Problem 170 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following sets of quantum numbers represents the highest energy of an atom? (a) \(\mathrm{n}=3, l=2, m=1, \mathrm{~s}=+1 / 2\) (b) \(\mathrm{n}=4, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (c) \(\mathrm{n}=3, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (d) \(\mathrm{n}=3, l=1, \mathrm{~m}=1, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (a) represents the highest energy set.

Step 1: Understand Quantum Numbers

Quantum numbers describe electrons in an atom. They include: - Principal Quantum Number ( "): Indicates the electron's energy level or shell. Higher values correspond to higher energy levels. - Azimuthal Quantum Number ( l"): Defines the subshell type (s, p, d, f, etc.) and contributes to the energy level. - Magnetic Quantum Number ( m"): Relates to the orientation of the orbital in space but doesn't affect energy directly. - Spin Quantum Number ( s"): Is either +1/2 or -1/2, indicates electron spin, and does not affect energy directly.

Step 2: Identify Principal and Azimuthal Quantum Numbers

Examine the quantum numbers in each option: - (a) = 3, = 2 - (b) = 4, = 0 - (c) = 3, = 0 - (d) = 3, = 1 Principal and azimuthal quantum numbers decide the energy level of an electron: higher or indicates higher energy.

Step 3: Compare Energy Levels Using n and l

The electron's energy depends on a combination of and quantum numbers, often calculated as + l. Calculate for each option: - (a) + l = 3 + 2 = 5 - (b) + l = 4 + 0 = 4 - (c) + l = 3 + 0 = 3 - (d) + l = 3 + 1 = 4 The higher the sum + l, the higher the energy.

Step 4: Determine the Highest Energy Set

Comparing the sums from Step 3, option (a) has the highest sum of + l = 5, which indicates that it represents the highest energy level among the given options.

Key Concepts

Principal Quantum Number

When studying atomic structure, it's vital to understand what the principal quantum number, denoted as \(n\), represents. This quantum number reveals which shell an electron occupies in an atom and is fundamental to describing its energy. The principal quantum number can have positive integer values of 1, 2, 3, and so on.

The value of \(n\) determines the overall size and energy of the electron's orbit. A larger \(n\) value signifies an electron that resides farther from the nucleus, thus having higher energy. This is crucial as electrons in higher shells require more energy to overcome the attractive pull of the nucleus. In simple terms, the further an electron is from the nucleus, the higher its potential energy.

To consider an example from the exercise, an \(n\) value of 4 (found in option b) would suggest the electron is in the fourth energy level or shell, which is generally a higher energy situation compared to \(n = 3\). However, the full energy is determined when combined with the azimuthal quantum number, which we will explore next.

Azimuthal Quantum Number

The azimuthal quantum number \(l\) contributes to understanding the subshell and shape of the orbital in which the electron resides. In terms of quantum numbers, \(l\) defines different subshells labeled as s, p, d, f, and so on.

The values that \(l\) can take are integers from 0 up to \(n-1\). These numbers not only depict the shape of the electron cloud but also play a role in determining the energy levels of the subshells themselves. For instance, when \(l = 0\), it corresponds to an s subshell, while \(l = 1\) corresponds to a p subshell.

In the exercise, option (a) has \(l = 2\), indicating a d subshell, one of the highest energy states compared to s and p subshells. This shows how the azimuthal quantum number, in combination with the principal quantum number \(n\), can provide a deep insight into the energetic configuration within an atom.

Electron Energy Levels

Electron energy levels are an elegant expression of the quantum mechanical model of the atom. Each electron in an atom occupies a specific energy state, which can be calculated with the sum of its principal and azimuthal quantum numbers \(n + l\). This approach predicts the energy hierarchy of electron levels within an atom.

The calculated \(n + l\) value offers a systematic way to determine which electron state is energetically favorable or higher. For example, an \(n + l\) value of 5 implies more energy than a sum of 4 or 3, irrespective of the actual values of \(n\) and \(l\) individually.

Returning to this logic, we see in the exercise example that option (a) with \(n + l = 5\) ranks the highest, thus confirming it has the highest energy level. This concept is fundamental for predicting electron configurations and understanding chemical reactivity.