Question

Which of the following orbital designations is(are) possible? a. \(1 s\) b. \(2 p\) c. \(2 d\) d. \(4 f\)

Short answer

The possible orbitals are \(1s\), \(2p\), and \(4f\).

Step-by-step solution

Check orbital 'a': 1s

For this orbital, n = 1 and l = 0 (since s is for l = 0). Since l can have values from 0 to n - 1, this is a possible orbital.

Check orbital 'b': 2p

For this orbital, n = 2 and l = 1 (since p is for l = 1). Given that l can have values from 0 to n - 1, this is also a possible orbital.

Check orbital 'c': 2d

For this orbital, n = 2 and l = 2 (since d is for l = 2). In this case, l = n, which violates the rule that l should range from 0 to n - 1. Therefore, this is not a possible orbital.

Check orbital 'd': 4f

For this orbital, n = 4 and l = 3 (since f is for l = 3). Given that l can have values from 0 to n - 1, this is a possible orbital. So, the possible orbitals are 1s, 2p, and 4f.

Key concepts

Quantum Numbers

Quantum numbers are like the address system for electrons within an atom, describing where they are likely positioned around the nucleus. They help in identifying the shape, orientation, and size of atomic orbitals. Each electron in an atom is described by a set of four quantum numbers:

• Principal Quantum Number (n): This number indicates the energy level or shell that an electron occupies. It ranges from 1 to infinity. The higher the principal quantum number, the farther the electron is from the nucleus.
• Azimuthal Quantum Number (l): Also known as the angular momentum quantum number, this one defines the shape of the orbital. It can take integer values ranging from 0 to -1 (where n is the principal quantum number). For example, for s orbitals l = 0, for p orbitals l = 1, for d orbitals l = 2, and for f orbitals l = 3.
• Magnetic Quantum Number (m_l): This number gives us information about the orientation of the orbital in space. It can range from -l to +l, including zero.
• Spin Quantum Number (m_s): This describes the intrinsic spin of the electron within the orbital, with possible values of +\(\frac{1}{2}\) or -\(\frac{1}{2}\).

Understanding these quantum numbers is crucial because they not only describe the electron's position but also ensure that no two electrons in an atom have the same set of quantum numbers, thanks to the Pauli Exclusion Principle.

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals, following certain rules. The idea is to fill the lowest energy orbitals first before moving to higher ones, a concept known as the Aufbau principle. Here's a simple way to understand it:

• Order of Filling: Electrons fill orbitals in a specific order. The sequence generally starts from 1s, moving through to 2s, 2p, 3s, and so forth.
• Pauli Exclusion Principle: Each orbital can hold a maximum of two electrons with opposite spins. This helps in predicting electron arrangement in atoms.
• Hund's Rule: Before any orbital in a sublevel is filled with a second electron, all orbitals within that sublevel are singly occupied first. This minimizes repulsion among electrons.

For example, the electron configuration of an atom like Carbon would be 1s\(^2\) 2s\(^2\) 2p\(^2\), indicating that the first two electrons occupy the 1s orbital, the next two fill the 2s, and the remaining two go into separate orbitals within the 2p sublevel.

Possible Orbitals

The concept of possible orbitals boils down to understanding which ones can actually exist according to the rules of quantum numbers. The principal quantum number (n) and the azimuthal quantum number (l) are primarily used to determine this. These are the guidelines:

• Possible values for l: For any given principal quantum number n, the azimuthal quantum number l can range from 0 to n-1. This means, for n = 1, l can only be 0; for n = 2, l can be 0 or 1, and so forth.
• Typical Orbitals: The common designations are s for l = 0, p for l = 1, d for l = 2, and f for l = 3. Based on this rule, orbitals like 1s, 2p, and 4f are possible because they follow the n and l value rules.
• Impossible Combinations: If l equals n or exceeds it, the orbital is not possible. For instance, a 2d orbital would not be possible because l (2) equals n (2).

Knowing about possible and impossible orbitals is key for understanding electron placement and the fundamental structure of atoms.