Problem 43 - 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.

Step 1: Identify Principal Quantum Number

The principal quantum number \(n\) describes the energy level of an electron in an atom. Here, \(n = 4\). This means we are looking for an orbital in the fourth energy level.

Step 2: Identify Angular Momentum Quantum Number

The angular momentum quantum number \(\ell\) describes the shape of the orbital. It can take integer values from 0 to \(n-1\). For \(\ell = 2\), the possible designation is \(d\).

Step 3: Match Quantum Numbers to Orbital

Combine \(n = 4\) and \(\ell = 2\) to determine the orbital designation. The principal quantum number \(n = 4\) indicates the energy level, and \(\ell = 2\) means \(d\) orbital. Hence, the correct orbital designation is \(4d\).

Step 4: Verify Against Options

Review the given options: (a) 3p, (b) 4p, (c) 5d, and (d) 4d. Compare with our findings. Only option (d) 4d matches the \(n=4, \ell=2\) combination.

Key Concepts

Principal Quantum Number

In the fascinating world of quantum mechanics, the principal quantum number, represented as \( n \), plays a critical role in determining an electron's position within an atom. This number is integral to understanding the energy levels where electrons are likely to be found. Specifically, \( n \) signifies the main energy shell around the nucleus of an atom.
\( n \) can have positive integer values starting from 1, such as \( 1, 2, 3, \) and so forth.

• As \( n \) increases, the electron occupies a shell farther from the nucleus.
• Energy levels are not continuous; electrons can only exist in these specific shells.

In simpler terms, if you think of an atom like a building, \( n \) is akin to the floor number, indicating how close or far an electron is from the ground floor (nucleus). This number significantly influences the amount of energy the electron possesses, with higher values of \( n \) generally correlating with more energy.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \( \ell \), is another foundational principle in quantum mechanics that describes the shape of an electron's orbital. Think of it as defining the style or type of room an electron can occupy within a building (energy level) defined by \( n \).
\( \ell \) can have integer values ranging from 0 to \( n-1 \).

• Each value of \( \ell \) corresponds to a different type of orbital, namely: \( \ell = 0 \) (s orbital), \( \ell = 1 \) (p orbital), \( \ell = 2 \) (d orbital), and \( \ell = 3 \) (f orbital).
• The value of \( \ell \) also relates to the surface complexity of electron clouds; higher values mean more complex patterns.

For example, in our exercise, when \( \ell = 2 \), it indicates a d orbital. Thus, the angular momentum quantum number helps us recognize and predict the spatial arrangement and geometry of electron orbitals within an atom, intrinsically linking it to the atom's chemical properties and binding characteristics.

Electron Orbitals

Electron orbitals are the 'rooms' where electrons 'live' within an atom, specifically within the energy levels determined by the principal quantum number \( n \) and the shape defined by the angular momentum quantum number \( \ell \). Orbital designations therefore provide a detailed map of where an electron is likely to be found.
Orbital naming combines these quantum numbers, like in our exercise, \( n=4 \) and \( \ell=2 \), leading to the term "4d orbital."

• The number "4" comes from the principal quantum number, indicating it's in the fourth energy level.
• The "d" follows from \( \ell=2 \), representing the shape of the orbital.

It’s important to remember that electrons don't actually 'orbit' in neat paths like planets around the sun. Instead, they exist within these orbitals, which are regions in space where there is a high probability of finding an electron. Understanding this stochastic nature of electron placement is key to mastering the basics of quantum chemistry and the behavior of atoms under various conditions.