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Chapter 7: Problem 81

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

Step by step solution

01

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.
02

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)
03

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#
04

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).
05

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}$
06

Identify the possible values of ℓ for given n

The possible values of ℓ for n=5 are 0,1,2,3, and 4.
07

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)
08

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#
09

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.
10

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#
11

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.
12

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

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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. \(n=2 \rightarrow n=1\)

A carbon-oxygen double bond in a certain organic molecule absorbs radiation that has a frequency of \(6.0 \times 10^{13} \mathrm{~s}^{-1}\). a. What is the wavelength of this radiation? b. To what region of the spectrum does this radiation belong? c. What is the energy of this radiation per photon? per mole of photons? d. A carbon-oxygen bond in a different molecule absorbs radiation with frequency equal to \(5.4 \times 10^{13} \mathrm{~s}^{-1} .\) Is this radiation more or less energetic?

For each of the following pairs of elements \(\begin{array}{ll}(\mathrm{Mg} \text { and } \mathrm{K}) & (\mathrm{F} \text { and } \mathrm{Cl})\end{array}\) pick the atom with a. more favorable (exothermic) electron affinity. b. higher ionization energy. c. larger size.

Complete and balance the equations for the following reactions. a. \(\mathrm{Li}(s)+\mathrm{N}_{2}(g) \rightarrow\) b. \(\mathrm{Rb}(s)+\mathrm{S}(s) \rightarrow\)

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