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Question: If n and \(\mathcal{l}\) are respectively the principal and azimuthal quantum numbers, then the expr...

If n and l\mathcal{l} are respectively the principal and azimuthal quantum numbers, then the expression for calculating the total number of electrons in any orbit is –

A

∑l=1l=n2(2l+1)\sum_{\mathcal{l =}1}^{\mathcal{l =}n}{2\left( 2\mathcal{l +}1 \right)}

B

∑l=1l=n−12(2l+1)\sum_{\mathcal{l} = 1}^{\mathcal{l} = n - 1}{2\left( 2\mathcal{l} + 1 \right)}

C

∑l=0l=n+12(2l+1)\sum_{\mathcal{l =}0}^{\mathcal{l =}n + 1}{2\left( 2\mathcal{l +}1 \right)}

D

∑l=0l=n−12(2l+1)\sum_{\mathcal{l} = 0}^{\mathcal{l} = n - 1}{2\left( 2\mathcal{l} + 1 \right)}

Answer

∑l=0l=n−12(2l+1)\sum_{\mathcal{l} = 0}^{\mathcal{l} = n - 1}{2\left( 2\mathcal{l} + 1 \right)}

Explanation

Solution

Total number of electrons in an orbital = 2 (2l\mathcal{l} +1).

The value of l varies from 0 to n – 1.

∴\therefore Total numbers of electrons in any orbit = ∑l=0l=n−12(2l+1)\sum_{\mathcal{l} = 0}^{\mathcal{l} = n - 1}{2\left( 2\mathcal{l} + 1 \right)}