Problem 10 - A Complete Resource Book in Chemistry for JEE Main

Question

The correct set of quantum numbers is (a) \(\mathrm{n}=2, l=1, \mathrm{~m}=-2, \mathrm{~s}=0\) (b) \(\mathrm{n}=2, l=-2, \mathrm{~m}=1, \mathrm{~s}=+1 / 2\) (c) \(\mathrm{n}=2, l=2, \mathrm{~m}=-1, \mathrm{~s}=-1 / 2\) (d) \(\mathrm{n}=2, l=1, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (d) is the correct set of quantum numbers.

Step 1: Understanding Quantum Numbers

The quantum numbers describe the properties of an electron in an atom and include the principal quantum number \( n \), the azimuthal (angular momentum) quantum number \( l \), the magnetic quantum number \( m \), and the spin quantum number \( s \). The constraints are: \( n \geq 1 \), \( 0 \leq l < n \), \( -l \leq m \leq l \), and \( s = \pm\frac{1}{2} \).

Step 2: Analyzing Option A

For option (a) \( n = 2, l = 1, m = -2, s = 0 \):- \( n = 2 \) is valid.- \( l = 1 \) is valid since \( 0 \leq l < n \).- \( m = -2 \) is invalid because \( m \) must satisfy \( -l \leq m \leq l \), so \( m \) should be between -1 and 1.- \( s = 0 \) is invalid since \( s \) must be \( \pm\frac{1}{2} \).

Step 3: Analyzing Option B

For option (b) \( n = 2, l = -2, m = 1, s = +\frac{1}{2} \):- \( n = 2 \) is valid.- \( l = -2 \) is invalid because \( l \) must be \( \geq 0 \).- \( m = 1 \) could be valid if \( l \) were correct.- \( s = +\frac{1}{2} \) is valid.

Step 4: Analyzing Option C

For option (c) \( n = 2, l = 2, m = -1, s = -\frac{1}{2} \):- \( n = 2 \) is valid.- \( l = 2 \) is invalid because \( l \) must be \( < n \), so the maximum \( l \) for \( n = 2 \) is 1.- \( m = -1 \) could be valid if \( l \) were correct.- \( s = -\frac{1}{2} \) is valid.

Step 5: Analyzing Option D

For option (d) \( n = 2, l = 1, m = 0, s = +\frac{1}{2} \):- \( n = 2 \) is valid.- \( l = 1 \) is valid.- \( m = 0 \) is valid as \( -1 \leq 0 \leq 1 \).- \( s = +\frac{1}{2} \) is valid.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is the first quantum number, and it plays a crucial role in determining the primary energy level or shell of an electron in an atom. It's essentially the big picture number when it comes to understanding where an electron lives. This number can be any positive integer: 1, 2, 3, and so on. The larger \( n \), the larger the orbit, and thus, the greater the energy level and distance from the nucleus.

For example, in the solution: option (d), where \( n = 2 \), implies that the electron is in the second energy level. It's like being on the second floor of a building; you're farther from the ground (the nucleus). Remember, the principal quantum number also indicates the total number of subshells within that particular energy level, corresponding to azimuthal quantum numbers. Hence, \( n \) defines the main layer of an electron's location.

Azimuthal Quantum Number

The azimuthal quantum number, \( l \), also known as the angular momentum quantum number, defines the shape of the electron's orbital within a given energy level. Its value depends on the principal quantum number, \( n \), and ranges from 0 up to \( n-1 \).

Here's an easy way to picture this: if \( n \) is the shell, \( l \) is the type or shape of the orbital within that shell (like different shaped rooms on a floor). For instance, if \( n = 2 \), \( l \) can be either 0 or 1, corresponding to \( s \) and \( p \) orbitals. In our correct solution option (d), \( l = 1 \) indicates a "p" orbital present in the second energy level. These orbitals have distinct shapes, impacting the electron's possible locations.

Magnetic Quantum Number

The magnetic quantum number, \( m \), provides more detail about the orientation of the electron's orbital within a given subshell, described by \( l \). It takes integer values ranging from \( -l \) to \( +l \). Hence, \( m \) helps pinpoint the exact orientation in three-dimensional space, akin to how turning your head in different directions gives you varied perspectives.

For example, in option (d) with \( l = 1 \), the possible \( m \) values are -1, 0, and 1. The solution where \( m = 0 \) implies that the orbital has a specific orientation relative to the nucleus. This orientation dictates how orbitals stack within particular subshells, influencing the magnetic properties of the atom as a whole.

Spin Quantum Number

The spin quantum number, often represented as \( s \), determines the direction of the electron's intrinsic spin (like the way a top spins). It can only take one of two possible values: \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This is comparable to flipping a coin: heads or tails, only here it's clockwise or counterclockwise.

Spin is a fundamental property of electrons, and its inclusion helps explain why no two electrons in an atom can share the same set of quantum numbers. This is known as the Pauli Exclusion Principle. In option (d), \( s = +\frac{1}{2} \), meaning the electron's spin contributes to the overall magnetic moment of the atom in a specific way. Understanding this concept is essential for studying concepts like magnetism, chemical bonding, and the quantum behavior of electrons.