Problem 39 - Chemistry and Chemical Reactivity

Question

State which of the following orbitals cannot exist according to the quantum theory: \(2 s, 2 d, 3 p, 3 f, 4 f\) and \(5 s .\) Briefly explain your answers.

Step-by-Step Solution

Answer

The orbitals "2d" and "3f" do not exist.

Step 1: Understanding Atomic Orbitals

Atomic orbitals are defined by quantum numbers. The principal quantum number, \(n\), determines the energy level and size of the orbital and is a positive integer (\(n = 1, 2, 3, \ldots\)). The azimuthal quantum number, \(l\), defines the shape of the orbital and can take on values from 0 to \(n-1\). The orbitals are labeled as \(s, p, d, f, \ldots\) corresponding to \(l = 0, 1, 2, 3, \ldots\).

Step 2: Analyzing Each Orbital

Let's analyze each orbital: 1. **2s:** For \(n=2\), \(l\) could be 0 (s orbital) so \(2s\) is valid.2. **2d:** For \(n=2\), \(l\) cannot be 2 as \(l\) can be at most \(n-1\) which is 1 here. Therefore, \(2d\) is not valid.3. **3p:** For \(n=3\), \(l\) can be 0, 1 (p orbital) so \(3p\) is valid.4. **3f:** For \(n=3\), \(l\) can be at most 2, so \(3f\) is not valid.5. **4f:** For \(n=4\), \(l\) can be 3 (f orbital) so \(4f\) is valid.6. **5s:** For \(n=5\), \(l\) could be 0 (s orbital) so \(5s\) is valid.

Step 3: Conclusion

Based on the quantum numbers, the orbitals that do not exist are \(2d\) and \(3f\). This is because the azimuthal quantum number \(l\) cannot be equal to or greater than \(n\) for a given energy level.

Key Concepts

Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where the probability of finding an electron is highest. These orbitals come in various shapes and sizes, each described by a set of quantum numbers. Imagine atomic orbitals as clouds around the nucleus where electrons move in complex patterns.

• The shape and orientation of these orbitals determine the chemical properties and reactions of the atoms.
• Orbitals are often depicted as simple shapes such as spheres or dumbbells to make understanding easier, but they can be more complicated.
• They are categorized by letters like s, p, d, and f, which are derived from the azimuthal quantum number, one of the four quantum numbers.

In simple terms, you can think of orbitals as the 'rooms' electrons live in within an atom.

Quantum Numbers

Quantum numbers are like the address for electrons in an atom, providing specific information about where an electron is likely to be found. There are four quantum numbers, each offering a different piece of information about the electron's position and motion:

• **Principal Quantum Number (n):** Indicates the main energy level of the electron, related to the size and energy of the orbital.
• **Azimuthal Quantum Number (l):** Defines the shape of the orbital, with each shape corresponding to a different letter (s, p, d, f).
• Other quantum numbers include the magnetic quantum number (m_l) and the spin quantum number (m_s), which further refine the address of the electron within an atom.

When considering which orbitals can exist, the interplay between these numbers helps determine their shapes and limits.

Principal Quantum Number

The principal quantum number, denoted as \( n \), correlates to the energy level of an electron within an atom. It is a positive integer starting from 1, and as \( n \) increases, the electron's average distance from the nucleus also increases. Here are some key points about \( n \):

• \( n \) values such as 1, 2, 3, etc., determine the electron shell. The larger \( n \), the higher the energy level and the larger the orbital.
• Orbits with the same \( n \) value can have different shapes and are also known as shells. These shells contain all possible orbitals for the given principal quantum number.
• The energy associated with these shells increases, making electrons in the higher \( n \) levels easier to remove.

Understanding \( n \) helps explain why certain orbitals like "2d" or "3f" do not exist, as the shapes are not allowed at those energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, represented as \( l \), provides a mathematical way to determine the orbital's shape. It can take on any integer value from 0 to \( n-1 \), where \( n \) is the principal quantum number:

• \( l = 0 \) corresponds to "s" orbitals, \( l = 1 \) corresponds to "p" orbitals, \( l = 2 \) to "d" orbitals, and \( l = 3 \) to "f" orbitals.
• This means for a principal quantum number \( n \), only certain shapes or types of orbitals are allowed.
• The relationship between \( n \) and \( l \) helps identify which orbitals can exist at specific energy levels. For example, a "2d" orbital is impossible because \( l \) would need to be 2, which exceeds the maximum allowed \( n-1 \) value for \( n = 2 \).

Thus, the azimuthal quantum number plays a crucial role in ruling out orbitals that defy these mathematical boundaries.