Problem 88 - A Complete Resource Book in Chemistry for JEE Main

Question

Correct set of four quantum numbers for the valence (outermost) electron of rubidium \((Z=37)\) is (a) \(5,0,0, \pm 1 / 2\) (b) \(6,0,0,+1 / 2\) (c) \(5,1,1, \pm 1 / 2\) (d) \(5,1,0, \pm 1 / 2\)

Step-by-Step Solution

Answer

The correct set is (a) \(5,0,0, \pm 1 / 2\).

Step 1: Identify the Electron Configuration

Rubidium (Rb) has the atomic number 37. The electron configuration up to rubidium is: \[ 1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^{10}, 4p^6, 5s^1 \]This tells us that the valence electron is in the 5s orbital.

Step 2: Understand Quantum Numbers

Quantum numbers describe an electron in an atom. The four quantum numbers are:- Principal quantum number \(n\): energy level (for 5s orbital, \(n = 5\)).- Azimuthal quantum number \(l\): describes the subshell (for s orbital, \(l = 0\)).- Magnetic quantum number \(m_l\): orientation of the orbital (for \(l = 0\), \(m_l = 0\)).- Spin quantum number \(m_s\): describes electron spin, which can be \(+1/2\) or \(-1/2\).

Step 3: Identify the Correct Set of Quantum Numbers

The correct set of quantum numbers for rubidium's valence electron is based on the 5s orbital:- \(n = 5\)- \(l = 0\)- \(m_l = 0\)- \(m_s = \pm 1/2\)Therefore, the correct option is (a): \(5, 0, 0, \pm 1/2\).

Key Concepts

Electron Configuration

In chemistry, the electron configuration of an atom describes how electrons are distributed among the various orbitals around the nucleus. For rubidium, with an atomic number of 37, the electron configuration is written as:

• \[ 1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^{10}, 4p^6, 5s^1 \]

This notation tells us where each of the 37 electrons resides. The superscripts show the number of electrons in each subshell; here, the valence electron is in the 5s orbital. This configuration helps predict the atom's chemical properties and behavior, such as bonding and reactivity. Remember, the valence electrons are those found in the outermost shell, making them significant in determining how elements interact.

Principal Quantum Number

The principal quantum number, denoted by \( n \), is crucial in quantum mechanics for identifying the energy level of an electron in an atom. The principal quantum number can be any integer from 1 to 7 in most cases today. So, if some electron lies in the fifth shell, as it does for the valence electron of rubidium, the principal quantum number would be \( n = 5 \).
This value not only indicates the energy level but also gives a rough idea of the distance from the nucleus – the larger the \( n \), the further away the electron tends to be. As electrons occupy higher energy levels, they usually possess higher potential energy relative to those closer to the nucleus.

Azimuthal Quantum Number

The azimuthal quantum number, often referred to as the angular momentum quantum number and symbolized as \( l \), is responsible for determining the shape of an electron's orbital. Each principal energy level contains several subshells, labeled s, p, d, and f, with corresponding \( l \) values:

• s subshell: \( l = 0 \)
• p subshell: \( l = 1 \)
• d subshell: \( l = 2 \)
• f subshell: \( l = 3 \)

For the 5s orbital of rubidium, \( l = 0 \) because it resides in an s subshell. This information tells us that the orbital is spherical.

Magnetic Quantum Number

The magnetic quantum number, noted as \( m_l \), specifies the orientation of the orbital in space. It is dependent on the azimuthal quantum number, so for any given \( l \), the magnetic quantum number can adopt integer values ranging from \(-l\) to \(+l\).
For rubidium's valence electron in the 5s orbital, where \( l = 0 \), the range of \( m_l \) is simply 0 because there's only one value. This indicates there is only one orientation in this case since the s subshell is spherical.

Spin Quantum Number

The spin quantum number, \( m_s \), is a fundamental property of the electron and describes its intrinsic spin, which can be thought of as the electron's "twist" or "rotation" around its axis. Electrons have two possible spin states:

• \( m_s = +1/2 \)
• \( m_s = -1/2 \)

These states correspond to what are known as "spin-up" and "spin-down" configurations. For each electron, the spin quantum number helps to distinguish it from other electrons in the same orbital that share the same three other quantum numbers (\( n, l, \) and \( m_l \)). In the 5s orbital of rubidium, the electron could have either of these spins, resulting in the flexibility of choice \( m_s = \pm 1/2 \). This flexibility plays a critical role in aligning and pairing electrons in chemical bonds.