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.

Step 1: Understand Quantum Numbers

Quantum numbers define the state of an electron in an atom. They are: \(n\) (principal quantum number), \(l\) (azimuthal quantum number), and \(m\) (magnetic quantum number). \(n\) can be any positive integer. \(l\) ranges from 0 to \(n-1\). \(m\) ranges from \(-l\) to \(+l\).

Step 2: Analyze Option (a)

For option (a), \(n=2\), \(l=1\), \(m=0\). This configuration is possible because \(l=1\) is within the range of 0 to \(n-1=1\), and \(m=0\) is within the range of \(-l\) to \(+l\).

Step 3: Analyze Option (b)

For option (b), \(n=2\), \(l=0\), \(m=-1\). This configuration is not possible. Here, \(l=0\) allows only a single value for \(m\), which is 0. Thus, \(m=-1\) is not possible.

Step 4: Analyze Option (c)

For option (c), \(n=3\), \(l=0\), \(m=0\). This configuration is possible as \(l=0\) has \(m=0\) within the allowable range \(-l\) to \(+l\).

Step 5: Analyze Option (d)

For option (d), \(n=3\), \(l=1\), \(m=-1\). This configuration is possible. Here \(l=1\) can have \(m\) values of -1, 0, or 1, making \(m=-1\) valid.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a fundamental quantum number in the atomic world. It plays a key role in determining the energy level or shell of an electron within an atom. Higher values of \( n \) correspond to electrons being further from the nucleus and typically at higher energy levels.

• \( n \) is a positive integer (e.g., 1, 2, 3, ...).
• Each increase in \( n \) signposts a further principal shell, like peeling layers of an onion.
• Electrons in shells with higher principal quantum numbers possess more energy and are typically further from the attractive pull of the nucleus.

Remember, the principal quantum number sets not only energy levels but also the size of orbitals. It's a stepping stone to understanding the atom's structure.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), provides insights into the shape of electron clouds around the nucleus, otherwise known as orbitals. It's also referred to as the angular momentum quantum number.

• For any given principal quantum number \( n \), \( l \) can range from 0 up to \( n-1 \).
• The value of \( l \) dictates the shape of the orbital:
  • \( l=0 \): s-orbital (spherical shape)
  • \( l=1 \): p-orbital (dumbbell shape)
  • \( l=2 \): d-orbital (clover shape)
  • \( l=3 \): f-orbital (complex shape)
• The azimuthal quantum number influences not only the shape but also the number of angular nodes within the orbital.

The azimuthal quantum number is essential for visualizing where an electron might be around the nucleus, contributing greatly to our understanding of chemical bonding and molecular geometry.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m \), specifies the orientation of an electron's orbital around the nucleus. While the azimuthal quantum number defines the shape, \( m \) illuminates how this shape is oriented in three-dimensional space.

• For each \( l \) value, \( m \) can range from \(-l\) to \(+l\). For example, if \( l = 1 \), \( m \) can be -1, 0, or 1.
• The magnetic quantum number is crucial in determining how orbitals are arranged in an atom, particularly when external magnetic fields are present.
• This quantum number explains the existence of multiple orientations for p, d, and f orbitals, which is critical when electrons interact with external fields.

Understanding the magnetic quantum number helps one anticipate electron position and behavior in the presence of magnetic influence, pivotal in atomic chemistry and physics.