Question

What is the maximum number of electrons that can be identified with each of the following sets of quantum numbers? In some cases, the answer may be "none" In such cases, explain why "none" is the correct answer. (a) \(n=4, \ell=3, m_{\ell}=1\) (b) \(n=6, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=-1 / 2\) (c) \(n=3, \ell=3, m_{c}=-3\)

Short answer

(a) 2 electrons, (b) 1 electron, (c) none.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers define the state of an electron in an atom. The principal quantum number, \(n\), describes the energy level of the electron. The azimuthal quantum number, \(\ell\), defines the subshell: 0 for s, 1 for p, 2 for d, and 3 for f. The magnetic quantum number, \(m_\ell\), specifies the orientation of the orbital within its subshell, and can take values from \(-\ell\) to \(\ell\). The spin quantum number, \(m_s\), specifies the spin of the electron and can be \(+1/2\) or \(-1/2\).

Evaluating Condition Set (a)

For set (a) \(n=4, \ell=3, m_{\ell}=1\): The quantum numbers are beneficial since \(\ell = 3\) corresponds to an 'f' subshell, and \(m_{\ell} = 1\) is within the range of valid values \(-3, -2, -1, 0, 1, 2, 3\) for an 'f' orbital. Since no \(m_s\) is given, we consider two possible electrons matching this set for two spins: \(+1/2\) and \(-1/2\). Thus, there can be 2 electrons.

Evaluating Condition Set (b)

For set (b) \(n=6, \ell=1, m_{\ell}=-1, m_s=-1/2\): The quantum numbers are valid. \(\ell=1\) indicates a 'p' subshell, whose \(m_{\ell}\) values range from \(-1\) to \(+1\). Here, \(m_{\ell}=-1\) and \(m_s=-1/2\) precisely describe one electron in these conditions, leading to a conclusion of one electron possible here.

Evaluating Condition Set (c)

For set (c) \(n=3, \ell=3, m_{\ell}=-3\): This set is invalid. With \(n=3\), the possible values for \(\ell\) are 0, 1, or 2. Thus, \(\ell=3\) is not possible in the principal energy level 3. Therefore, no electrons can have this set of quantum numbers.

Key concepts

Energy Levels

In atomic physics, energy levels refer to the different layers within an atom where electrons can be found. Each energy level is represented by the principal quantum number, denoted as \(n\). This number indicates how far an electron is from the nucleus. The larger the value of \(n\), the higher the energy level and the further the electron is from the nucleus.

It's like climbing a staircase: the higher the step, the further you are from the ground.

• \(n = 1\) is the closest energy level to the nucleus.
• \(n = 2\), \(n = 3\), and so forth, are progressively farther.

This is crucial for understanding electron configuration and how electrons fill available energy levels in an atom.

Subshells

Within each energy level, electrons occupy smaller regions called subshells. These subshells are determined by the azimuthal quantum number, \(\ell\), and they provide details about the shape of the electron clouds. The value of \(\ell\) determines which type of subshell is present. Each type has distinctive properties.

• \(\ell = 0\) refers to an 's' subshell.
• \(\ell = 1\) indicates a 'p' subshell.
• \(\ell = 2\) means a 'd' subshell.
• \(\ell = 3\) stands for an 'f' subshell.

These subshells can hold different numbers of electrons, and the subshells determine the orbital shape and energy distinctions within a given energy level.

Electron Configuration

Electron configuration is like the map that shows where all the electrons are located in an atom. It describes how electrons are distributed among the different energy levels and subshells. This distribution may seem complex, but it follows a specific order so that electrons fill the lowest available energy level first, known as the Aufbau principle.

For instance:

• Electrons fill the 's' subshell before moving to the 'p', 'd', or 'f' subshells at the same energy level.
• Within the configuration, numbers and letters (e.g., \(1s^2\)) show which subshells are filled and how many electrons are in each.

This arrangement tells chemists about the atom's stability and reactivity.

Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_\ell\), explains the specific orientation of an electron's orbital within its subshell. This defines how an orbital is positioned in space relative to the other orbitals.

It takes on integer values ranging from \(-\ell\) to \(\ell\). Therefore, an 'f' subshell (where \(\ell = 3\)) has magnetic quantum numbers from \(-3\) to \(3\). These orientations describe the distinct paths electrons take around the nucleus, helping us map out how each electron moves.

This is an essential part of quantum mechanics that allows us to predict how atoms will behave in magnetic fields and interact with other particles.