Problem 80 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement(s) are correct? (1) the electronic configuration of \(\mathrm{Cr}\) is \([\mathrm{Ar}] 3 \mathrm{~d}^{5} 4 \mathrm{~s}^{1}\) (atomic number of \(\mathrm{Cr}=24\) ) (2) the magnetic quantum number may have a negative value (3) in silver atom, 23 electrons have a spin of one type and 24 of the opposite type (atomic number of \(\mathrm{Ag}=47\) )

Step-by-Step Solution

Answer

All given statements are correct.

Step 1: Understanding Chromium's Configuration

The atomic number of Chromium (\(\mathrm{Cr}\)) is 24. Typically, the electron configuration would be written as \([\mathrm{Ar}] 3d^4 4s^2\). However, due to stability reasons, one electron from the 4s orbital is promoted to the 3d orbital, resulting in the configuration \([\mathrm{Ar}] 3d^5 4s^1\). This configuration provides a half-filled d-subshell, which is more stable.

Step 2: Evaluating the Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_l\), can take integral values from \(-l\) to \(+l\), where \(l\) is the azimuthal quantum number. Thus, \(m_l\) may indeed have negative values. For example, if \(l = 2\), then \(m_l\) can be \(-2, -1, 0, 1,\) or \(2\).

Step 3: Analyzing Electron Spin in Silver

Silver (\(\mathrm{Ag}\)) has an atomic number of 47, meaning it has 47 electrons. In a filled or half-filled subshell, electrons will pair up with opposite spins. Since there is an odd number of electrons in silver, one electron will be unpaired. Therefore, half will have one spin type, and the other half plus one electron will have the opposite type. Specifically, 23 electrons have one type of spin and 24 electrons have the opposite type.

Key Concepts

Electronic Configuration

The concept of electronic configuration is fundamental in understanding how electrons are arranged within an atom. Electrons are filled into atomic orbitals in a specific order that usually follows the Aufbau principle, where lower-energy orbitals are filled first. For instance, Chromium (Cr) is an interesting example where the typical order gets adjusted for stability.

In the case of Chromium, which has an atomic number of 24, one might expect the configuration to be \( \text{[Ar]} 3d^4 4s^2 \). However, due to electron-electron interactions and stabilization effects, a slightly unusual arrangement occurs. To achieve stability, one electron from the 4s orbital moves to the 3d orbital, resulting in the actual configuration of \( \text{[Ar]} 3d^5 4s^1 \). This configuration ensures that the d-subshell becomes half-filled, which is energetically favorable.

This rearrangement exemplifies what's known as the 'half-filled and fully-filled' stability rule.

• Half-filled subshells have symmetrical electron distribution, minimizing electron repulsions.
• These configurations often exhibit lower energy and lead to greater stability.

Magnetic Quantum Number

Each electron in an atom is described by a set of quantum numbers, and one key part of this set is the magnetic quantum number denoted as \( m_l \). This quantum number provides us with information on the orientation of the orbital in space relative to the other orbitals.

The possible values of \( m_l \) range from \(-l\) to \(+l\), where \(l\) is the azimuthal quantum number, indicating the shape of the orbital. For example, if \(l = 2\) (corresponding to a d-orbital), \( m_l \) could be any integer value: \(-2, -1, 0, +1, +2\).

This means that magnetic quantum numbers can be negative, zero, or positive, allowing orbitals to be oriented at various angles in a magnetic field. Understanding \( m_l \'s\) possible values helps in predicting electron behavior in atoms subjected to magnetic interactions, which is crucial when deriving magnetic properties in compounds.

Electron Spin

Electron spin is a fundamental property of electrons, defining their intrinsic angular momentum. Each electron has a spin quantum number, represented by \( s\), which can take values of \(+\frac{1}{2} \) or \(-\frac{1}{2} \). This quantization is essential in explaining the magnetic behavior of atoms as well as determining how electrons occupy orbitals.

In most scenarios, electrons are paired within an orbital, each with opposite spins. This pairing essentially cancels out the magnetic effects of the electrons. However, in atoms like Silver (i3;)), which has an odd number of electrons (47 in this case), there will be an unpaired electron.

• This results in 23 electrons having one type of spin.
• 24 electrons will have the opposite type of spin.

The behavior of electron spin and its effects can also explain many macro-properties, like why materials exhibit magnetism or the absence thereof. Spin considerations must be carefully managed when determining the electron configuration of atoms, as they impact the atom's chemical and physical properties.