Problem 127 - A Complete Resource Book in Chemistry for JEE Main

Question

Pick out the wrong statements. (a) \(\mathrm{Fe}^{3+}\) ion is more stable than \(\mathrm{Fe}^{2+}\) ion in the gaseous state. (b) For an electron in a \(4 \mathrm{p}\)-orbital, the quantum numbers are \(\mathrm{n}=4, l=1, \mathrm{~m}=2, \mathrm{~s}=+1 / 2\) (c) Angular momentum of 3 s electron is one. (d) \((\mathrm{n}+I)\) rule is followed for determining the orbital of the lowest energy state.

Step-by-Step Solution

Answer

Statements (b) and (c) are wrong.

Step 1: Evaluate statement (a)

Statement (a) says \(\mathrm{Fe}^{3+}\) ion is more stable than \(\mathrm{Fe}^{2+}\) ion in the gaseous state. Typically, \(\mathrm{Fe}^{3+}\) is more stable in aqueous solutions due to higher effective nuclear charge and stronger hydration energy, but not necessarily in the gaseous state. Consider other contexts for specific checks.

Step 2: Analyze statement (b)

In statement (b), the quantum numbers mentioned for an electron in a \(4p\)-orbital are \(n=4, l=1, m=2, s=+1/2\). The magnetic quantum number \(m\) should range from \(-l\) to \(+l\), so for \(l=1\), \(m\) can only be -1, 0, or +1. Therefore, the given value \(m=2\) is incorrect.

Step 3: Check statement (c)

Statement (c) claims the angular momentum of a \(3s\) electron is one. For any \(s\)-orbital, \(l=0\), so the angular momentum \(\sqrt{l(l+1)}\hbar = 0\). Hence, this statement is incorrect.

Step 4: Verify statement (d)

Statement (d) discusses the \((n+l)\) rule, which is a valid principle for determining the order of energy levels of orbitals. Hence, this statement is correct.

Key Concepts

Stability of Ions

The stability of ions is a fundamental concept in chemistry, particularly when discussing ions like \( \text{Fe}^{2+} \) and \( \text{Fe}^{3+} \). Stability often depends on several factors, including effective nuclear charge, electronic configuration, ionization energy, and the environment.

• **Effective Nuclear Charge**: \( \text{Fe}^{3+} \) has a higher effective nuclear charge compared to \( \text{Fe}^{2+} \) because it has one fewer electron, allowing the remaining electrons to be held more tightly by the nucleus.
• **Electronic Configuration**: For example, \( \text{Fe}^{3+} \) has the electron configuration [Ar] 3d⁵. A half-filled \(3d\) subshell is considered stable because of symmetry and exchange energy.
• **Environmental Influence**: While \( \text{Fe}^{3+} \) is more stable in aqueous solutions due to stronger hydration energy, in the gaseous state this hydration energy isn't a factor. Therefore, gas phase stability is not straightforward to determine based solely on common stability rules.

Understanding these factors assists in determining in which state an ion is more stable, although practical measurements are often required for concrete conclusions.

Angular Momentum

Angular momentum is an important concept in quantum mechanics, particularly in the context of electrons in an atom. It is derived from quantum numbers and plays a crucial role in defining the shape and orientation of an electron's orbital.

• **Quantum Number \( l \)**: The angular momentum of an electron is primarily determined by the azimuthal quantum number \( l \). This quantum number describes the shape of the orbital and can take values from 0 up to \( n-1 \), where \( n \) is the principal quantum number.
• **Formula for Angular Momentum**: The angular momentum of an electron in an orbital is given by \( \sqrt{l(l+1)}\hbar \), where \( \hbar \) is the reduced Planck's constant. For instance, in a \( 3s \) orbital, since \( l = 0 \), the angular momentum is \( 0 \), which shows that \( s \)-orbitals are spherical and have no angular momentum.
• **Role in Chemical Properties**: Angular momentum influences several chemical properties. Higher angular momentum indicates more complex orbital shapes and can affect how atoms interact and bond.

Understanding angular momentum helps in visualizing and predicting the behavior of electrons, crucial for interpreting electron configurations and predicting chemical interactions.

The n+l Rule

The \( (n+l) \) rule is an essential heuristic in quantum chemistry that helps predict the order of filling of atomic orbitals. It is a simple but powerful tool to identify which orbitals electrons will occupy in atoms.

• **Definition**: According to the rule, the atomic orbitals are filled in order of increasing \( n+l \) values. \( n \) refers to the principal quantum number, and \( l \) is the azimuthal quantum number.
• **Higher Values, Higher Energy**: Orbitals with a lower \( n+l \) value are filled first because they have lower energy. If two orbitals have the same \( n+l \) value, the one with the lower \( n \) is filled first, as its electrons are closer to the nucleus.
• **Predicting Electron Configuration**: This rule is crucial for predicting electron configurations of elements. For instance, an electron would fill a \( 4s \) orbital (\( n+l = 4 \)) before a \( 3d \) orbital (\( n+l = 5 \)).

Utilizing the \( (n+l) \) rule allows chemists to establish the building blocks for understanding atomic structure and the periodic table.