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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
3p: \(n=3, l=1\); 2s: \(n=2, l=0\); 4f: \(n=4, l=3\); 5d: \(n=5, l=2\).

Step by step solution

01

Understanding the Quantum Numbers

In atomic physics, each orbital designation consists of a principal quantum number, denoted as \( n \), and an azimuthal quantum number, denoted as \( l \). The principal quantum number \( n \) indicates the energy level of an electron within an atom and is a positive integer (1, 2, 3,...). The azimuthal quantum number \( l \) indicates the shape of the orbital and must satisfy \( 0 \leq l < n \). The value of \( l \) determines the type of orbital: \( s \) for \( l = 0 \), \( p \) for \( l = 1 \), \( d \) for \( l = 2 \), and \( f \) for \( l = 3 \).
02

Identify Quantum Numbers for 3p

For the orbital designation \(3p\), the principal quantum number \( n \) is 3. Since the letter \( p \) corresponds to the azimuthal quantum number of 1, \( l = 1 \). Hence, for the \(3p\) orbital, \( n = 3 \) and \( l = 1 \).
03

Identify Quantum Numbers for 2s

For the orbital designation \(2s\), the principal quantum number \( n \) is 2. The letter \( s \) indicates the azimuthal quantum number is 0, hence \( l = 0 \). Thus, for the \(2s\) orbital, \( n = 2 \) and \( l = 0 \).
04

Identify Quantum Numbers for 4f

In the \(4f\) orbital designation, the principal quantum number \( n \) is 4. The letter \( f \) corresponds to an azimuthal quantum number of 3, hence \( l = 3 \). Consequently, for the \(4f\) orbital, \( n = 4 \) and \( l = 3 \).
05

Identify Quantum Numbers for 5d

For the \(5d\) orbital designation, the principal quantum number \( n \) is 5. The letter \( d \) indicates an azimuthal quantum number of 2, so \( l = 2 \). Thus, for the \(5d\) 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 is a fundamental component in understanding the arrangement of electrons in an atom. Designated by the symbol \( n \), it mainly determines the energy level of an electron and corresponds to the shells surrounding the nucleus. These shells can be thought of as the various floors in a multi-story building where each floor can accommodate a certain number of electrons. The principal quantum number is always a positive integer (
  • \( n = 1, 2, 3, \ldots \)
) and as \( n \) increases, the energy and size of the electron cloud also increase.
View the principal quantum number as the address of the electron's neighborhood, within which electrons occupy specific energy levels. For example, in a 3p orbital, the principal quantum number \( n = 3 \), indicating it is on the third energy level.
Azimuthal Quantum Number
The azimuthal quantum number, often denoted by \( l \), signifies the subshells of an electron's orbital and determines the shape of the electron cloud. Additionally, it provides insight into the angular momentum of the electron. It is defined based on the principal quantum number, with values ranging from 0 to \( n-1 \). Each value of \( l \) corresponds to a different type of sublevel or orbital shape:
  • \( s \) for \( l = 0 \)
  • \( p \) for \( l = 1 \)
  • \( d \) for \( l = 2 \)
  • \( f \) for \( l = 3 \)
For instance, in a 3p orbital, since the letter \( p \) corresponds to \( l = 1 \), you know the shape of the electron cloud has a particular shape denoted as 'p' in chemistry.
Orbital Designation
An orbital designation is a shorthand notation used to specify the position and type of an electron within an atom. This notation combines the principal quantum number and the azimuthal quantum number to provide a complete description. It first lists the principal quantum number \( n \), followed by the letter representing the azimuthal quantum number \( l \) (i.e., \( s, p, d, \) or \( f \)). This is like the full address for an electron within an atomic structure.
The notation is crucial for identifying how electrons fill themselves in an atom, which in turn affects chemical bonding and material properties. For example:
  • In \( 3p \), the 3 indicates the third energy level, and \( p \) signifies the type of orbital (\( l = 1 \)).
  • Similarly, in \( 4f \), the 4 denotes the fourth energy level, and \( f \) represents an \( l = 3 \) configuration.
Understanding these notations helps in predicting chemical behavior and interacting forces in atomic interactions.

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Most popular questions from this chapter

The following electron configurations represent excited states. Identify the element and write its ground-state condensed electron configuration. (a) \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\) (b) \([\mathrm{Ne}] 3 s^{1} 3 p^{4} 4 p^{1}\) (c) \([\mathrm{Ar}] 4 s^{2} 3 d^{6} 4 p^{1}\).

Molybdenum metal must absorb radiation with an energy higher than \(7.22 \times 10^{-19} \mathrm{~J}\left({ }^{\text {" }}\right.\) 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 following do not represent valid ground-state electron configurations for an atom either because they violate the Pauli exclusion principle or because orbitals are not filled in order of increasing energy. Indicate which of these two principles is violated in each example. (a) \([\mathrm{Ne}] 3 s^{2} 3 p^{6} 3 d^{5}\) (c) \(1 s^{2} 3 s^{1}\) (b) \([\mathrm{Xe}] 6 s^{3}\)

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

In the television series Star Trek, the transporter beam is a device used to "beam down" people from the Starship Enterprise to another location, such as the surface of a planet. The writers of the show put a "Heisenberg compensator" into the transporter beam mechanism. Explain why such a compensator (which is entirely fictional) would be necessary to get around Heisenberg's uncertainty principle.
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