Question

How many orbitals in an atom can have the designation \(5 p\), \(3 d_{z^{2}}, 4 d, n=5, n=4 ?\)

Short answer

The total number of orbitals for the given designations is 50, calculated as follows: 3 (5p) + 1 (3d_{z^2}) + 5 (4d) + 25 (n=5) + 16 (n=4).

Step-by-step solution

1. Find the number of 5p orbitals

In the 5p designation, the principal quantum number n = 5 and the angular momentum quantum number l = 1 (for p orbitals). The magnetic quantum number (m_l) can have values in the range of -l to +l, in this case, from -1 to +1. Thus, for 5p orbitals, there are three possible values of m_l: -1, 0, and +1. This means there are three 5p orbitals.

2. Find the number of 3dz^2 orbitals

For 3d_{z^2}, the principal quantum number n = 3 and the angular momentum quantum number l = 2 (for d orbitals). Since the magnetic quantum number (m_l) is specifically designated as being z^2, that means there's just one orbital corresponding to this case, the dz^2 orbital.

3. Find the number of 4d orbitals

In the case of 4d, the principal quantum number n = 4 and the angular momentum quantum number l = 2 (for d orbitals). The magnetic quantum number (m_l) can have values in the range of -l to +l, in this case, from -2 to +2. Thus, for 4d orbitals, there are five possible values of m_l: -2, -1, 0, +1, and +2. This means there are five 4d orbitals.

4. Find the number of orbitals for n=5

For n = 5, the allowed values of the angular momentum quantum number l are 0, 1, 2, 3, and 4. For each value of l, there will be 2l+1 possible values for m_l. To find the total number of orbitals for n=5, we can sum the number of orbitals for each value of l. Orbitals for n = 5 = (2×0+1) + (2×1+1) + (2×2+1) + (2×3+1) + (2×4+1) = 1+3+5+7+9 = 25

5. Find the number of orbitals for n=4

For n = 4, the angular momentum quantum number l can have values from 0 to 3. Following the same formula as for n=5, we calculate the total number of orbitals for n=4: Orbitals for n = 4 = (2×0+1) + (2×1+1) + (2×2+1) + (2×3+1) = 1+3+5+7 = 16

6. Calculate the total number of orbitals

Now we can combine the number of orbitals for each designation to find the total number of orbitals for the given exercise: Total orbitals = 3 (5p) + 1 (3d_{z^2}) + 5 (4d) + 25 (n=5) + 16 (n=4) = 50 So, there are 50 orbitals in total for the given designations.

Key concepts

Orbitals

Orbitals are regions in an atom where there is a high probability of finding electrons. They have unique sizes, shapes, and orientations in space, which are described by quantum numbers. It is important to remember that each orbital can hold up to two electrons with opposite spins.
Different types of orbitals include:

• **s orbitals** – spherical in shape and found in each energy level.
• **p orbitals** – dumbbell-shaped and appear in sets of three (px, py, pz) in energy levels n=2 and higher.
• **d orbitals** – more complex shapes and appear in sets of five (such as dyz, dzx, dxy, dx^2-y^2, and dz^2) in energy levels n=3 and higher.
• **f orbitals** – even more complex shapes and appear in sets of seven in energy levels n=4 and higher.

These orbitals define the electron configuration of elements and affect chemical bonding and interactions.

Principal Quantum Number

The principal quantum number, denoted by "n," is crucial as it determines the energy level of an electron in an atom. It takes positive integer values (n=1, 2, 3,...), where a higher "n" value generally means the electron is further from the nucleus and has more energy.
The principal quantum number also defines the size of the atomic orbital; a higher "n" value means a larger orbital. As a result, the principal quantum number plays a significant role in determining the overall energy and location of an electron within an atom. Energy levels, or shells, associated with the principal quantum number help indicate where electrons reside and how easily they can interact with electrons from other atoms in chemical reactions.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized by "l," describes the shape of an orbital. It can take integer values from 0 to n-1. Each value of "l" corresponds to a particular type of orbital:

• l=0 represents s orbitals
• l=1 represents p orbitals
• l=2 represents d orbitals
• l=3 represents f orbitals

The shape of the orbital influences how electrons are distributed around the nucleus and can impact an atom's chemical characteristics. For example, spherical s orbitals allow electrons to be evenly distributed, while the more complex shapes of p, d, and f orbitals can create regions of high electron density, influencing the way atoms interact in bonding.

Magnetic Quantum Number

The magnetic quantum number, represented by "m_l," specifies the orientation of an orbital within a subshell. It can have integer values from -l to +l, where "l" is the angular momentum quantum number.
For instance, if l=1 (for p orbitals), m_l can take values of -1, 0, and +1, corresponding to the orientations px, py, and pz, respectively. Similarly, for d orbitals (where l=2), m_l can be -2, -1, 0, +1, and +2.
The different orientations reflect the spatial distribution of electron clouds within the atom, which affects how atoms align and interact with neighboring atoms during chemical reactions. It's essential for determining how electrons are arranged in an atom and affects the atom's magnetic properties.