Question

How many unique combinations of the quantum numbers \(l\) and \(m_{l}\) are there when (a) \(n=1,(\mathbf{b}) n=5 ?\)

Short answer

For \(n=1\), there is only 1 unique combination of the quantum numbers \(l\) and \(m_l\), where \(l = 0\) and \(m_l = 0\). For \(n=5\), there are 25 unique combinations of the quantum numbers \(l\) and \(m_l\).

Step-by-step solution

Determine the range of l for each given value of n

For (a) when n = 1, l ranges from 0 to (n-1), which is 0. For (b) when n = 5, l ranges from 0 to (n-1), which is 0, 1, 2, 3, 4.

Calculate the total unique combinations for each l value

(a) For n = 1, there is only one possible l value, which is 0. In this case, m_l can only be 0, giving us just one unique combination: l = 0 and m_l = 0. (b) For n = 5, there are five possible l values: 0, 1, 2, 3, 4. For each of these l values, we will calculate the unique combinations for m_l. - For l = 0, there is only one possible value for m_l, and that is 0. So, for l = 0, we have one unique combination: m_l = 0. - For l = 1, m_l can range from -1 to 1, giving us three unique combinations: m_l = -1, 0, 1. - For l = 2, m_l can range from -2 to 2, giving us five unique combinations: m_l = -2, -1, 0, 1, 2. - For l = 3, m_l can range from -3 to 3, giving us seven unique combinations: m_l = -3, -2, -1, 0, 1, 2, 3. - For l = 4, m_l can range from -4 to 4, giving us nine unique combinations: m_l = -4, -3, -2, -1, 0, 1, 2, 3, 4.

Sum the total unique combinations

Now we need to calculate the sum of all the unique combinations for each l value when n = 5. Total unique combinations for n = 5 = 1 (for l=0) + 3 (for l=1) + 5 (for l=2) + 7 (for l=3) + 9 (for l=4) = 25. So, when n = 1, the total number of unique combinations for l and m_l is 1, and when n = 5, the total number of unique combinations for l and m_l is 25.

Key concepts

Angular Momentum Quantum Number

The angular momentum quantum number, often represented by the letter \( l \), is a crucial part of understanding electron configurations within atoms. It helps describe the shape of the electron's orbit around the nucleus. This quantum number is also called the azimuthal or subsidiary quantum number.

It determines the general form of the orbital and has integral values that range from 0 to \( n-1 \), where \( n \) is the principal quantum number. For instance, if \( n = 5 \), then \( l \) can be 0, 1, 2, 3, or 4.

Each value of \( l \) corresponds to a specific type of orbital:

• \( l = 0 \) corresponds to an s orbital
• \( l = 1 \) corresponds to a p orbital
• \( l = 2 \) corresponds to a d orbital
• \( l = 3 \) corresponds to an f orbital

Understanding \( l \) allows us to gain insight into the energy distribution and the path of an electron's orbit, which is vital for predicting chemical bonding and behavior.

Magnetic Quantum Number

The magnetic quantum number, designated as \( m_{l} \), reveals more about the orientation of an electron's orbital in space relative to an external magnetic field. It is intimately related to the angular momentum quantum number \( l \).

The value of \( m_{l} \) ranges from \( -l \) to \( +l \). This means if \( l = 2 \), \( m_{l} \) can be -2, -1, 0, 1, or 2, encompassing five possible orientations. Each orientation corresponds to a distinct energy level when an external field is applied.

The significance of \( m_{l} \) lies in its ability to split degenerate orbitals (orbitals of the same energy) under the influence of an external magnetic field, a phenomenon called the Zeeman effect. Thus, \( m_{l} \) not only indicates the number of orbitals per \( l \) value but also adds richness to the electron's spatial properties.

Principal Quantum Number

The principal quantum number, represented as \( n \), is fundamental in atomic theory. It denotes the main energy level of an electron in an atom and is typically a positive integer. The larger the value of \( n \), the greater the energy level and the further the electron is from the nucleus.

This quantum number directly influences other quantum numbers, setting limits on their possible values. For instance, \( n \) dictates the range for \( l \) (angular momentum quantum number), where \( l \) can be any integer from 0 to \( n-1 \).

The principal quantum number is critical in determining the electron's potential energy and the atomic shell in which the electron resides. If you think of energy levels as floors in a building, \( n \) tells you which floor you're on. As \( n \) increases, the energy spacing between levels becomes closer, and electrons tend to occupy higher energy levels, affecting chemical reactivity and atomic size.