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 :

Options

A. `sum_(l=1)^(l=n)2(2l+1)`
B. `sum_(l=1)^(l=n-1)2(2l+1)`
C. `sum_(l=0)^(l=n+1)2(2l+1)`
D. `sum_(l=0)^(l=n-1)2(2l+1)`

Answer

To solve the problem of calculating the total number of electrons in any energy level given 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 or shell.
- The azimuthal quantum number \( l \) indicates the subshell (s, p, d, f, etc.) and can take values from \( 0 \) to \( n-1 \).

### Step 2: Determine the Maximum Electrons in Each Subshell
- Each subshell can hold a specific number of electrons:
- For \( l = 0 \) (s subshell): 2 electrons
- For \( l = 1 \) (p subshell): 6 electrons
- For \( l = 2 \) (d subshell): 10 electrons
- For \( l = 3 \) (f subshell): 14 electrons

### Step 3: Generalize the Formula for Total Electrons
- The number of electrons in a subshell can be expressed as \( 2(2l + 1) \).
- Therefore, the total number of electrons in all subshells for a given principal quantum number \( n \) can be calculated by summing over all possible values of \( l \) (from \( 0 \) to \( n-1 \)).

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

### Step 5: Simplify the Expression
- This summation can be simplified:
\[ = 2 \sum_{l=0}^{n-1} (2l + 1) \]
- The sum \( \sum_{l=0}^{n-1} (2l + 1) \) can be calculated as:
- The first term is \( 1 \) (when \( l = 0 \)), and the last term is \( 2(n-1) + 1 = 2n - 1 \).
- The number of terms in the summation is \( n \).

### Step 6: Calculate the Final Result
- The total number of electrons in an energy level can be expressed as:
\[ \text{Total Electrons} = 2n^2 \]

### Conclusion
Thus, the expression for calculating the total number of electrons in any energy level is \( 2n^2 \).

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