Question

Which of the following represent impossible combinations of \(n\) and \(l\) ? (a) \(1 p\), (b) \(4 s\), (c) \(5 f\), (d) \(2 d\)

Short answer

The impossible combinations are (a) \(1p\) and (d) \(2d\), as the given \(l\) values are outside the allowable range for their respective \(n\) values.

Step-by-step solution

Determine the possible values of \(l\) for each \(n\)

For each given value of \(n\), identify the range of possible values for \(l\). Recall that \(l\) can be any integer value from 0 to \(n-1\).

Check if given \(l\) values are within allowable range

Compare the given \(l\) values for each combination with the possible range of values for that respective \(n\). If the given \(l\) values are within the range of possible values, then the combination is possible; otherwise, it's impossible.

Evaluate each combination

(a) \(1p\): For \(n=1\), \(l\) can only have a value of 0. However, the given \(l\) value is 1, so this combination is impossible. (b) \(4s\): For \(n=4\), \(l\) can have values of 0, 1, 2, or 3. The given \(l\) value is 0, which is within the allowable range, thus this combination is possible. (c) \(5f\): For \(n=5\), \(l\) can have values of 0, 1, 2, 3, or 4. The given \(l\) value is 3, which is within the allowable range, thus this combination is possible. (d) \(2d\): For \(n=2\), \(l\) can have values of 0 or 1. However, the given \(l\) value is 2, so this combination is impossible. So, the impossible combinations are (a) \(1p\) and (d) \(2d\).

Key concepts

Principal Quantum Number (n)

The Principal Quantum Number, represented as \(n\), is one of the four quantum numbers that describe the unique quantum state of an electron in an atom. The value of \(n\) determines the main energy level, or shell, of an electron and influences its average distance from the nucleus.

• \(n\) is always a positive integer: 1, 2, 3, and so on.
• As \(n\) increases, the electron's energy and its average distance from the nucleus increase.
• Each value of \(n\) corresponds to a different electron shell (K, L, M, N... etc.).

Understanding \(n\) is crucial since it sets the stage for the other quantum numbers, especially the azimuthal quantum number which depends on \(n\). In problems dealing with quantum numbers, \(n\) helps determine the range of possible shapes for the electron's orbital and ultimately influences chemical bonding and reactivity.

Azimuthal Quantum Number (l)

The Azimuthal Quantum Number, denoted as \(l\), provides insight into the shape of an electron's orbital. It comes second in the hierarchy after the principal quantum number \(n\) and helps define the angular momentum of an electron's orbit.

• \(l\) can take on any integer value from 0 to \(n-1\).
• Values of \(l\) correspond to particular sublevels or subshells characterized as s (0), p (1), d (2), f (3), and beyond.
• Different \(l\) values indicate different orbital shapes, with each shape contributing differently to the atom's electron cloud.

For any atomic electron, knowing \(l\) is essential to visualizing where the electron might reside relative to the nucleus. This understanding is important when considering the distribution of electrons in chemical reactions and how they affect the atom's physical properties.

Electron Configuration

Electron configuration is a shorthand description of how electrons are distributed in an atom's orbitals, an arrangement dictated by the quantum numbers. How these electrons fill different orbitals can explain a huge array of chemical behavior and periodic trends.

• Electron configurations are written using the principal quantum number and the azimuthal quantum number to determine how electrons fill available subshells, as in 1s, 2p, 3d, etc.
• Pauli Exclusion Principle, Hund's Rule, and Aufbau Principle govern electron filling order.
• The correct representation shows electrons filling the lowest energy orbitals first, in an order given by increasing energy levels.

Mastering electron configurations provides essential insights into an element's phenomenological traits such as magnetic properties, ionization energies, and reaction tendencies. It sketches a clearer picture of atomic behavior across the periodic table.