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 \(b\)

Step-by-Step Solution

Answer

Options c and d have the same energy.

Step 1: Understanding Atomic Orbitals

In an atom, the energy of an electron in an orbital is primarily determined by its principal quantum number (n) in the absence of external fields. Electrons with the same principal quantum number have similar energies. The other quantum numbers (l and m) describe the shape and orientation of orbitals, but in a multielectron atom, the energy is primarily affected by n.

Step 2: Compare Quantum Numbers

Examine the given options and group them based on their principal quantum number: - Option a: n=1, Option b: n=2, Options c, d, e: n=3. Since energy is primarily determined by n, compare options with n=3 to find which orbitals can have the same energy.

Step 3: Identify Orbitals with Same Principal Quantum Number (n)

From Step 2, notice that orbitals in options c, d, and e share the same principal quantum number n=3. This means they will have the same energy levels, given no external perturbations.

Step 4: Verify Correct Option

The options (c) and (d) represent orbitals from answer c, which mentions options c and d (n=3, l=1, m=1) and (n=3, l=2, m=1) respectively. Therefore, these two orbitals will have the same energy.

Key Concepts

Quantum Numbers

Atoms have special numbers called quantum numbers that tell us about the behavior and placement of electrons within the atom. There are four quantum numbers, but the ones often involved are the **principal quantum number** ( ext{n} ), the **azimuthal quantum number** ( ext{l} ), and the **magnetic quantum number** ( ext{m} ).
The principal quantum number ( ext{n} ) tells us the energy level and size of the electron's orbit. It's like saying which floor of a building you're on - the higher the floor, the more energy or space you have.
The azimuthal quantum number ( ext{l} ) is about the shape of the path the electron travels. It's like choosing a room on a floor of the building with different shapes. Lastly, the magnetic quantum number ( ext{m} ) describes the orientation of the orbital in space.
These numbers are crucial in understanding **how electrons are arranged** in an atom and how they behave in different energy states and environments.

Multielectron Atoms

Most elements that we deal with are multielectron atoms, meaning they have more than one electron orbiting their nucleus. This makes things a bit more complex because electrons not only interact with the nucleus, but they also interact with each other. In multielectron atoms, the distribution of electrons among the atoms' orbitals is influenced not only by their energy levels but also by the repulsive forces between electrons.
Due to electron-electron repulsion, these atoms do not strictly follow the same energy patterns found in hydrogen-like or single electron systems. For example, in a multielectron environment, the energy of orbitals also depends on both the principal quantum number and the azimuthal quantum number rather than just one. This means that the energy is not just a simple outcome of one quantum number but is instead a combination of interactions and various quantum properties.

Energy Levels

When we talk about energy levels in atoms, we often refer to them as shells or layers where electrons are likely to be found. These shells correspond to the principal quantum number ( ext{n} ) and are usually labeled as K, L, M, etc., with K being the closest shell to the nucleus.
In a multielectron atom, the energy of an electron in a particular orbital (also called energy level) is not just determined by its **principal quantum number**. Different factors, such as electron shielding and penetration, also affect these energy levels. As electrons fill the inner orbitals, they can shield outer electrons from the full effect of the nuclear charge, impacting the energy of the other electrons.

• Lower ext{n} values represent lower energy levels, often closer to the nucleus.
• Higher ext{n} values represent higher energies and further distances from the nucleus.

Understanding these energy levels helps in predicting electron arrangements and chemical properties of elements.

Principal Quantum Number

The principal quantum number is one of the main components for defining the electron's energy level in an atom. Represented by ** ext{n} **, it also signifies the size of the electron cloud area. ext{n} is always a positive integer (1, 2, 3, etc.), and each increase represents an increase in an orbit's size and potential energy.
The principal quantum number not only dictates the overall energy and size but also determines how far away the electron cloud is generally from the nucleus. For example, an electron with ext{n} = 3 is typically further away from the nucleus than one with ext{n} = 1. This is important when analyzing atomic structures because orbital energies in multi-electron atoms—and therefore their chemical behaviors—are heavily influenced by these differences in ext{n} values.
Therefore, in a set of options like those in exercises, identifying shared ext{n} values allows you to quickly hypothesize about similar energy levels for the orbitals involved. This understanding is crucial for looking at options and determining equivalences and differences in atomic and molecular energy considerations.