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

In summary, there are 100 orbitals in total that can have the designation \(5p\), \(3d_{z^2}\), \(4d\), \(n=5\), and \(n=4\) in an atom.

Step-by-step solution

Identify the four quantum numbers

For each designation, we need to identify the four quantum numbers (n, l, m_l, and m_s) involved: - n (Principal Quantum Number): Represents the energy level of the electron and the relative distance of the electron from the nucleus. Allowed integer values are: n = 1, 2, 3, ... - l (Azimuthal Quantum Number): Represents the shape of the orbital and has integer values from 0 to (n-1). 0 = s, 1 = p, 2 = d, 3 = f, ... - m_l (Magnetic Quantum Number): Represents the orientation of the orbital and has integer values from -l to +l. For example, if l = 1, then m_l can be -1, 0, +1. - m_s (Spin Quantum Number): Represents the spin of an electron and has two values: +1/2 (up) and -1/2 (down).

Determine orbitals for each designation

Now we will work with each specified designation: 1. \(5p\): For this designation, n = 5 and l = 1. As a result, m_l can have values -1, 0, and +1. The spin quantum number (m_s) will double the number of orbitals since each orbital can hold 2 electrons (up and down). So, there are 2 x 3 = 6 orbitals in this case. 2. \(3d_{z^2}\): For this designation, n = 3 and l = 2. Since the question specifies the \(d_{z^2}\) orbital, we only need to consider one particular m_l value, which in this case is 0. There are 2 possible spins (up and down) for the electrons in this orbital. Therefore, there are 2 x 1 = 2 orbitals. 3. \(4d\): For this designation, n = 4 and l = 2. For l = 2, there are 5 possible m_l values: -2, -1, 0, +1, and +2. As before, there are 2 possible spins (up and down) for the electrons in each of these orbitals. Therefore, there are 2 x 5 = 10 orbitals in this case. 4. \(n=5\): For this designation, since only n is specified, we need to consider all possible combinations of l and m_l for n = 5. For n = 5, there can be l = 0, 1, 2, 3, 4. For each value of l, we can calculate the number of orbitals as: \[ \begin{aligned} \text{orbitals} & = 2 \cdot (1 + 3 + 5 + 7 + 9) \\ & = 2 \cdot (25) \\ & = 50 \end{aligned} \] 5. \(n=4\): Similarly, for n = 4, we will consider all possible combinations of l and m_l. For n = 4, there can be l = 0, 1, 2, 3. The calculation for the number of orbitals is: \[ \begin{aligned} \text{orbitals} & = 2 \cdot (1 + 3 + 5 + 7) \\ & = 2 \cdot (16) \\ & = 32 \end{aligned} \]

Sum the orbitals for each designation

Now, we can add up the number of orbitals for each designation: Total orbitals = Orbitals for 5p + Orbitals for 3d_{z^2} + Orbitals for 4d + Orbitals for n=5 + Orbitals for n=4 = 6 + 2 + 10 + 50 + 32 = 100 So, there are 100 orbitals in total that can have the designation \(5p\), \(3d_{z^2}\), \(4d\), \(n=5\), and \(n=4\) in an atom.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a key concept in quantum mechanics. It defines the main energy level, or shell, of an electron in an atom. It gives us an idea of the average distance of the electron from the nucleus. The larger the value of \( n \), the higher the energy level and the farther the electron is likely to be from the nucleus.
This number is crucial because it also determines the total number of subshells in a given energy level. For example:

• \( n = 1 \) can only have the \( s \) subshell.
• \( n = 2 \) can have both \( s \) and \( p \) subshells.
• \( n = 3 \) can have \( s \), \( p \), and \( d \) subshells.

The principal quantum number can take any positive integer value such as \( 1, 2, 3, \ldots \). It's essential for predicting the electron configuration and understanding the organization of electrons in multi-electron atoms.

Azimuthal Quantum Number

The azimuthal quantum number, signified by \( l \), tells us the shape of the subshell or the type of orbital that an electron occupies. Each value of \( l \) corresponds to a specific type of subshell. The azimuthal quantum number can have integer values ranging from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number.
Here's a quick breakdown of what each value of \( l \) corresponds to:

• \( l = 0 \): \( s \) subshell
• \( l = 1 \): \( p \) subshell
• \( l = 2 \): \( d \) subshell
• \( l = 3 \): \( f \) subshell

These subshells vary not only in shape but also in complexity, from the simple spherical \( s \) orbital to the more complex \( f \) orbitals. Each type of orbital defines the probability distribution of locating an electron and the capacity for holding electrons in that orbital.

Magnetic Quantum Number

The magnetic quantum number, represented by \( m_l \), describes the orientation of an orbital in space. This comes into play especially when we consider atoms in magnetic fields or when discussing angular momentum. It can take on integer values that range from \(-l\) to \(+l\), inclusive.
For instance, when \( l = 1 \) (a \( p \) subshell), the possible values for \( m_l \) are

• -1
• 0
• +1

Each unique value of \( m_l \) corresponds to a different orientation of the orbital in space. This is why the \( p \) orbital, for example, can orient along the x, y, or z-axis, defined as \( p_x \), \( p_y \), and \( p_z \). The possible number of \( m_l \) values directly influences the number of orbitals available within a given subshell.

Spin Quantum Number

The spin quantum number, denoted by \( m_s \), is perhaps one of the most intriguing quantum numbers because it relates to an electron's intrinsic property — its spin. Unlike the other quantum numbers, \( m_s \) is not derived from solutions to the Schrödinger equation but is a fundamental property of electrons that explains their magnetic moment.
There are only two possible values for the spin quantum number:

• +1/2 (spin-up)
• -1/2 (spin-down)

Each electron in an atom has one of these spins, and this characteristic allows each orbital to accommodate up to two electrons with opposite spins. This is a fundamental principle in quantum mechanics known as the Pauli exclusion principle, which asserts that no two electrons in an atom can have the same set of all four quantum numbers \( n, l, m_l, \) and \( m_s \). The spin is what completes the quantum mechanical description of an electron's state within an atom.