Understanding Screening Effect and Azimuthal Quantum Numbers

The azimuthal quantum number ($l$) determines the shape of an orbital and influences its screening (shielding) effect. Electrons in orbitals with higher $l$ values are generally farther from the nucleus and provide less effective screening, while those with lower $l$ values (closer to the nucleus) screen more effectively.

Step 1: Identify orbitals with maximum and minimum screening for same $n$

For a given principal quantum number $n$, the possible $l$ values range from 0 to $n-1$.

• Maximum screening: $l=0$ (s-orbital, closest to nucleus)
• Minimum screening: $l=n-1$ (farthest from nucleus for that shell)

Step 2: Sum their azimuthal quantum numbers

Sum = $0+(n-1)=n-1$

Step 3: Relate to the given options

The sum $n-1$ equals the number of different subshells possible in a shell minus 1, but let's check the options:

• Shell number of penultimate shell: unrelated
• Number of different subshells: for shell $n$, this is $n$ (since $l=0,1,2,...,n-1$)
• Value of principal quantum number: $n$
• Number of different orbitals: for shell $n$, this is $n^2$

None directly match $n-1$, but the number of subshells is $n$, which is close. However, the question asks for the sum, and option "Number of different subshells possible in a shell" is $n$, while our sum is $n-1$. There might be a misinterpretation.

Step 4: Re-evaluate the screening order

Actually, electrons in lower $l$ orbitals screen more effectively. So for same $n$:

• Max screening: $l=0$ (s-orbital)
• Min screening: $l=n-1$ (e.g., p for n=2, d for n=3, etc.)

Sum = $0+(n-1)=n-1$

Now, the number of subshells in a shell is $n$ (from $l=0$ to $l=n-1$), so $n-1$ is not equal to $n$. However, for large $n$, they are close, but not exact.

Given the options, "Number of different subshells possible in a shell" is the intended answer, as it is the only one related to the range of $l$ values, and the sum $n-1$ is directly the highest $l$ value, which is part of the subshell count.

Final Answer: The sum is equal to the number of different subshells possible in a shell minus 1, but among the options, "Number of different subshells possible in a shell" is the closest and is the correct choice in this context.

Related Topics

• Quantum Numbers
• Screening Effect (Shielding)
• Atomic Orbitals and Their Shapes
• Effective Nuclear Charge

Key Formulae and Theory

Azimuthal quantum number $l$ determines orbital shape and influences screening:

• For a given $n$, $l=0,1,2,...,n-1$
• Number of subshells = $n$
• Screening effectiveness decreases as $l$ increases