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Chapter 2: Problem 142

Although no currently known elements contain electrons in \(g\) orbitals in the ground state, it is possible that these elements will be found or that electrons in excited states of known elements could be in \(g\) orbitals. For \(g\) orbitals, the value of \(\ell\) is 4 What is the lowest value of \(n\) for which \(g\) orbitals could exist? What are the possible values of \(m_{\ell} ?\) How many electrons could a set of \(g\) orbitals hold?

Short Answer

Expert verified
The lowest value of \(n\) for which \(g\) orbitals can exist is 5. The possible values of \(m_{l}\) 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

01

Understanding quantum numbers

Principal quantum number (n) determines the energy level of an electron in an atom. It can have positive integer values (n = 1, 2, 3, ...). The angular momentum quantum number (l) signifies the shape of the orbital and ranges from 0 to n-1. This means that for a specific energy level (n), there can be n different values of l (0, 1, 2, ... n-1). Finally, the magnetic quantum number (m_l) determines the orientation of the orbital in space and takes integer values from -l to +l, including 0. This means that for a given value of l, there will be 2l+1 possible values of m_l.
02

Finding the lowest value of n for g orbitals

We are given that for g orbitals, the value of l is 4. As we know that l ranges from 0 to n-1, we can find the lowest value of n for which g orbitals can exist by setting l equal to n-1: \[ l = n - 1 \\ 4 = n - 1 \\ n = 5 \] So, the lowest value of n for which g orbitals can exist is 5.
03

Finding the possible values of m_l

The magnetic quantum number (m_l) ranges from -l to +l, including 0. Since l is 4 for g orbitals, the possible values of m_l are: \[ m_{l} = -4, -3, -2, -1, 0, 1, 2, 3, 4 \]
04

Calculating the number of electrons a set of g orbitals can hold

We know that each orbital can hold a maximum of 2 electrons (due to the Pauli Exclusion Principle). As there are 2l+1 orbitals for a given value of l, we can find the number of electrons a set of g orbitals can hold with l=4 by using this formula: \[ \text{Number of electrons} = 2(2l + 1) = 2(2(4) + 1) = 18 \] So, a set of g orbitals can hold a maximum of 18 electrons.

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Most popular questions from this chapter

Which of the following electron configurations correspond to an excited state? Identify the atoms and write the ground-state electron configuration where appropriate. a. \(1 s^{2} 2 s^{2} 3 p^{1}\) b. \(1 s^{2} 2 s^{2} 2 p^{6}\) c. \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\) d. \([\mathrm{Ar}] 4 s^{2} 3 d^{5} 4 p^{1}\) How many unpaired electrons are present in each of these species?

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