Question

No currently known elements contain electrons in g \((\ell=4)\) orbitals in the ground state. If an element is discovered that has electrons in the g orbital, what is the lowest value for \(\mathbf{n}\) in which these \(\mathrm{g}\) orbitals could exist? What are the possible values of \(\mathbf{m}_{\ell} ?\) How many electrons could a set of \(\mathrm{g}\) orbitals hold?

Short answer

Answer: The lowest value of the principal quantum number (n) where g orbitals can exist is 5. The possible values of the magnetic quantum number (mₗ) for g orbitals are -4, -3, -2, -1, 0, 1, 2, 3, and 4. A set of g orbitals can hold a maximum of 18 electrons.

Step-by-step solution

Identify the relationship between quantum numbers for g orbitals

For an electron to occupy a g (\(\ell = 4\)) orbital, we know that the value of the orbital angular momentum quantum number (\(\ell\)) must be 4. Next, we will find the minimum value for the principal quantum number (n) in which these g orbitals could exist.

Find the minimum value of the principal quantum number (n)

The relationship between the principal quantum number (n) and the orbital angular momentum quantum number (\(\ell\)) is such that \(n > \ell\). Therefore, the minimum value for n in which g orbitals could exist is when \(n = 5\).

Find the possible values of the magnetic quantum number (\(\mathbf{m}_{\ell}\))

The magnetic quantum number (\(\mathbf{m}_{\ell}\)) values range from \(-\ell\) to \(+\ell\). For g orbitals where \(\ell = 4\), the possible values for \(\mathbf{m}_{\ell}\) are: $$\mathbf{m}_{\ell} = -4, -3, -2, -1, 0, 1, 2, 3, 4$$

Calculate the maximum number of electrons a set of g orbitals can hold

To find the maximum number of electrons a set of g orbitals can hold, we need to know the number of orbitals in a set of g orbitals and the maximum number of electrons per orbital. The number of orbitals for a given \(\ell\) value is \(2\ell + 1\). For g orbitals (\(\ell = 4\)), this results in: $$\text{Number of orbitals} = 2(4) + 1 = 9$$ Each orbital can hold a maximum of 2 electrons due to the electron spin quantum number (s) which can take on two values (+1/2 and -1/2). Therefore, the maximum number of electrons a set of g orbitals can hold is: $$\text{Max number of electrons} = 9 \times 2 = 18$$ In summary, the lowest value for \(\mathbf{n}\), where g orbitals can exist is when \(n = 5\), the possible values for \(\mathbf{m}_{\ell}\) are -4, -3, -2, -1, 0, 1, 2, 3, and 4, and a set of g orbitals can hold a maximum of 18 electrons.