Quantum Mechanics: Orbitals and Quantum Numbers

Quantum Mechanics: Orbitals and Quantum Numbers

Principal Quantum Number

  • Definition: The principal quantum number, denoted as n, indicates the main energy level occupied by an electron within an atom. It plays a crucial role in determining the size and energy of atomic orbitals.
  • n = 3: Refers to the third energy level of an atom, where specific subshells and orbitals are associated.

Total Number of Orbitals for n = 3

  • For the principal quantum number n, the number of orbitals can be calculated as follows:
  • Formula: The total number of orbitals in any given principal energy level n can be derived by the formula:
    extTotalNumberofOrbitals=n2ext{Total Number of Orbitals} = n^2
  • Calculation for n = 3:
    • Substituting into the formula gives:
      extTotalNumberofOrbitals=32=9ext{Total Number of Orbitals} = 3^2 = 9
  • Therefore, when n = 3, there are a total of 9 orbitals available for electron occupancy.

Subshells Associated with n = 3

  • The principal quantum number n = 3 corresponds to three types of subshells, which are:

    1. s subshell
    • Number of Orbitals: 1
    1. p subshell
    • Number of Orbitals: 3
    1. d subshell
    • Number of Orbitals: 5

  • Total Orbital Calculation from Subshells:

    • Adding up the orbitals from each subshell:
      1(s)+3(p)+5(d)=9extorbitals1 (s) + 3 (p) + 5 (d) = 9 ext{ orbitals}

Conclusion

  • In summary, there are a total of 9 orbitals in the principal quantum number n = 3, derived from the subshells s, p, and d as explained. Each subshell contributes a specific number of orbitals which sums to the total count.