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Chapter 7: Problem 76

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. \(n=1, \ell=0, m_{\ell}=0\)

Short Answer

Expert verified
a. 0 electrons b. 1 electron c. 9 electrons d. 0 electrons e. 2 electrons

Step by step solution

01

a. \(n=0\), \(\ell=0\), \(m_\ell=0\)

The principal quantum number n must be a positive integer (n > 0), but here n is equal to 0. Therefore, this quantum number set is not valid, and there are no electrons associated with these quantum numbers.
02

b. \(n=2\), \(\ell=1\), \(m_\ell=-1\), \(m_s=-\frac{1}{2}\)

These quantum numbers are valid according to the rules. Since the spin quantum number (m_s) is given and uniquely defines an electron, there can be only one electron with these quantum numbers in an atom.
03

c. \(n=3\), \(m_s=+\frac{1}{2}\)

In this case, the angular momentum quantum number (\(\ell\)) and magnetic quantum number (\(m_\ell\)) are not provided. Based on the rules, for a given principal quantum number n, there can be \(n^2\) orbitals and 2 electrons per orbital (one for each spin). For n=3, there are \(3^2 = 9\) orbitals. Since \(m_s\) is specified, each orbital can accommodate only one electron with this spin. Therefore, the maximum number of electrons with these quantum numbers is 9.
04

d. \(n=2\), \(\ell=2\)

The angular momentum quantum number (\(\ell\)) has a maximum value of n-1. In this case, \(\ell\) is equal to n, which is not valid. Thus, there are no electrons associated with these quantum numbers.
05

e. \(n=1\), \(\ell=0\), \(m_\ell=0\)

These quantum numbers are valid according to the rules. Since the magnetic quantum number (m_ell) and the angular momentum quantum number (l) are both given and zero, there is only one orbital associated with these quantum numbers. Each orbital can accommodate 2 electrons (one with spin +1/2 and one with spin -1/2). Therefore, the maximum number of electrons with these quantum numbers is 2.

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Most popular questions from this chapter

The four most abundant elements by mass in the human body are oxygen, carbon, hydrogen, and nitrogen. These four elements make up about \(96 \%\) of the human body. The next four most abundant elements are calcium, phosphorus, magnesium, and potassium. Write the expected ground-state electron configurations for these eight most abundant elements in the human body.

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 forbitals be able to hold?

Order the atoms in each of the following sets from the least exothermic electron affinity to the most. a. \(\mathrm{N}, \mathrm{O}, \mathrm{F}\) b. \(\mathrm{Al}, \mathrm{Si}, \mathrm{P}\)

As the weapons officer aboard the Starship Chemistry, it is your duty to configure a photon torpedo to remove an electron from the outer hull of an enemy vessel. You know that the work function (the binding energy of the electron) of the hull of the enemy ship is \(7.52 \times 10^{-19} \mathrm{~J}\). a. What wavelength does your photon torpedo need to be to eject an electron? b. You find an extra photon torpedo with a wavelength of 259 \(\mathrm{nm}\) and fire it at the enemy vessel. Does this photon torpedo do any damage to the ship (does it eject an electron)? c. If the hull of the enemy vessel is made of the element with an electron configuration of \([\mathrm{Ar}] 4 s^{1} 3 d^{10}\), what metal is this?

An electron is excited from the \(n=1\) ground state to the \(n=3\) state in a hydrogen atom. Which of the following statements are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\). d. The wavelength of light emitted when the electron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\). e. For \(n=3\), the electron is in the first excited state.
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