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Chapter 3: Problem 83

Determine the allowed values of the angular quantum number, \(l\), when the principal quantum number is 4 . Describe the difference between orbitals that have the same principal quantum number and different angular quantum numbers.

Short Answer

Expert verified
For \( n = 4 \), allowed \( l \) values are 0, 1, 2, 3; they account for s, p, d, and f orbitals.

Step by step solution

01

Understanding Quantum Numbers

Quantum numbers are values that describe the unique quantum state of an electron in an atom. The principal quantum number, denoted by \( n \), specifies the energy level of an electron. The angular quantum number, denoted by \( l \), defines the shape of the orbital. The values of \( l \) depend on the value of \( n \).
02

Determining Allowed Values of l

The angular quantum number \( l \) can take any integer value from 0 to \( n - 1 \). Therefore, when the principal quantum number \( n \) is 4, the possible values of \( l \) are 0, 1, 2, and 3.
03

Describing Orbitals with Different l Values

Orbitals with the same principal quantum number \( n \) but different angular quantum numbers \( l \) have different shapes. For example, if \( n = 4 \), the following are the types:- \( l = 0 \): **s-orbital** (spherical shape)- \( l = 1 \): **p-orbital** (dumbbell shape)- \( l = 2 \): **d-orbital** (cloverleaf shape)- \( l = 3 \): **f-orbital** (complex shape). The difference in shape results in different energies and chemical properties for the electrons in those orbitals.

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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, represented by the symbol \( n \), is a fundamental idea in quantum mechanics that helps define an electron's position around an atom's nucleus. Every electron in an atom has a unique set of quantum numbers, which dictate where it can be found. The principal quantum number is particularly important because:
  • It determines the main energy level or shell an electron occupies.
  • The value of \( n \) starts from 1 and goes up as energy levels increase鈥攈igher \( n \) indicates electrons are farther from the nucleus.
  • It influences the size of the electron cloud surrounding the nucleus; as \( n \) increases, the electron's wavelength becomes larger.
Understanding this number helps in predicting the energy and distance of an electron's presence from the nucleus, which is crucial for explaining chemical interactions and behaviors.
Orbital Shapes
The concept of orbital shapes is directly linked to the angular quantum number \( l \). Each different value of \( l \) corresponds to a distinct shape of an electron cloud, often called an "orbital." These shapes help determine where an electron is likely to be found around an atom. Here are the possible shapes:
  • For \( l = 0 \): **s-orbital** - These are spherical in shape and represent the simplest form of an orbital.
  • For \( l = 1 \): **p-orbital** - With a dumbbell shape, these orbitals are more complex and extend further out into space.
  • For \( l = 2 \): **d-orbital** - Known for their cloverleaf shape, they represent even more complex structures.
  • For \( l = 3 \): **f-orbital** - Characterized by very intricate shapes, they are found in the highest energy levels.
Understanding these shapes is vital, as they dictate the chemical properties and reactions of the elements by influencing how electrons interact.
Electron Orbitals
Electron orbitals describe regions within an atom where electrons are most likely to be. These orbitals are defined by quantum numbers, which together specify the electron's state. For the case where the principal quantum number \( n = 4 \), there can be several different electron orbitals, each with unique characteristics. Here鈥檚 how they vary:
  • Orbitals with the same principal quantum number \( n \) can have different angular quantum numbers \( l \), resulting in varying shapes.
  • These shapes affect how orbitals overlap in multiple atoms, impacting molecular bonding.
  • Higher \( n \) values allow more complex orbitals (e.g., s, p, d, f), leading to unique patterns of electron distribution.
This fundamental understanding of electron orbitals is key for predicting and relating to the properties of materials, the formation of chemical bonds, and their subsequent behaviors in chemical reactions.

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Most popular questions from this chapter

Which of the following atoms has the smallest radius? (a) \(\mathrm{Na}\) (b) \(\mathrm{Mg}\) (c) \(\mathrm{Al}\) (d) \(\underline{K}\) (e) Ca

According to the shell model, why do first ionization energies decrease down a column (group) of the periodic table?

Which of the following is a legitimate set of \(n, l, m_{l}\) and \(m_{s}\) quantum numbers? (a) \(4,-2,-1,1 / 2\) (b) \(4,2,3,1 / 2\) (c) 4,3,0,1 (d) \(4,0,0,-1 / 2\)

What is the relationship among the core charge, number of valence electrons, and group number?

Determine the allowed values of the spin quantum number, \(m_{s},\) when \(n=5, l=2,\) and \(m_{l}=-1\).
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