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Chapter 6: Problem 57

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

Short Answer

Expert verified
For the given orbital designations, the numerical values of n and l are as follows: (a) 3p orbital: n = 3 and l = 1. (b) 2s orbital: n = 2 and l = 0. (c) 4f orbital: n = 4 and l = 3. (d) 5d orbital: n = 5 and l = 2.

Step by step solution

01

(a) 3p orbital

For the 3p orbital, the principal quantum number is already provided, which is n = 3. The letter "p" corresponds to the azimuthal quantum number value l = 1. Thus, for a 3p orbital, n = 3 and l = 1.
02

(b) 2s orbital

For the 2s orbital, the principal quantum number is given as n = 2. The letter "s" corresponds to the azimuthal quantum number value l = 0. Thus, for a 2s orbital, n = 2 and l = 0.
03

(c) 4f orbital

For the 4f orbital, the principal quantum number is given as n = 4. The letter "f" corresponds to the azimuthal quantum number value l = 3. Thus, for a 4f orbital, n = 4 and l = 3.
04

(d) 5d orbital

For the 5d orbital, the principal quantum number is given as n = 5. The letter "d" corresponds to the azimuthal quantum number value l = 2. Thus, for a 5d orbital, n = 5 and l = 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Quantum Number
The principal quantum number, symbolized by the letter \( n \), is a fundamental physical constant that helps to define the size and energy level of an electron orbital. Imagine it as a kind of address system for electrons within an atom. Electrons reside in specific energy levels, which are determined by this number. The principal quantum number can take positive integer values such as 1, 2, 3, and so on. The larger the value of \( n \), the higher the energy level and further the electron is generally from the nucleus.

In the orbital designation "3p," the number "3" in front is the principal quantum number, which shows us that the electron is in the third energy level. The principal quantum number is crucial because it not only indicates the energy level but also the general size of the electron's orbit, with larger \( n \) values indicating larger orbitals.
Azimuthal Quantum Number
The azimuthal quantum number, represented by the letter \( l \), determines the shape of the electron's orbital, and sometimes people refer to it as the angular momentum quantum number. This quantum number can have integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number.

Let's look at the alphabetic designation that represents different \( l \) values:
  • "s" means \( l = 0 \)
  • "p" means \( l = 1 \)
  • "d" means \( l = 2 \)
  • "f" means \( l = 3 \)
As an example, in a "3p" orbital, the "p" indicates that the azimuthal quantum number \( l \) is 1. This tells us the shape of the orbital. "s" orbitals are spherical, "p" orbitals have a dumbbell shape, and "d" and "f" orbitals have more complex shapes.
Orbital Designations
Orbital designations play an important role in describing the specific location and energy of an electron within an atom's shell or energy level. These designations combine numbers and letters to convey this information succinctly.

In an orbital designation like "3p," "3" represents the principal quantum number \( n \), while "p" designates the azimuthal quantum number \( l \).

Here's how to break down some common orbital designations:
  • **3p**: \( n=3 \), \( l=1 \)
  • **2s**: \( n=2 \), \( l=0 \)
  • **4f**: \( n=4 \), \( l=3 \)
  • **5d**: \( n=5 \), \( l=2 \)
Knowing these designations allows you to predict the behavior and properties of electrons. Each designation refers to a unique combination of energy level and orbital shape, essential for understanding the structure of an atom.

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Most popular questions from this chapter

The series of emission lines of the hydrogen atom for which \(n_{1}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series - those for which \(n_{1}=4,5,\) and \(6 .\)

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{nm},\) whereas UV-B radiation has wavelengths in the range of \(290-320 \mathrm{nm} .\) (a) Calculate the frequency of light that has a wavelength of 320 \(\mathrm{nm}\) . (b) Calculate the energy of a mole of 320 -nm photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a greater cause of sunburn in humans than is UV-A radiation. Is this observation consistent with your answer to part (c)?

Identify the specific element that corresponds to each of the following electron configurations and indicate the number of unpaired electrons for each: (a) \(1 s^{2} 2 s^{2},(\mathbf{b}) 1 s^{2} 2 s^{2} 2 p^{4}\) (c) \([\operatorname{Ar}] 4 s^{1} 3 d^{5},(\mathbf{d})[\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\)

(a) What are the similarities of and differences between the 1s and 2\(s\) orbitals of the hydrogen atom? (b) In what sense does a 2\(p\) orbital have directional character? Compare the "directional" characteristics of the \(p_{x}\) and \(d_{x^{2}-y^{2}}\) orbitals. (That is, in what direction or region of space is the electron density concentrated? (c) What can you say about the average distance from the nucleus of an electron in a 2 orbital as compared with a 3\(s\) orbital? (d) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p .\)

Label each of the following statements as true or false. For those that are false, correct the statement. (a) Visible light is a form of electromagnetic radiation. (b) Ultraviolet light has longer wavelengths than visible light. (c) \(X\) rays travel faster than microwaves. (d) Electromagnetic radiation and sound waves travel at the same speed.
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