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Chapter 2: Problem 98

Give a possible set of values of the four quantum numbers for the \(4 s\) and \(3 d\) electrons in titanium.

Short Answer

Expert verified
For the 4s electrons in titanium, the possible sets of quantum numbers are (n = 4, l = 0, m_l = 0, m_s = +1/2) and (n = 4, l = 0, m_l = 0, m_s = -1/2). For the 3d electrons, the possible sets of quantum numbers are: n = 3, l = 2, and m_l ranges from -2 to 2 with m_s being either +1/2 or -1/2, yielding ten possible sets of values in total.

Step by step solution

01

Identify the electronic configuration

To determine the set of quantum numbers for the 4s and 3d electrons, we first need to find the electronic configuration of titanium. Titanium (Ti) has an atomic number of 22. Therefore, it has 22 electrons. The electronic configuration for titanium is: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d² Now, we'll focus on the 4s and 3d electrons.
02

Determine the principal quantum number (n)

The principal quantum number (n) is directly related to the energy level of the electron. In this case, the 4s electrons have n = 4, while the 3d electrons have n = 3.
03

Determine the azimuthal quantum number (l)

The azimuthal quantum number (l) is related to the shape of the orbital. The values for l depend on the value of n and range from 0 to (n-1). Here, we already know the orbitals: s and d. The s orbital corresponds to l = 0, and the d orbital corresponds to l = 2. So for 4s electrons, l = 0, and for 3d electrons, l = 2.
04

Determine the magnetic quantum number (m_l)

The magnetic quantum number (m_l) describes the orientation of the orbital. The values for m_l depend on the value of l and range from -l to +l, including 0. For 4s electrons (l = 0), there is only one possible value for m_l: 0. For 3d electrons (l = 2), there are five possible values for m_l: -2, -1, 0, 1, and 2.
05

Determine the spin quantum number (m_s)

The spin quantum number (m_s) describes the intrinsic angular momentum of the electron. There are two possible values for m_s: +1/2 and -1/2, representing spin-up and spin-down electrons, respectively.
06

Find the possible sets of quantum numbers

Now, we can combine the information to find the possible sets of quantum numbers for the 4s and 3d electrons in titanium. For 4s electrons: 1. n = 4, l = 0, m_l = 0, m_s = +1/2 2. n = 4, l = 0, m_l = 0, m_s = -1/2 For 3d electrons: 1. n = 3, l = 2, m_l = -2, m_s = +1/2 2. n = 3, l = 2, m_l = -2, m_s = -1/2 3. n = 3, l = 2, m_l = -1, m_s = +1/2 4. n = 3, l = 2, m_l = -1, m_s = -1/2 5. n = 3, l = 2, m_l = 0, m_s = +1/2 6. n = 3, l = 2, m_l = 0, m_s = -1/2 7. n = 3, l = 2, m_l = 1, m_s = +1/2 8. n = 3, l = 2, m_l = 1, m_s = -1/2 9. n = 3, l = 2, m_l = 2, m_s = +1/2 10. n = 3, l = 2, m_l = 2, m_s = -1/2 These are the possible sets of values for the four quantum numbers for the 4s and 3d electrons in titanium.

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Most popular questions from this chapter

One bit of evidence that the quantum mechanical model is "correct" lies in the magnetic properties of matter. Atoms with unpaired electrons are attracted by magnetic fields and thus are said to exhibit paramagnetism. The degree to which this effect is observed is directly related to the number of unpaired electrons present in the atom. Consider the ground-state electron configurations for Li, N, Ni, Te, Ba, and Hg. Which of these atoms would be expected to be paramagnetic, and how many unpaired electrons are present in each paramagnetic atom?

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An excited hydrogen atom with an electron in the \(n=5\) state emits light having a frequency of \(6.90 \times 10^{14} \mathrm{s}^{-1}\). Determine the principal quantum level for the final state in this electronic transition.

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 \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\).
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