If n and `l` are respectively the principal and azimuthal quantum numbers , then the expression for calculating the total number of electrons in any energy level is :
(a)`underset(l=0)overset(l=n)sum2(2l+1)`
(b)`underset(l=1)overset(l=n)sum2(2l+1)`
(c)`underset(l=0)overset(l=n)sum2(2l+1)`
(d)`underset(l=0)overset(l=n-1)sum2(2l+1)`

Options

A. `underset(l=0)overset(l=n)sum2(2l+1)`
B. `underset(l=1)overset(l=n)sum2(2l+1)`
C. `underset(l=0)overset(l=n)sum2(2l+1)`
D. `underset(l=0)overset(l=n-1)sum2(2l+1)`

Answer

To solve the question regarding the total number of electrons in any energy level based on the principal quantum number \( n \) and the azimuthal quantum number \( l \), we can follow these steps:

### Step 1: Understand the Quantum Numbers
- The principal quantum number \( n \) indicates the energy level of an electron in an atom.
- The azimuthal quantum number \( l \) indicates the subshell (s, p, d, f) and can take values from \( 0 \) to \( n-1 \).

### Step 2: Determine the Formula for Total Electrons
- The total number of electrons in a subshell is given by the formula:
\[ \text{Total Electrons} = 2(2l + 1) \]
- Here, \( 2l + 1 \) represents the number of orbitals in a subshell, and multiplying by 2 accounts for the two possible spin states of electrons.

### Step 3: Identify the Range of \( l \)
- Since \( l \) can take values from \( 0 \) to \( n-1 \), we need to sum the contributions from all possible values of \( l \):
\[ \text{Total Electrons} = \sum_{l=0}^{n-1} 2(2l + 1) \]

### Step 4: Write the Expression
- The correct expression for calculating the total number of electrons in any energy level is:
\[ \sum_{l=0}^{n-1} 2(2l + 1) \]

### Step 5: Analyze the Options
- Now, let's analyze the given options:
- (a) \( \sum_{l=0}^{n} 2(2l + 1) \) - Incorrect, as \( l \) cannot equal \( n \).
- (b) \( \sum_{l=1}^{n} 2(2l + 1) \) - Incorrect, as it starts from \( l=1 \) and also includes \( n \).
- (c) \( \sum_{l=0}^{n} 2(2l + 1) \) - Incorrect, as \( l \) cannot equal \( n \).
- (d) \( \sum_{l=0}^{n-1} 2(2l + 1) \) - Correct, as it correctly sums from \( l=0 \) to \( n-1 \).

### Conclusion
- The correct answer is option (d):
\[ \sum_{l=0}^{n-1} 2(2l + 1) \]