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Chapter 6: Problem 93

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=3, l=3, m_{l}=2, m_{5}=+\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{\mathrm{s}}=+\frac{1}{2}\) (c) \(n=3, l=1, m_{l}=2, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{\mathrm{s}}=0\) (e) \(n=2, l=1, m_{l}=1, m_{s}=-\frac{1}{2}\)

Short Answer

Expert verified
(a) Not valid, l is invalid. (b) Valid. (c) Not valid, \(m_l\) is invalid. (d) Not valid, \(m_s\) is invalid. (e) Valid.

Step by step solution

01

(a) Check the set for n = 3, l = 3, m_l = 2, m_s = +1/2

For n = 3, l must range from 0 to 2 (0 ≤ l < n). However, l = 3 is given, which is outside this range. Therefore, this set of quantum numbers is not valid due to an incorrect value for `l`.
02

(b) Check the set for n = 4, l = 3, m_l = -3, m_s = +1/2

For n = 4, l must range from 0 to 3 (0 ≤ l < n). Since l = 3, it is valid. Now, for l = 3, m_l must range from -3 to +3 (-3 ≤ m_l ≤ 3). Since m_l = -3, it is also valid. Finally, the spin quantum number m_s can take values of either +1/2 or -1/2, so m_s = +1/2 is valid. Therefore, this set of quantum numbers is valid.
03

(c) Check the set for n = 3, l = 1, m_l = 2, m_s = +1/2

For n = 3, l must range from 0 to 2 (0 ≤ l < n). Since l = 1, it is valid. However, for l = 1, m_l must range from -1 to +1 (-1 ≤ m_l ≤ 1). Since m_l = 2 is given, which is outside this range, this set of quantum numbers is not valid due to an incorrect value for `m_l`.
04

(d) Check the set for n = 5, l = 0, m_l = 0, m_s = 0

For n = 5, l must range from 0 to 4 (0 ≤ l < n). Since l = 0, it is valid. For l = 0, m_l must be equal to 0, so m_l = 0 is also valid. However, the spin quantum number m_s can take values of either +1/2 or -1/2; m_s = 0 is not a valid value. Therefore, this set of quantum numbers is not valid due to an incorrect value for `m_s`.
05

(e) Check the set for n = 2, l = 1, m_l = 1, m_s = -1/2

For n = 2, l must range from 0 to 1 (0 ≤ l < n). Since l = 1, it is valid. For l = 1, m_l must range from -1 to +1 (-1 ≤ m_l ≤ 1). Since m_l = 1, it is also valid. Finally, since m_s = -1/2, it is also valid (as m_s can take values of either +1/2 or -1/2). Therefore, this set of quantum numbers is valid.

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Most popular questions from this chapter

Identify the group of elements that corresponds to each of the following generalized electron configurations and indicate the number of unpaired electrons for each: (a) [noble gas]ns \(^{2} n p^{5}\) (b) \([\) noble gas \(] n s^{2}(n-1) d^{2}\) (c) \([\) noble gas \(] n s^{2}(n-1) d^{10} n p^{1}\) (d) \([\) noble \(\operatorname{gas}] n s^{2}(n-2) f^{6}\)

For each element, indicate the number of valence electrons, core electrons, and unpaired electrons in the ground state: (a) sodium, (b) sulfur, (c) fluorine.

Titanium metal requires light with a maximum wavelength of \(286 \mathrm{nm}\) to emit electrons. (a) What is the minimum energy of the photons necessary to emit electrons from titanium via the photoelectric effect? (b) What is the frequency of this radiation? (c) Is it possible to eject electrons from titanium metal using infrared light? (d) If titanium is irradiated with light of wavelength \(276 \mathrm{nm}\), what is the maximum possible kinetic energy of the emitted electrons?

(a) A green laser pointer emits light with a wavelength of \(532 \mathrm{nm}\). What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of \(532-\mathrm{nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Place the following transitions of the hydrogen atom in order from shortest to longest wavelength of the photon emitted: \(n=5\) to \(n=2, n=4\) to \(n=3, n=8\) to \(n=4,\) and \(n=4\) to \(n=2\).
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