Problem 165 - A Complete Resource Book in Chemistry for JEE Main

Question

In a multielectron atom, which of the following orbitals described by the three quantum numbers will have the same energy in the absence of magnetic field and electric fields? (a) \(\mathrm{n}=1, l=0, \mathrm{~m}=0\) (b) \(\mathrm{n}=2, l=0, \mathrm{~m}=0\) (c) \(\mathrm{n}=3, l=1, \mathrm{~m}=1\) (d) \(\mathrm{n}=3, l=2, \mathrm{~m}=1\) (e) \(\mathrm{n}=3, l=2, \mathrm{~m}=0\) (a) \(\mathrm{b}\) and \(\mathrm{c}\) (b) \(\mathrm{d}\) and \(\mathrm{e}\) (c) \(\mathrm{c}\) and \(\mathrm{d}\) (d) a and \(\mathrm{b}\)

Step-by-Step Solution

Answer

In a multielectron atom, orbitals (d) and (e) will have the same energy as they share the same principal quantum number \( n=3 \).

Step 1: Understand Orbital Energy within Principle Energy Level

In the absence of magnetic and electric fields, orbitals within the same principal quantum number, \( n \), have the same energy. This is because the energy of an electron in a multielectron atom is primarily determined by its principal quantum number \( n \), not by the angular momentum \( l \) or the magnetic quantum number \( m \).

Step 2: Analyze Options Based on Principal Quantum Numbers

We compare the orbital sets: - (b) \( n=2 \) and (c) \( n=3 \) do not match in \( n \), so they are not the same energy level.- (d) \( n=3 \) and (e) \( n=3 \) have the same principal quantum number, so they are at the same energy level.

Step 3: Select the Correct Pair Based on Analysis

Given we are looking for orbitals with the same energy, we select the pairs which share the same \( n \). Therefore, the correct answer is option (b), which gives the pair (d) and (e), both \( n=3 \).

Key Concepts

Multi-electron Atoms

In multi-electron atoms, electrons don't just hang around randomly. They reside in structured energy levels known as orbitals. These atoms are more complex than hydrogen atoms, which contain just one electron. This complexity arises because electrons in multi-electron atoms experience not just the attraction from the positively charged nucleus, but also repulsion from other electrons. Each electron is thus affected by the electric field produced by the nucleus and the other electrons.

• This interaction modifies the energy levels of the electrons.
• It leads to energy splitting of orbitals that might otherwise have the same energy in an isolated hydrogen atom.

When understanding multi-electron atoms, it’s crucial to remember that the electrons are not isolated — they influence each other, affecting their movement and energy.

Quantum Numbers

Quantum numbers are values that describe the unique quantum state of an electron in an atom. These numbers give us important information about the electron’s position and energy in an atom, similar to how a postal code tells us the precise location of a house within a city.

• The principal quantum number, \( n \), indicates the main energy level of an electron and its relative size.
• The angular momentum quantum number, \( l \), defines the shape of the orbital, ranging from \( 0 \) to \( n-1 \).
• The magnetic quantum number, \( m \), specifies the orientation of the orbital, which can range from \( -l \) to \( +l \).

These numbers work together to provide a complete description of an electron's probable location and energy, allowing us to predict how an atom behaves physically and chemically.

Orbital Energy Levels

Orbital energy levels, or electron shells, refer to the fixed energies an electron can have within an atom. In multi-electron atoms, these levels don't only depend on the principal quantum number, \( n \). This situation is different from hydrogen atoms, where energy levels only depend on \( n \).

• In multi-electron atoms, electron-electron repulsion and the effects of quantum mechanics mean that energy levels also depend on the angular momentum quantum number, \( l \).
• The \( s \), \( p \), \( d \), and \( f \) orbitals have different energies, even if they share the same principal quantum number.

Thus, electrons fill the lowest available energy states first, following the Aufbau principle, resulting in a specific electron configuration that influences chemical properties.

Principal Quantum Number

The principal quantum number, \( n \), is a crucial part of understanding an atom's electron arrangement. It essentially tells us the "shell" or energy level in which the electron resides. A higher \( n \) value means the electron is farther from the nucleus and has higher energy.

• \( n \) can be any positive integer starting from 1 (1, 2, 3, ...).
• Electrons in a lower \( n \) value are closer to the nucleus and typically more tightly bound.

In multi-electron atoms, electrons with the same \( n \) value are in the same "shell", which influences how they share the same energy without external magnetic and electric fields. This concept is vital in identifying orbital energies and patterns within an atom, and essential in predicting an atom’s behavior and properties.