Tell which of the following combinations of quantum numbers are not allowed. Explain your answers. (a) n = 3, l = 0, ml = -1 (b) n = 3, l = 1, ml = 1 (c) n = 4, l = 4, ml = 0

Verified step by step guidance

1. Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. The four quantum numbers are: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s).
2. The principal quantum number (n) can be any positive integer (1, 2, 3, ...). It indicates the energy level and size of the orbital.
3. The azimuthal quantum number (l) can be any integer from 0 to n-1. It defines the shape of the orbital. For example, if n = 3, then l can be 0, 1, or 2.
4. The magnetic quantum number (m_l) can be any integer from -l to +l, including zero. It describes the orientation of the orbital in space.
5. Let's evaluate each set of quantum numbers: (a) n = 3, l = 0, m_l = -1: Here, l = 0, so m_l can only be 0. Therefore, this combination is not allowed. (b) n = 3, l = 1, m_l = 1: Here, l = 1, so m_l can be -1, 0, or 1. This combination is allowed. (c) n = 4, l = 4, m_l = 0: Here, l cannot be equal to n, so this combination is not allowed.

Key Concepts

Quantum Numbers

Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. There are four quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms). Each quantum number has specific rules governing its possible values, which are essential for determining the allowed states of electrons.

Principal Quantum Number (n) and Azimuthal Quantum Number (l)

The principal quantum number (n) indicates the energy level of an electron and can take positive integer values (1, 2, 3, ...). The azimuthal quantum number (l) defines the shape of the orbital and can take values from 0 to (n-1). Therefore, for a given n, l must always be less than n, which is crucial for determining the validity of the quantum number combinations.

Magnetic Quantum Number (ml)

The magnetic quantum number (ml) specifies the orientation of the orbital in space and can take integer values ranging from -l to +l, including zero. This means that for each value of l, there are (2l + 1) possible values for ml. Understanding these restrictions is vital for identifying which combinations of quantum numbers are permissible.