Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

The following quantum numbers are possible for how many orbital? \(\mathrm{n}=3, l=2, \mathrm{~m}=+2\) (a) 1 (b) 2 (c) 3 (d) 4

Step-by-Step Solution

Answer

1 orbital is possible for the given quantum numbers.

Step 1: Understanding Quantum Numbers

The quantum number \(n\) refers to the principal quantum number and determines the energy level of an electron in an atom. Here, \(n=3\) indicates the third energy level.

Step 2: Analyzing Orbital Angular Momentum

The quantum number \(l\) represents the angular momentum quantum number, which determines the shape of the orbital. The value \(l=2\) corresponds to a 'd' orbital.

Step 3: Magnetic Quantum Number Specification

The magnetic quantum number \(m\) identifies the orientation of the orbital and can have integer values between \(-l\) and \(+l\), inclusive. Given \(l=2\), \(m\) can be \(-2, -1, 0, +1, +2\). Thus, \(m=+2\) specifies one particular orientation of the \(d\) orbital.

Step 4: Conclusion from Quantum Number Constraints

Since all quantum numbers \(n=3\), \(l=2\), and \(m=+2\) are fully specified, the quantum numbers describe exactly one possible orbital within this configuration.

Key Concepts

Principal Quantum Number

The principal quantum number, often represented as \(n\), is a key component in the quantum mechanical model of an atom. It primarily determines the energy level or shell in which an electron resides. The values that \(n\) can take are positive integers: 1, 2, 3, and so forth. Each number indicates a specific energy level and, consequently, relative distance from the nucleus.
For instance, an electron with \(n = 3\) is in the third energy level, which is farther out than those in the \(n = 1\) or \(n = 2\) levels. As \(n\) increases:

• The electron's energy increases.
• The electron's probable distance from the nucleus increases.
• The number of orbitals in the energy level increases, providing more room for electrons.

This number doesn't exclusively define the physical size of the shell but rather its potential capacity to hold electrons, with higher energy levels hosting more complex orbital types.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \(l\), determines the shape and type of orbital where the electron is likely found. It's crucial for defining the orbital's geometry and is always a non-negative integer, ranging from 0 to \(n-1\). For example, if \(n = 3\), the potential values for \(l\) are 0, 1, and 2. These correspond to different orbital types:

• \(l = 0\) is an 's' orbital, spherical in shape.
• \(l = 1\) is a 'p' orbital, dumbbell-shaped with three different orientations.
• \(l = 2\) is a 'd' orbital, which is more complex with five different orientations.

In the given scenario, \(l = 2\) specifies a 'd' orbital, indicating a complex shape with five possible orientations, which significantly influences the electron cloud's distribution around the nucleus.

Magnetic Quantum Number

The magnetic quantum number, indicated by \(m\), has a critical role in defining the orientation of an orbital within a magnetic field. It can be any integer value ranging from \(-l\) to \(+l\), including zero. This means for each angular momentum quantum number \(l\), there are \(2l+1\) possible \(m\) values. For \(l = 2\) as in our problem, \(m\) could be -2, -1, 0, +1, or +2. These different \(m\) values align with the five orientations of the 'd' orbitals.
In the specific case presented, the magnetic quantum number \(m = +2\) points to one distinct orientation out of the five possible for a 'd' orbital when \(l = 2\). By fixing \(m\) to a single value, such as +2, we identify a unique orientation, thereby specifying a precise orbital configuration within the \(n = 3\) energy level. This specification contributes to the unique description of exactly one orbital in this context.