The maximum number of electrons in the shell with principal quantum number (n) is equal to :

Question

Accepted Answer

The maximum number of electrons that can be accommodated in a shell with a given principal quantum number (n) is determined by the formula $2n^2$. This formula arises from the quantum mechanical model of the atom, which describes how electrons are distributed in an atom's electron shells.

1. **Explanation for Option 3 ($2n^2$)**:
   - The principal quantum number $n$ defines the shell or energy level of an electron in an atom. Each shell can hold a maximum number of electrons, which is calculated using the formula $2n^2$.
   - This formula is derived from the fact that each shell consists of subshells (s, p, d, f, etc.), and each subshell can hold a specific number of electrons. The total number of electrons in a shell is the sum of electrons in all its subshells.
   - For example, for $n = 1$, the maximum number of electrons is $2(1)^2 = 2$; for $n = 2$, it is $2(2)^2 = 8$; for $n = 3$, it is $2(3)^2 = 18$, and so on.
   - Therefore, Option 3 is correct because it accurately represents the maximum electron capacity of a shell based on the principal quantum number.

2. **Explanation for why other options are incorrect**:
   - **Option 1 ($n^2$)**: This formula would suggest that the maximum number of electrons in a shell is equal to the square of the principal quantum number. However, this does not account for the fact that each orbital can hold two electrons (one with spin-up and one with spin-down), which is why the correct formula is $2n^2$, not $n^2$.
   - **Option 2 ($n+l$)**: The term $n+l$ is used in the context of the Aufbau principle to determine the order of filling of orbitals, but it does not relate to the maximum number of electrons in a shell. It combines the principal quantum number $n$ and the azimuthal quantum number $l$ to prioritize energy levels, not electron capacity.
   - **Option 4 ($(n+l)^2$)**: Similar to Option 2, this formula is not relevant to the maximum electron capacity of a shell. It incorrectly combines the $n+l$ rule with squaring, which does not reflect the principles of electron distribution in atomic shells.

In conclusion, Option 3 ($2n^2$) is the correct choice as it accurately describes the maximum number of electrons that can be accommodated in a shell with a given principal quantum number $n$.