength (λ) of the spectral line produced can be determined using the Rydberg formula:
  1 λ

  = R_H × Z² · [

  1 n₁²

  −

  1 n₂²

  ]
  Where:
  n₂ = Higher/Outer energy level (orbit from which electron drops)
  n₁ = Lower/Inner energy level (orbit into which electron falls)
  R_H = Rydberg constant ≈ 109,678 cm⁻¹ (or 1.097 × 10⁷ m⁻¹)
  Z = Atomic number (Z = 1 for Hydrogen atom)