Problem 41 - Chemistry and Chemical Reactivity

Question

A particular orbital has \(n=4\) and \(\ell=2 .\) What must this orbital be: (a) \(3 p,\) (b) \(4 p,\) (c) \(5 d,\) or (d) \(4 d ?\)

Step-by-Step Solution

Answer

The orbital is \(4d\), option (d).

Step 1: Identify the Principal Quantum Number

The symbol \(n\) represents the principal quantum number, which determines the energy level of the orbital. In this exercise, \(n = 4\). This means we're looking for an orbital that belongs to the fourth energy level.

Step 2: Determine the Azimuthal Quantum Number

The symbol \(\ell\) represents the azimuthal or angular momentum quantum number, which determines the shape of the orbital. Here, \(\ell = 2\). The azimuthal quantum numbers correspond to subshells: \(\ell = 0\) for \(s\), \(\ell = 1\) for \(p\), \(\ell = 2\) for \(d\), and \(\ell = 3\) for \(f\). So, \(\ell = 2\) indicates a \(d\) orbital.

Step 3: Match the Quantum Numbers to the Available Options

Now we compare the quantum numbers with the given options: - (a) \(3p\): \(n=3\), \(\ell=1\) - (b) \(4p\): \(n=4\), \(\ell=1\) - (c) \(5d\): \(n=5\), \(\ell=2\) - (d) \(4d\): \(n=4\), \(\ell=2\)Matching this with our values (\(n=4\) and \(\ell=2\)), we find that option (d) \(4d\) is the correct choice.

Key Concepts

Principal Quantum Number

The Principal Quantum Number is denoted by the symbol \( n \). It is a fundamental concept in quantum mechanics, serving the key purpose of defining the main energy level or shell in which an electron resides. For instance, if we have \( n = 1 \), we are looking at the first energy level. This quantum number can have integer values starting from 1, and each increment signifies a higher energy level and further distance from the nucleus.

• The principal quantum number gives us an idea about the size of the electron cloud; higher \( n \) means a larger cloud.
• It also influences the energy; as \( n \) increases, the energy level gets higher.

Each energy level can hold a specific number of electrons, calculated using the formula \( 2n^2 \). For example, the second energy level (when \( n=2 \)) can hold up to 8 electrons.

Azimuthal Quantum Number

The Azimuthal Quantum Number, represented by \( \ell \), provides more detailed information about the electron's position and is closely linked to the shape of the electron's orbital. This quantum number can range from 0 to \( n-1 \) for any given value of \( n \).

• \( \ell = 0 \) refers to an s orbital, which is spherical.
• \( \ell = 1 \) indicates a p orbital, which is dumbbell-shaped.
• \( \ell = 2 \) stands for a d orbital, which has a more complex, cloverleaf shape.
• \( \ell = 3 \) indicates an f orbital, known for having even more complex shapes.

This quantum number doesn't just define the shape but also contributes to the angular momentum of an electron within the atom.

Energy Levels

Energy Levels, often described through the principal quantum number \( n \), are the different states where electrons can exist in an atom. These levels are quantized, meaning electrons reside at specific energy levels, not between them. This quantization is a fundamental aspect of quantum mechanics.

• Electrons move to higher energy levels when they gain energy, typically through absorption of photons.
• Conversely, they release energy, often as light, when they return to lower energy levels.

Each energy level can be thought of as a "shell," and within these shells, subshells (represented by \( \ell \)) further define the electron's probable locations.

Electron Orbitals

Electron Orbitals are regions within an atom where electrons are most likely to be found. Each orbital can hold a specific number of electrons, determined by rules that stem from the quantum numbers.These orbitals vary by type, based on the azimuthal quantum number (\( \ell \)), and each type has a unique shape and capacity:

• s-orbitals: Spherical in shape and can hold up to 2 electrons.
• p-orbitals: Dumbbell-shaped and can hold up to 6 electrons across three orientations.
• d-orbitals: Cloverleaf-shaped and can hold up to 10 electrons across five orientations.
• f-orbitals: Complex shapes and can hold up to 14 electrons across seven orientations.

Understanding these configurations is crucial for visualizing how electrons are arranged in an atom and predicting chemical behavior.