Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement is/ are correct? (a) The number of unpaired electrons in both \(\mathrm{Fe}^{2+}\) and \(\mathrm{Mn}^{2+}\) are five. (b) In silver atom, 23 electrons have a spin of one type and 24 of the opposite (atomic number of \(\mathrm{Ag}=\) 47). (c) The azimuthal quantum number may have a negative value. (d) The electronic configuration of \(\mathrm{Cr}\) is \([\mathrm{Ar}] 3 \mathrm{~d}^{5} 4 \mathrm{~s}^{\mathrm{l}}\) (atomic number of \(\mathrm{Cr}=24\) )

Step-by-Step Solution

Answer

Statements (b) and (d) are correct.

Step 1: Analyzing Statement (a)

The electron configuration for \(\mathrm{Fe}^{2+}\) is \([\mathrm{Ar}] 3d^6\), which means it has 4 unpaired electrons. For \(\mathrm{Mn}^{2+}\), the electron configuration is \([\mathrm{Ar}] 3d^5\), indicating 5 unpaired electrons. Thus, this statement is incorrect because \(\mathrm{Fe}^{2+}\) has 4 unpaired electrons, not 5.

Step 2: Analyzing Statement (b)

The atomic number of Ag is 47, meaning it has 47 electrons. Using the electron spin parity rule, 46 electrons will be paired (each pair having opposite spins), and 1 (the unpaired electron) can have a spin in either direction. Therefore, 23 electrons will have one type of spin, and 24 electrons will have the opposite spin. This statement is correct.

Step 3: Analyzing Statement (c)

The azimuthal quantum number \((l)\) represents the angular momentum of an electron and is always a non-negative integer from 0 to \(n-1\), where \(n\) is the principal quantum number. Thus, it cannot have a negative value. This statement is incorrect.

Step 4: Analyzing Statement (d)

The electronic configuration of chromium (\(\mathrm{Cr}\)) is an exception to the typical electron filling order. Its configuration is indeed \([\mathrm{Ar}] 3d^5 4s^1\), explaining the stability provided by a half-filled d-subshell. This statement is correct.

Key Concepts

Quantum Numbers

Quantum numbers are essential for describing the unique quantum state of electrons in an atom. They provide a roadmap of where an electron is likely to be found within an atom. There are four main quantum numbers, each describing a different property of the electron:

• The Principal Quantum Number ( n ): This tells us the electron's main energy level or shell. It is a positive integer (1, 2, 3, ...), and higher numbers mean electrons are further from the nucleus and possess more energy.
• The Azimuthal Quantum Number ( l ): This number deals with the electron's orbital type, defining the shape of the electron cloud. It ranges from 0 to n-1 , where each value correlates to a particular shape (s, p, d, f).
• The Magnetic Quantum Number ( m_l ): Ranging between -l and +l , this number indicates the orientation of an electron's orbital in space.
• The Spin Quantum Number ( m_s ): This is either +1/2 or -1/2 , representing the two possible "spin" states of an electron. Spin is an intrinsic form of angular momentum carried by electrons.

Understanding these quantum numbers helps chemists and physicists predict and explain properties like magnetic behavior and spectral lines. It's worth noting that each electron in an atom must have a unique set of quantum numbers, a principle known as the Pauli Exclusion Principle.

Unpaired Electrons

Unpaired electrons play a crucial role in determining the magnetic properties of an element or compound. Electrons tend to pair up in orbitals, with each pair having opposite spins, minimizing the overall spin value. When an electron does not have a partner, it is considered 'unpaired'
Unpaired electrons contribute to a material's magnetism because they have intrinsic angular momentum, or spin, that produces a magnetic moment. More unpaired electrons mean stronger magnetic properties.
For example, in the case of Mn^{2+} mentioned in the exercise, it has five unpaired electrons because its electron configuration ends in 3d^5 . Each of these five electrons occupies an individual d orbital, aligned in parallel spins, leading to significant magnetic behavior. In contrast, Fe^{2+} has one fewer unpaired electron, resulting in four unpaired electrons in 3d^6 configuration. Consequently, Fe^{2+} exhibits slightly different magnetic properties compared to Mn^{2+} .

Spin Parity Rule

The Spin Parity Rule helps in understanding how electrons are arranged within an atom or ion, particularly focusing on electron spins. One of the fundamental principles in chemistry and quantum physics is that electron spins are quantized, meaning electrons can either spin "up" ( +1/2 ) or "down" ( -1/2 ).
In neutral atoms, electrons fill orbitals in pairs. According to Hund's rule, each orbital in a subshell must receive one electron with parallel spins first, before any orbital gets a second electron with an opposite spin. This rule aims at maximizing the even distribution of electrons.
The practical application of the spin parity rule is seen in calculating total spin for magnetic studies. For silver (Ag), with atomic number 47, there are 23 electrons with one spin type and 24 with the opposite, contributing to its overall spin state. This property is crucial in explaining magnetic behavior and electron arrangements in both natural and synthesized materials.