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Chapter 7: Problem 75

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

Expert verified
a. 32 electrons b. 2 electrons c. 25 electrons d. 10 electrons e. 6 electrons

Step by step solution

01

Calculate total number of orbitals

For n = 4, we can have: ℓ = 0, 1, 2, 3 For ℓ = 0, m_ℓ = 0, m_s = +1/2, -1/2 (one s-orbital) For ℓ = 1, m_ℓ = -1, 0, 1, m_s = +1/2, -1/2 (three p-orbitals) For ℓ = 2, m_ℓ = -2, -1, 0, 1, 2, m_s = +1/2, -1/2 (five d-orbitals) For ℓ = 3, m_ℓ = -3, -2, -1, 0, 1, 2, 3, m_s = +1/2, -1/2 (seven f-orbitals)
02

Count maximum electrons

Each orbital can accommodate two electrons (m_s = +1/2 and -1/2). So: 1s: 2 electrons 3p: 6 electrons 5d: 10 electrons 7f: 14 electrons Total number of electrons = 2 + 6 + 10 + 14 = 32 electrons #b. n = 5, m_ℓ = +1#
03

Determine possible values of ℓ and m_s

Given m_ℓ = 1, ℓ can only be 1 (p-orbital), as values for m_ℓ are from -ℓ to +ℓ. m_s has two possible values: +1/2 and -1/2
04

Count maximum electrons

As ℓ = 1, it means that we are referring to a p-orbital. A p-orbital can hold a maximum of 2 electrons with different spin quantum numbers (m_s = +1/2 and -1/2). Total number of electrons = 2 electrons #c. n = 5, m_s = +1/2#
05

Determine total number of orbitals

For n = 5, we can have: ℓ = 0, 1, 2, 3, 4 Since m_s value is constant, all orbitals can accommodate only one electron with a given m_s value.
06

Count maximum electrons

For each value of ℓ, we can count the number of electrons: ℓ = 0 (s): 1 electron ℓ = 1 (p): 3 electrons ℓ = 2 (d): 5 electrons ℓ = 3 (f): 7 electrons ℓ = 4 (g): 9 electrons Total number of electrons: 1 + 3 + 5 + 7 + 9 = 25 electrons #d. n = 3, ℓ = 2#
07

Determine possible values of m_ℓ and m_s

Since ℓ = 2, we'll have m_ℓ values from -2 to 2. m_ℓ = -2, -1, 0, 1, 2 m_s has two possible values: +1/2 and -1/2
08

Count maximum electrons

As ℓ = 2, it means that we are referring to a d-orbital. A d-orbital can hold a maximum of 10 electrons (2 electrons for each m_ℓ value). Total number of electrons = 10 electrons #e. n = 2, ℓ = 1#
09

Determine possible values of m_ℓ and m_s

Since ℓ = 1, we'll have m_ℓ values from -1 to 1. m_ℓ = -1, 0, 1 m_s has two possible values: +1/2 and -1/2
10

Count maximum electrons

As ℓ = 1, it means that we are referring to a p-orbital. A p-orbital can hold a maximum of 6 electrons (2 electrons for each m_ℓ value). Total number of electrons = 6 electrons

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Most popular questions from this chapter

Calculate the wavelength of light emitted when each of the following transitions occur in the hydrogen atom. What type of electromagnetic radiation is emitted in each transition? a. \(n=3 \rightarrow n=2\) b. \(n=4 \rightarrow n=2\) c. \(n=2 \rightarrow n=1\)

Write the expected ground-state electron configuration for each of the following. a. the lightest halogen atom b. the alkali metal with only \(2 p\) and \(3 p\) electrons c. the Group \(3 \mathrm{~A}\) element in the same period as \(\mathrm{Sn}\) d. the nonmetallic elements in Group \(4 \mathrm{~A}\)

For each of the following pairs of elements \((\mathrm{C}\) and \(\mathrm{N}) \quad(\mathrm{Ar}\) and \(\mathrm{Br})\) pick the atom with a. more favorable (exothermic) electron affinity. b. higher ionization energy. c. larger size.

Assume that a hydrogen atom's electron has been excited to the \(n=5\) level. How many different wavelengths of light can be emitted as this excited atom loses energy?

A photon of ultraviolet (UV) light possesses enough energy to mutate a strand of human DNA. What is the energy of a single UV photon and a mole of UV photons having a wavelength of \(25 \mathrm{~nm} ?\)
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