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Chapter 3: Problem 79

What are the possible values of quantum number \(\ell\) when \(n=4 ?\)

Short Answer

Expert verified
Answer: The possible values of \(\ell\) when \(n=4\) are 0, 1, 2, and 3.

Step by step solution

01

Understanding quantum numbers

In this problem, we're dealing with two quantum numbers: the principal quantum number (\(n\)) and the angular momentum quantum number (\(\ell\)). The principal quantum number \(n\) represents the energy level of the electron in an atom and can be any positive integer. The angular momentum quantum number \(\ell\) describes the shape of the electron's orbital and can have integral values ranging from \(0\) to \(n-1\).
02

Set up the equation for the range of \(\ell\)

We are given the principal quantum number \(n=4\). The range of possible values of the angular momentum quantum number \(\ell\) can be determined using the rule that \(\ell\) ranges from \(0\) to \(n-1\). So, we can write this relationship as: 0 鈮� \(\ell\) 鈮� \(n-1\).
03

Find the possible values of \(\ell\)

Now we will substitute the given value of \(n\) into the equation to find the range of possible values of \(\ell\). For \(n = 4\), we have: 0 鈮� \(\ell\) 鈮� \(4-1\). Therefore, 0 鈮� \(\ell\) 鈮� 3. This means the possible values of \(\ell\) when \(n=4\) are: \(\ell = 0, 1, 2, 3\).

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Most popular questions from this chapter

By absorbing different wavelengths of light, an electron in a hydrogen atom undergoes a transition from \(n=2\) to \(n=3\) and then from \(n=3\) to \(n=4\) a. Are the wavelengths for the two transitions additive that is, does \(\lambda_{2 \rightarrow 4}=\lambda_{2 \rightarrow 3}+\lambda_{3 \rightarrow 4} ?\) b. Are the energies of the two transitions additive-that is, does \(E_{2 \rightarrow 4}=E_{2 \rightarrow 3}+E_{3 \rightarrow 4} ?\)

In what ways should the cmission spectra of \(\mathrm{H}\) and \(\mathrm{He}^{+}\) be alike, and in what ways should they be different?

Which of the following electron configurations represent an excited state? a. \([\mathrm{He}] 2 s^{1} 2 p^{5}\) b. \(\left[\mathrm{K}_{\mathrm{r}}\right] 4 d^{10} 5 s^{2} 5 p^{1}\) c. \([\mathrm{Ar}] 3 d^{10} 4 s^{2} 4 p^{5}\) d. \([\mathrm{Ne}] 3 s^{2} 3 p^{2} 4 s^{1}\)

What are the electron configurations of \(\mathrm{Li}^{+}, \mathrm{Ca}, \mathrm{F}^{-}, \mathrm{Mg}^{2+}\) and \(A 1^{3+} ?\)

Identify the symbols in the de Broglie relation, \(\lambda=b / m u\) and explain how the relation links the properties of a particle to those of a wave.
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