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Chapter 6: Problem 60

A hydrogen atom orbital has \(n=4\) and \(m_{l}=-2\). (a) What are the possible values of \(l\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

Short Answer

Expert verified
(a) The possible values of \(l\) for this orbital are 2 and 3. (b) The possible values of \(m_s\) for the orbital are always +1/2 and -1/2.

Step by step solution

01

(a) Possible values of l

To find the possible values of the angular momentum quantum number l, we need to recall the relationship between n, l and m_l: \(l\) can have integer values from 0 to n-1, and \(m_{l}\) can have integer values ranging from -l to +l. In our case, n = 4, and \(m_{l} = -2\). Since m_l = -2, we know that l must be at least 2, since m_l can only take values from -l to +l. Therefore, we will try each possible value of l = 2, 3, and check if \(m_l = -2\) is a valid option. For l = 2, \(m_l\) can take values -2, -1, 0, 1, 2, which includes \(m_l = -2\). For l = 3, \(m_l\) can take values -3, -2, -1, 0, 1, 2, 3, which also includes \(m_l = -2\). So, the possible values of l for this orbital are 2 and 3.
02

(b) Possible values of m_s

To find the possible values of the spin quantum number \(m_s\), we recall that for an electron in an orbital, there are only two possible spin orientations available: 1. Spin-up (represented as +1/2) 2. Spin-down (represented as -1/2) So, for any electron in an orbital, the possible values of \(m_s\) are always +1/2 and -1/2.

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Most popular questions from this chapter

(a) The average distance from the nucleus of a 3 s electron in a chlorine atom is smaller than that for a \(3 p\) electron. In light of this fact, which orbital is higher in energy? (b) Would you expect it to require more or less energy to remove a 3 s electron from the chlorine atom, as compared with a $2 p$ electron?

Molybdenum metal must absorb radiation with an energy higher than $7.22 \times 10^{-19} \mathrm{~J}$ ( "energy threshold") before it can eject an electron from its surface via the photoelectric effect. (a) What is the frequency threshold for emission of electrons? (b) What wavelength of radiation will provide a photon of this energy? (c) If molybdenum is irradiated with light of wavelength of \(240 \mathrm{nm}\), what is the maximum possible velocity of the emitted electrons?

The hydrogen atom can absorb light of wavelength \(1094 \mathrm{nm}\). (a) In what region of the electromagnetic spectrum is this absorption found? (b) Determine the initial and final values of \(n\) associated with this absorption.

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p\), (b) \(2 s,(\mathbf{c}) 4 f,\) (d) $5 d$.

The first 25 years of the twentieth century were momentous for the rapid pace of change in scientists' understanding of the nature of matter. (a) How did Rutherford's experiments on the scattering of \(\alpha\) particles by a gold foil set the stage for Bohr's theory of the hydrogen atom? (b) In what ways is de Broglie's hypothesis, as it applies to electrons, consistent with J. J. Thomson's conclusion that the electron has mass? In what sense is it consistent with proposals preceding Thomson's work that the cathode rays are a wave phenomenon?
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