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Chapter 2: Problem 73

Which of the following sets of quantum numbers are not allowed in the hydrogen atom? For the sets of quantum numbers that are incorrect, state what is wrong in each set. a. \(n=3, \ell=2, m_{c}=2\) b. \(n=4, \ell=3, m_{\ell}=4\) c. \(n=0, \ell=0, m_{\ell}=0\) d. \(n=2, \ell=-1, m_{c}=1\)

Short Answer

Expert verified
In conclusion: a. Allowed in the hydrogen atom. b. Not allowed, 𝑚_𝑙=4 is not within the range of -3 to 3. c. Not allowed, n=0 is not a positive integer (n must be greater than 0). d. Not allowed, 𝑙=-1 is not a non-negative integer.

Step by step solution

01

Check the principal quantum number (n).

Only positive integers are allowed for n. a. n=3, valid. b. n=4, valid. c. n=0, invalid. (n must be greater than 0) d. n=2, valid.
02

Check the azimuthal quantum number (𝑙).

0 ≤ 𝑙 ≤ n-1 and it is an integer. a. 𝑙=2, valid. (0 ≤ 2 ≤ 3-1) b. 𝑙=3, valid. (0 ≤ 3 ≤ 4-1) c. 𝑙=0, invalid. (Since n is already invalid, there's no need to check 𝑙) d. 𝑙=-1, invalid. (𝑙 must be non-negative)
03

Check the magnetic quantum number (𝑚_𝑙).

-𝑙 ≤ 𝑚_𝑙 ≤ 𝑙 and it is an integer. a. 𝑚_𝑐=2, valid. (-2 ≤ 2 ≤ 2) b. 𝑚_𝑙=4, invalid. (-3 ≤ 4 ≤ 3) c. 𝑚_𝑙=0, invalid. (Since n and 𝑙 are already invalid, there's no need to check 𝑚_𝑙) d. 𝑚_𝑐=1, invalid. (Since 𝑙 is invalid, there's no need to check 𝑚_𝑐) In conclusion: a. Allowed in the hydrogen atom. b. Not allowed, 𝑚_𝑙=4 is not within the range of -3 to 3. c. Not allowed, n=0 is not a positive integer (n must be greater than 0). d. Not allowed, 𝑙=-1 is not a non-negative integer.

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Most popular questions from this chapter

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers \(p, q, r,\) and \(s .\) The rules for these quantum numbers are as follows: \(p=1,2,3,4,5, \dots\) \(q\) takes on positive odd integers and \(q \leq p\) \(r\) takes on all even integer values from \(-q\) to \(+q\). (Zero is considered an even number.) \(s=+\frac{1}{2}\) or \(-\frac{1}{2}\) a. Sketch what the first four periods of the periodic table will look like in this universe. b. What are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the first four rows, of ionic compounds with the formulas XY, XY \(_{2}, X_{2} Y, X Y_{3}\) and \(\mathrm{X}_{2} \mathrm{Y}_{3}\) d. How many electrons can have \(p=4, q=3 ?\) e. How many electrons can have \(p=3, q=0, r=0 ?\) f. How many electrons can have \(p=6 ?\)

Calculate the de Broglie wavelength for each of the following. a. an electron with a velocity \(10 . \%\) of the speed of light b. a tennis ball \((55 \mathrm{g})\) served at \(35 \mathrm{m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})\)

The first ionization energies of As and Se are 0.947 and \(0.941 \mathrm{MJ} / \mathrm{mol},\) respectively. Rationalize these values in terms of electron configurations.

Calculate the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^{2} \mathrm{nm}\) and \(1.0 \mathrm{nm} .\)

Does a photon of visible light \((\lambda=400 \text { to } 700 \mathrm{nm}\) ) have sufficient energy to excite an electron in a hydrogen atom from the \(n=1\) to the \(n=5\) energy state? From the \(n=2\) to the \(n=\) 6 energy state?
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