Question

How many unique combinations of the quantum numbers \(l\) and \(m_{l}\) are there when (a) \(n=3,\) (b) \(n=4 ?\)

Short answer

For \(n=3\), there are 9 unique combinations of \(l\) and \(m_l\), and for \(n=4\), there are 16 unique combinations of \(l\) and \(m_l\).

Step-by-step solution

Finding unique combinations for n = 3

1. Identify the possible values of l: For n = 3, the possible values for l = 0, 1, 2. 2. Identify the possible values of m_l for each value of l: - For l = 0: m_l can only be 0 (1 value). - For l = 1: m_l = -1, 0, 1 (3 values). - For l = 2: m_l = -2, -1, 0, 1, 2 (5 values). 3. Calculate the total number of unique combinations: Sum of possible m_l values: 1 + 3 + 5 = 9 unique combinations

Finding unique combinations for n = 4

1. Identify the possible values of l: For n = 4, the possible values for l = 0, 1, 2, 3. 2. Identify the possible values of m_l for each value of l: - For l = 0: m_l can only be 0 (1 value). - For l = 1: m_l = -1, 0, 1 (3 values). - For l = 2: m_l = -2, -1, 0, 1, 2 (5 values). - For l = 3: m_l = -3, -2, -1, 0, 1, 2, 3 (7 values). 3. Calculate the total number of unique combinations: Sum of possible m_l values: 1 + 3 + 5 + 7 = 16 unique combinations Thus, for n = 3, there are 9 unique combinations of l and m_l, and for n = 4, there are 16 unique combinations of l and m_l.

Key concepts

Quantum Number n

Quantum number n, also known as the principal quantum number, plays a pivotal role in determining the energy level of an electron in an atom. It is the first of a set of quantum numbers which describe the unique state of an electron and can take on any positive integer value. The higher the value of n, the greater the energy and the farther away the electron is from the nucleus, residing in a higher energy level or shell. For instance, when n equals 1, the electron is in the closest orbit to the nucleus, which is the lowest energy state.

Quantum Number l

The quantum number l, known as the azimuthal or angular momentum quantum number, defines the shape of an electron's orbital. It can take on any integer value from 0 to n-1, where n is the principal quantum number. The values of l are often associated with specific orbital shapes; for l=0, we have s-orbitals which are spherical, p-orbitals with l=1 are dumbbell-shaped, d-orbitals for l=2 are more complex, and f-orbitals for l=3 are even more intricate. Thus, the possible values of l provide a means of identifying the type of orbital an electron occupies.

Magnetic Quantum Number ml

The magnetic quantum number m_l further refines the description of an electron's position within an orbital by indicating its orientation in space. This quantum number can take on any integer value between -l and +l, including zero. Each value of m_l corresponds to a specific orientation of an orbital, and thus a different magnetic behavior under an external magnetic field. For each value of l, there are 2l +1 possible m_l values, which means the number of allowed orientations exponentially increases with higher l values.

Unique Combinations of Quantum Numbers

The unique combinations of quantum numbers define the unique state an electron can occupy within an atom. These combinations arise from the possible values of n, l, and m_l for each electron. An electron is described by its set of quantum numbers, with no two electrons in the same atom having the same set (in accordance with Pauli's exclusion principle). The step-by-step solution to the given problem illustrates how, for a given value of n, the possible combinations of l and m_l can be determined and summed up to find the total number of unique combinations available for electrons to occupy.

Atomic Orbitals

Atomic orbitals are regions in space around an atom's nucleus where the probability of finding an electron is highest. Orbitals are not fixed paths, but rather fuzzy clouds where electrons are likely to be found. They are defined by the quantum numbers n, l, and m_l, with each set corresponding to a particular orbital shape and orientation. The principal quantum number n determines the shell of the orbital, l its shape, and m_l its orientation. For example, an electron in a p-orbital (l=1) could be found in one of three p-orbitals, each oriented differently in space, as defined by the three possible m_l values (-1, 0, and +1) at this level.