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Chapter 2: 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. Maximum number of electrons: 32 b. Maximum number of electrons: 4 c. Maximum number of electrons: 25 d. Maximum number of electrons: 10 e. Maximum number of electrons: 6

Step by step solution

01

Find the possible values of _ell

Since the principal quantum number n=4, the possible values of _ell are 0, 1, 2, and 3.
02

Find the possible values of m_ell for each _ell

For each value of _ell, the possible values of m_ell are as follows: - _ell = 0: m_ell = 0 - _ell = 1: m_ell = -1, 0, +1 - _ell = 2: m_ell = -2, -1, 0, +1, +2 - _ell = 3: m_ell = -3, -2, -1, 0, +1, +2, +3
03

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have: - _ell = 0: 2 electrons - _ell = 1: 6 electrons - _ell = 2: 10 electrons - _ell = 3: 14 electrons So, for n=4, the total number of electrons is \(2 + 6 + 10 + 14 = 32\). #a. Maximum number of electrons: 32# #b. n=5, m_ell=+1#
04

Find the possible values of _ell for m_ell=+1

Possible values of _ell for m_ell=+1 are _ell=1 and _ell=2.
05

Find the total number of electrons for each _ell

For each value of m_ell, there are two possible values of m_s: - _ell = 1: 2 electrons - _ell = 2: 2 electrons So, for n=5 and m_ell=+1, the total number of electrons is \(2 + 2 = 4\). #b. Maximum number of electrons: 4# #c. n=5, m_s=+1/2#
06

Calculate possible values of _ell and m_ell

For n=5, possible values of _ell are 0, 1, 2, 3, 4, and for each _ell, we can have their respective m_ell values.
07

Find the total number of electrons

For all of these cases, m_s=+1/2, so there will be only 1 electron for each of m_ell. Counting the possible combinations, we have a total of: 1 + 3 + 5 + 7 +9 = 25 electrons. #c. Maximum number of electrons: 25# #d. n=3, _ell=2#
08

Determine the possible values of m_ell

For _ell=2, the possible values of m_ell are: -2, -1, 0, +1, +2.
09

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have 5 × 2 = 10 electrons. #d. Maximum number of electrons: 10# #e. n=2, _ell=1#
10

Determine the possible values of m_ell

For _ell=1, the possible values of m_ell are: -1, 0, +1.
11

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have 3 × 2 = 6 electrons. #e. Maximum number of electrons: 6#

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Most popular questions from this chapter

It takes \(476 \mathrm{kJ}\) to remove 1 mole of electrons from the atoms at the surface of a solid metal. How much energy (in kJ) does it take to remove a single electron from an atom at the surface of this solid metal?

Identify the following three elements. a. The ground-state electron configuration is \([\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\). b. The ground-state electron configuration is \([\mathrm{Ar}] 4 s^{2} 3 d^{10} 4 p^{2}\). c. An excited state of this element has the electron configuration \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\).

Answer the following questions, assuming that \(m_{s}\) could have three values rather than two and that the rules for \(n, \ell,\) and \(m_{\ell}\) are the normal ones. a. How many electrons would an orbital be able to hold? b. How many elements would the first and second periods in the periodic table contain? c. How many elements would be contained in the first transition metal series? d. How many electrons would the set of \(4 f\) orbitals be able to hold?

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=0, \ell=0, m_{\ell}=0\) b. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-\frac{1}{2}\) c. \(n=3, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=2\) e. \(n=1, \ell=0, m_{\ell}=0\)

Which of the following statements is(are) true? a. The \(2 s\) orbital in the hydrogen atom is larger than the \(3 s\) orbital also in the hydrogen atom. b. The Bohr model of the hydrogen atom has been found to be incorrect. c. The hydrogen atom has quantized energy levels. d. An orbital is the same as a Bohr orbit. e. The third energy level has three sublevels, the \(s, p,\) and \(d\) sublevels.
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