Question

Determine the maximum number of electrons that can have each of the following designations: \(2 f, 2 d_{x y}, 3 p, 5 d_{y z},\) and \(4 p\)

Short answer

The maximum number of electrons for each designation are: 2f is an impossible configuration; 2d_xy can have 2 electrons; 3p can have 6 electrons; 5d_yz can have 2 electrons; and 4p can have 6 electrons.

Step-by-step solution

1. Recall rules for electron configuration and quantum numbers

Recall that electrons occupy atomic orbitals based on the following quantum numbers: - Principal quantum number (n): Denotes the energy level or shell of the electron (positive integer values starting from 1) - Azimuthal quantum number (l): Describes the shape of the orbital, from 0 to n-1 - Magnetic quantum number (m_l): Describes the orientation of the orbital in space, ranges from -l to +l, including 0. Additionally, the maximum number of electrons that can have the same designation is determined by the number of m_l values available for each orbital type. The maximum number of electrons in an orbital is given by the formula 2(2l + 1), where l is the azimuthal quantum number.

2. Determine the l values for each designation

For each of the electron designations, find the orbital type (l value) corresponding to the given electron configuration: - 2f: n=2, f implies l=3 (This is an impossible configuration since the value of l should be between 0 and n-1) - 2d_xy: n=2, d implies l=2 - 3p: n=3, p implies l=1 - 5d_yz: n=5, d implies l=2 - 4p: n=4, p implies l=1 With these l values, we can now determine the maximum number of electrons for each designation.

3. Calculate the maximum number of electrons for each designation

Using the above formula 2(2l + 1) and the respective l values, calculate the maximum number of electrons for each designation: - 2f: l=3, the configuration is impossible - 2d_xy: l=2, the orbital can accommodate 2(2*2 + 1) = 10 electrons, but since it is specified as d_xy (m_l = 0), it can only have 2 electrons (spin up and spin down) - 3p: l=1, the orbital can accommodate 2(2*1 + 1) = 6 electrons - 5d_yz: l=2, the orbital can accommodate 2(2*2 + 1) = 10 electrons, but since it is specified as d_yz (m_l = 1), it can only have 2 electrons (spin up and spin down) - 4p: l=1, the orbital can accommodate 2(2*1 + 1) = 6 electrons

4. Summarize the result

From the steps above, we have found the maximum number of electrons that can have each given designation: - 2f: impossible configuration - 2d_xy: 2 electrons - 3p: 6 electrons - 5d_yz: 2 electrons - 4p: 6 electrons

Key concepts

Quantum Numbers

Quantum numbers are essential for defining the position and energy of an electron within an atom. These numbers provide a set of rules that help describe each electron's characteristics in an atom's electron cloud. There are four key quantum numbers you need to understand:

• Principal Quantum Number (n): This number indicates the energy level or shell where the electron resides, starting from 1 and increasing in whole numbers.
• Azimuthal Quantum Number (l): Also known as the angular momentum quantum number, this describes the shape of the electron's orbital and can range from 0 to (n-1).
• Magnetic Quantum Number (m_l): This specifies the orientation in space of the orbital. Its values range from -l to +l, including 0.
• Spin Quantum Number (m_s): This denotes the electron's spin direction, with possible values of +1/2 or -1/2.

These quantum numbers collectively ensure that no two electrons in an atom have the same set of quantum numbers, following the Pauli exclusion principle, which is key to understanding electron configuration.

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. The shape and size of these orbitals are determined by the azimuthal quantum number (l). Here are the common orbital types:

• s-orbital (l=0): Spherical in shape, each s-type can hold up to 2 electrons.
• p-orbital (l=1): Shaped like dumbbells, there are three p orbitals per energy level, each oriented differently in space, and collectively they can hold up to 6 electrons.
• d-orbital (l=2): More complex in shape, five d orbitals in each energy level can hold up to 10 electrons together.
• f-orbital (l=3): The most complex, with seven forms found at higher energy levels, and up to 14 electrons can be accommodated.

Understanding atomic orbitals and their configurations is crucial as it determines how atoms interact in chemical reactions and how they form bonds with one another.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized by (l), is fundamental in determining the shape of the electron cloud or orbital. Each electron orbital's shape, ranging from simple spheres to complex lobes, is defined by this number. Here's a clearer look at what it represents:

• ​l = 0 (s-orbital): Orbital is spherical. Only one orientation is possible, explaining its spherical shape.
• ​l = 1 (p-orbital): Orbital is dumbbell-shaped. There are three possible orientations, as seen in the p_x, p_y, and p_z orbitals.
• l = 2 (d-orbital): Features clover-shaped orbitals with five orientations in space, accounting for the differing positions like d_xy, d_yz, etc.
• l = 3 (f-orbital): Exhibits even more complexity with seven orientations, appearing at higher energy levels.

The azimuthal quantum number is vital for predicting how many electrons an orbital can hold and how they are distributed in an atom, helping define the structure and reactivity of different elements.