Question

Which of the following orbital designations are incorrect: \(1, s$$1 p, 7 d, 9 s, 3 f, 4 f, 2 d ?\)

Short answer

The incorrect orbital designations are \(1p\), \(3f\), and \(2d\).

Step-by-step solution

Understand orbital designations

For the given orbital designations, the first number represents the principal quantum number (n) and the letter represents the subshell or the azimuthal quantum number (l). Here's the correspondence between the subshell letters and their respective l values: - s: l = 0 - p: l = 1 - d: l = 2 - f: l = 3

Evaluate the given orbital designations

We will now analyze each given orbital designation to check if it satisfies the two rules mentioned in the analysis: 1. \(1s\): - n = 1 and l = 0, which satisfies both rules. Therefore, this designation is correct. 2. \(1p\): - n = 1 and l = 1, but l should be less than n (0 to n - 1). This designation violates rule 2, so it is incorrect. 3. \(7d\): - n = 7 and l = 2, which satisfies both rules. Therefore, this designation is correct. 4. \(9s\): - n = 9 and l = 0, which satisfies both rules. Therefore, this designation is correct. 5. \(3f\): - n = 3 and l = 3, but l should be less than n (0 to n - 1). This designation violates rule 2, so it is incorrect. 6. \(4f\): - n = 4 and l = 3, which satisfies both rules. Therefore, this designation is correct. 7. \(2d\): - n = 2 and l = 2, but l should be less than n (0 to n - 1). This designation violates rule 2, so it is incorrect. Based on the analysis, the orbital designations \(1p\), \(3f\), and \(2d\) are incorrect.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as 'n', is fundamental in the realm of quantum mechanics. It specifies the energy level or shell to which an electron belongs within an atom. The value of 'n' is a positive integer starting from 1, meaning the closer an electron is to the nucleus, the lower the value of 'n' and the lower its energy state.

As 'n' increases, electrons reside in shells farther away from the nucleus and possess higher energy. Furthermore, the number of possible subshells within a shell is equal to the value of the principal quantum number. For instance, in the second energy level (n=2), there are two subshells (s and p). An important rule is that the azimuthal quantum number 'l' must be in the range of 0 to n-1 for any given energy level.

Azimuthal Quantum Number

The azimuthal quantum number, represented by 'l', determines the shape of the electron orbital and is closely related to the subshell designation (s, p, d, f). Its value ranges from 0 up to (n-1), where 'n' is the principal quantum number. Each value of 'l' correlates with different subshell types: for l=0, the subshell is 's'; for l=1, it's 'p'; l=2 corresponds to 'd'; and l=3 is for 'f' subshells.

It's these azimuthal numbers that help us understand why certain orbital designations, like '1p' or '3f', are incorrect – they violate the rule that 'l' should always be less than 'n'.

Electron Orbitals

Electron orbitals are regions around an atom's nucleus where electrons have the highest probability of being found. These orbitals are 3D representations of an electron's position and are classified as s, p, d, or f. Each type of orbital holds a different number of electrons: s orbitals hold up to 2, p orbitals can contain up to 6, d orbitals up to 10, and f orbitals up to 14 electrons.

The shape and complexity of these orbitals increase with their type; for example, s orbitals are spherical, p orbitals are dumbbell-shaped, and d and f orbitals have more complex shapes. These shapes are important as they dictate how electrons in these orbitals will interact with those in other orbitals, impacting the chemical behavior of the elements.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It underpins the scientific understanding of atoms and subatomic particles, and their behavior at the smallest scales.

In terms of electron orbitals, quantum mechanics explains how electrons can exist in only certain quantized energy states and how their position cannot be pinpointed exactly but rather described by probabilities. Concepts like the principal and azimuthal quantum numbers arise from this theory and dictate the rules we use to determine if an electron orbital designation is correct or not. Understanding these quantum numbers is essential for predicting electron configurations and, consequently, the chemical properties of elements.