Problem 37 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is not possible? (a) \(\mathrm{n}=2, l=1, \mathrm{~m}=0\) (b) \(\mathrm{n}=2, l=0, \mathrm{~m}=-1\) (c) \(\mathrm{n}=3, l=0, \mathrm{~m}=0\) (d) \(\mathrm{n}=3, l=1, \mathrm{~m}=-1\)

Step-by-Step Solution

Answer

Option (b) is not possible due to incorrect magnetic quantum number m.

Step 1: Review Quantum Number Rules

There are four quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number. For this problem, we focus on n, l, and m. The principal quantum number n must be a positive integer, the azimuthal quantum number l ranges from 0 to n-1, and the magnetic quantum number m ranges from -l to +l.

Step 2: Evaluate Option (a)

Given parameters are \( n=2, l=1, m=0 \). Check if these values follow quantum number rules:\- Here, \( l = 1 \), which is within the range \( 0 \leq l \leq 1 \) for \( n=2 \).\- The value \( m=0 \) is within the range \( -1 \leq m \leq 1 \) for \( l=1 \).\Since all conditions are satisfied, this configuration is possible.

Step 3: Evaluate Option (b)

Given parameters are \( n=2, l=0, m=-1 \). Check each value:\- \( l = 0 \) is appropriate since it ranges from 0 to 1 for \( n=2 \).\- However, for \( l=0 \), \( m \) should be only 0 (since \( -0 \leq m \leq 0 \)).\Here, \( m=-1 \) is not possible, making this configuration incorrect.

Step 4: Evaluate Option (c)

Given parameters are \( n=3, l=0, m=0 \). Verify each value:\- \( l = 0 \) is valid since it ranges from 0 to 2 for \( n=3 \).\- Since \( l=0 \), \( m \) is correctly 0.\The values satisfy all quantum number conditions, making this possible.

Step 5: Evaluate Option (d)

Given parameters are \( n=3, l=1, m=-1 \). Check compliance with quantum rules:\- \( l=1 \) is acceptable because it ranges from 0 to 2 for \( n=3 \).\- The value \( m=-1 \) is within the range \( -1 \leq m \leq 1 \) for \( l=1 \).\This set of numbers is possible according to the rules.

Key Concepts

Principal Quantum Number (n)

The principal quantum number, symbolized as \( n \), plays a crucial role in defining the structure of an atom. It determines the energy level or shell in which an electron resides. This number must always be a positive integer, such as 1, 2, 3, and so on. Since it describes the electron's shell, it also indicates the average distance of the electron from the nucleus.

• A higher principal quantum number means the electron is located further out from the nucleus and will have higher energy.
• The principal quantum number contributes to the overall size of the atom.

Thus, \( n \) gives us a basic idea of the electron's 'floor'—like the floors of a building, the higher the number \( n \), the further the electron is from the center (the nucleus). Understanding \( n \) helps predict electron configuration and how atoms will interact chemically.

Azimuthal Quantum Number (l)

The azimuthal quantum number, denoted as \( l \), is essential for understanding subshells or sublevels within a principal energy level. This number modifies \( n \) and is crucial in defining the shape of the electron cloud or orbital. It can take on integer values from 0 up to \( n-1 \).

• If \( n=1 \), then \( l \) can only be 0, relating to the s subshell.
• For \( n=2 \), \( l \) can be 0 or 1, which corresponds to the presence of both s and p subshells.
• These variations help define the orbital's shape, such as spherical (s), dumbbell (p), and more complex forms for higher \( l \) values.

The azimuthal quantum number is pivotal when considering the diversity of electron paths and their respective chemical properties. It's the identity card of each electron in terms of shape within its principal level.

Magnetic Quantum Number (m)

The magnetic quantum number \( m \) further refines our understanding of an electron's orbital by specifying its orientation in space. \( m \) is an integer ranging between \( -l \) and \( +l \), including zero.

• When \( l=0 \), there's only one permitted value: \( m=0 \).
• If \( l=1 \), \( m \) can be -1, 0, or +1, reflecting three possible orientations.

Each distinct \( m \) value represents a unique spatial orientation for the orbital, illustrating why certain electron arrangements are possible or impossible. For instance, when \( l=0 \), as in a s subshell, \( m \) must also be 0 due to the singular orientation.

Thus, the magnetic quantum number tells us how that shape or path is aligned in a three-dimensional space, adding depth to our understanding of electron behavior within an atom.