When the subsidiary quantum numbers (l) equals 1, the shape of the orbital is

1. Analyze the question

The question asks for the geometric shape of an atomic orbital when the subsidiary (azimuthal) quantum number, denoted by $l$, is equal to 1.

2. Explain the concept

In quantum mechanics, the subsidiary quantum number ($l$) determines the shape of the orbital where an electron is most likely to be found. The values of $l$ correspond to specific subshells:

• If $l = 0$: This represents an s-orbital. Its shape is spherical.
• If $l = 1$: This represents a p-orbital. Its shape is dumb-bell.
• If $l = 2$: This represents a d-orbital. Its shape is double dumb-bell (or complex).
• If $l = 3$: This represents an f-orbital. Its shape is complex.

3. Show the solution

• The question specifies that $l = 1$.
• Based on the rules of quantum numbers, $l = 1$ corresponds to the p-subshell.
• The p-subshell consists of three orbitals ($p_x, p_y, p_z$), all of which have a characteristic dumb-bell shape.

4. Identify the answer

Comparing this to the given options: A. spherical (corresponds to $l=0$) B. dumb-bell (corresponds to $l=1$) C. flat (not a standard orbital shape) D. round (a general term, but "spherical" is the scientific term for $l=0$)

The correct option is B.

5. Exam Tip

To remember this easily, use the mnemonic "S P D F" and match them to shapes:

• S = Spherical
• P = Peanut (a peanut looks like a dumb-bell!)
• D = Double dumb-bell
• F = Flower (complex shape)