Question

How many non-spherical subshell are possible that have atleast two maxima if a curve is plotted between radial probability distribution function versus radial distance for which principal quantum number:

n ≤ 4?

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

The number of maxima in the radial probability distribution function (RPDF) is given by $n-l$. For at least two maxima, we need $n-l \ge 2$. Non-spherical subshells imply $l > 0$. Given $n \le 4$, we need to find pairs $(n, l)$ satisfying:

1. $n \in \{1, 2, 3, 4\}$
2. $l \in \{0, 1, \dots, n-1\}$
3. $l > 0$
4. $l \le n-2$

Combining conditions 3 and 4, we need $1 \le l \le n-2$.

• For $n=1$: $1 \le l \le 1-2 = -1$ (No solutions)
• For $n=2$: $1 \le l \le 2-2 = 0$ (No solutions)
• For $n=3$: $1 \le l \le 3-2 = 1 \implies l=1$ (3p subshell). Here $n-l = 3-1 = 2$.
• For $n=4$: $1 \le l \le 4-2 = 2 \implies l=1$ (4p subshell) and $l=2$ (4d subshell). Here $n-l = 4-1 = 3$ and $n-l = 4-2 = 2$.

The valid subshells are 3p, 4p, and 4d. Thus, there are 3 such subshells.