The total number of orbitals in a principal shell is

General
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the total number of orbitals in a principal shell is

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The total number of orbitals in a principal shell is

Answer:

The total number of orbitals in a principal shell is determined by the principal quantum number ( n ). The principal quantum number ( n ) also defines the main energy level or shell of an electron in an atom.

Solution by Steps:

  1. Principal Quantum Number (n):

    • The principal quantum number ( n ) can take positive integer values: ( n = 1, 2, 3, \ldots ).

  2. Number of Subshells:

    • For a given principal quantum number ( n ), there are ( n ) subshells. Each subshell is characterized by the azimuthal quantum number ( l ), which can take integer values from 0 to ( n-1 ).

  3. Number of Orbitals in Each Subshell:

    • For a given subshell ( l ), the magnetic quantum number ( m_l ) can take integer values from ( -l ) to ( +l ), inclusive. Therefore, the number of orbitals in a subshell with quantum number ( l ) is ( 2l + 1 ).

  4. Total Number of Orbitals in a Principal Shell:

    • To find the total number of orbitals in the principal shell, we sum the number of orbitals for all subshells in that shell. This sum can be calculated as follows:

      \text{Total number of orbitals} = \sum_{l=0}^{n-1} (2l + 1)

  5. Calculating the Total:

    • The sum of the first ( n ) odd numbers is known to equal ( n^2 ). This can be derived and verified mathematically, but it is a known result.

      \text{Total number of orbitals in the } n \text{th principal shell} = n^2

Explanation for Different Principal Quantum Numbers:

  1. For ( n = 1 ):

    • Only one subshell (l = 0).

    • Number of orbitals = ( 2(0) + 1 = 1).

      So, total number of orbitals = ( 1^2 = 1 ).

  2. For ( n = 2 ):

    • Two subshells (l = 0, 1).

    • Number of orbitals for ( l=0 ) = 1, and for ( l=1 ) = 3.

      So, total number of orbitals = ( 1 + 3 = 4 ).

      Equivalently, ( 2^2 = 4 ).

  3. For ( n = 3 ):

    • Three subshells (l = 0, 1, 2).

    • Number of orbitals for ( l=0 ) = 1; for ( l=1 ) = 3; for ( l=2 ) = 5.

      So, total number of orbitals = ( 1 + 3 + 5 = 9 ).

      Equivalently, ( 3^2 = 9 ).

  4. For ( n = 4 ):

    • Four subshells (l = 0, 1, 2, 3).

    • Number of orbitals for ( l=0 ) = 1; for ( l=1 ) = 3; for ( l=2 ) = 5; for ( l=3 ) = 7.

      So, total number of orbitals = ( 1 + 3 + 5 + 7 = 16 ).

      Equivalently, ( 4^2 = 16 ).

Final Answer:
Therefore, the total number of orbitals in a principal shell is given by ( n^2 ), where ( n ) is the principal quantum number.