Question

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=4\) b. \(n=5, m_{\ell}=+1\) c. \(n=5, m_{s}=+\frac{1}{2}\) d. \(n=3, \ell=2\) e. \(n=2, \ell=1\)

Short answer

a. Answer: 32 electrons b. Answer: 4 electrons c. Answer: 25 electrons d. Answer: 10 electrons e. Answer: 6 electrons

Step-by-step solution

Identify the possible values of ℓ for given n

Since the azimuthal quantum number (ℓ) can have values ranging from 0 to (n-1), the possible values of ℓ for n=4 are 0,1,2, and 3.

Identify the number of orbitals for each ℓ value

For each value of ℓ, there are (2ℓ + 1) orbitals with magnetic quantum numbers (mℓ) ranging from -ℓ to +ℓ. So, for each ℓ value (0,1,2,3), there are the following numbers of orbitals. 1. ℓ = 0: 1 orbital (s-orbital) 2. ℓ = 1: 3 orbitals (p-orbitals) 3. ℓ = 2: 5 orbitals (d-orbitals) 4. ℓ = 3: 7 orbitals (f-orbitals)

Determine the total number of electrons

As there are two possible spin quantum numbers (ms = +1/2 or -1/2) for each electron, each orbital can accommodate up to 2 electrons. Thus, the total number of electrons in n=4 is given by (1+3+5+7) * 2 = 32 electrons. #a. Answer: 32 electrons #b. n=5, m_{ℓ}=+1#

Identify the possible values of ℓ for given mℓ

Since mℓ ranges from -ℓ to +ℓ, for mℓ = +1, the only possible values of the azimuthal quantum number (ℓ) are 1 (p-orbital) and 2 (d-orbital).

Determine the total number of electrons

As there are two possible values of ms for each electron, each orbital can accommodate up to 2 electrons. Since there is only one mℓ given, there will be one p-orbital and one d-orbital. The total number of electrons is 2 + 2 = 4 electrons. #b. Answer: 4 electrons #c. n=5, m_{s}=+\frac{1}{2}$

Identify the possible values of ℓ for given n

The possible values of ℓ for n=5 are 0,1,2,3, and 4.

Identify the number of orbitals for each ℓ value

For each ℓ value (0,1,2,3,4), there are the following numbers of orbitals: 1. ℓ = 0: 1 orbital (s-orbital) 2. ℓ = 1: 3 orbitals (p-orbitals) 3. ℓ = 2: 5 orbitals (d-orbitals) 4. ℓ = 3: 7 orbitals (f-orbitals) 5. ℓ = 4: 9 orbitals (g-orbitals)

Determine the number of electrons with given ms

Since ms is fixed (+1/2), every orbital has only one electron with this particular ms value. The total number of electrons with ms=+1/2 in n=5 is given by 1+3+5+7+9 = 25 electrons. #c. Answer: 25 electrons #d. n=3, ℓ=2#

Identify the number of orbitals for given ℓ

For ℓ=2 (d-orbitals), there are 5 orbitals with magnetic quantum numbers (mℓ) ranging from -2 to +2.

Determine the total number of electrons

As there are two possible values of ms for each electron, each orbital can accommodate up to 2 electrons. Thus, the total number of electrons in n=3 and ℓ=2 is 5 * 2 = 10 electrons. #d. Answer: 10 electrons #e. n=2, ℓ=1#

Identify the number of orbitals for given ℓ

For ℓ=1 (p-orbitals), there are 3 orbitals with magnetic quantum numbers (mℓ) ranging from -1 to +1.

Determine the total number of electrons

As there are two possible values of ms for each electron, each orbital can accommodate up to 2 electrons. Thus, the total number of electrons in n=2 and ℓ=1 is 3 * 2 = 6 electrons. #e. Answer: 6 electrons

Key concepts

Electron Configuration

In the world of quantum mechanics, understanding how electrons are distributed within an atom is vital. This arrangement is known as electron configuration. An electron configuration describes the arrangement of electrons within the energy levels and orbitals of an atom.

• The main energy levels of electrons are denoted by the principal quantum number, \( n \).
• The electrons are distributed in various orbitals such as s, p, d, and f, indicating their specific energy and location.
• The subshells (s, p, d, f) represent different types of orbitals within a shell.

The configuration guides how electrons fill these orbitals in a specific order, known as the Aufbau principle. According to this principle, electrons fill orbitals starting from the lowest energy level moving to the higher levels. For example, an atom with \( n=4 \) could have electrons distributed in 1 s orbital (2 electrons), 3 p orbitals (6 electrons), 5 d orbitals (10 electrons), and 7 f orbitals (14 electrons), adding up to 32 electrons.

Orbitals

In quantum chemistry, an orbital is a region of space where electrons are likely to be found. Each type of orbital has a unique shape and orientation.

• s-orbitals are spherical in shape with one orientation.
• p-orbitals have a dumbbell shape with three orientations: \( x, y, z \).
• d-orbitals are more complex with five orientations.
• f-orbitals have even more complex shapes with seven orientations.

Each orbital can hold a maximum of two electrons, which are noted by their spin. The existence of different types of orbitals at each energy level allows for the rich diversity of chemical behaviors observed in elements. For example, \( n=3, \ell=2 \) corresponds to d-orbitals, which can host a total of 10 electrons because each of the 5 orbitals can hold 2 electrons.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), arises from the orientation of orbitals in space. It specifies the particular orbital within a subshell where an electron is likely to be found.

• For every value of the azimuthal quantum number, \( \ell \), the magnetic quantum number has values ranging from \(-\ell \) to \(+\ell \).
• It helps determine how many orbitals are available in a subshell. For instance, for \( \ell=1 \), the values of \( m_\ell \) are -1, 0, +1, indicating three possible p-orbitals.

The magnetic quantum number does not impact the energy of the orbital unless in the presence of an external magnetic field. In an atom with \( n=5, m_\ell=+1 \), there could be contributions from both p and d orbitals, each accommodating distinct electron numbers.

Azimuthal Quantum Number

The azimuthal quantum number, \( \ell \), is crucial for understanding electron configuration. It describes the shape of the orbital and is also known as the angular momentum quantum number.

• It takes on values from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• Each value of \( \ell \) corresponds to a particular subshell: \( \ell=0 \) for s, \( \ell=1 \) for p, \( \ell=2 \) for d, and \( \ell=3 \) for f.

This quantum number significantly impacts the atom's chemical properties as it determines the shape and number of orbitals within a shell. For example, when \( n=2, \ell=1 \), it means that electrons occupy p-orbitals that can hold up to 6 electrons (in 3 orbitals). Understanding \( \ell \) helps predict electron distribution across the orbitals of an atom.