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Chapter 7: Problem 32

Imagine a world in which the rule for the \(l\) quantum number is that values start with 1 and go up to \(n\). The rules for the \(n\) and \(m_{l}\) quantum numbers are unchanged from those of our world. Write the quantum numbers for the first two shells (i.e., \(n=1\) and \(n=2\) ).

Short Answer

Expert verified
For \(n=1\), there are 6 combinations; for \(n=2\), there are 16 combinations of quantum numbers.

Step by step solution

01

Understanding Quantum Numbers

In the context of quantum mechanics, each electron in an atom is described by a set of four quantum numbers: the principal quantum number \( n \), the azimuthal quantum number \( l \), the magnetic quantum number \( m_l \), and the spin quantum number \( m_s \). The principal quantum number \( n \) describes the energy level or shell of the electron, and it is always a positive integer. The azimuthal quantum number \( l \) determines the shape of the electron's orbital, and in our problem, it ranges from 1 to \( n \). The magnetic quantum number \( m_l \) indicates the orientation of the orbital, with values ranging from \(-l\) to \(l\). The spin quantum number \( m_s \) is either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
02

Determining Quantum Numbers for \(n=1\)

When \( n = 1 \), the possible values for \( l \) are 1. With \( l = 1 \), \( m_l \) can take the values -1, 0, or 1. The values for each set of quantum numbers, including the two possibilities for \( m_s \), would therefore be: 1. \( (n=1, l=1, m_l=-1, m_s=\frac{1}{2}) \)2. \( (n=1, l=1, m_l=-1, m_s=-\frac{1}{2}) \)3. \( (n=1, l=1, m_l=0, m_s=\frac{1}{2}) \)4. \( (n=1, l=1, m_l=0, m_s=-\frac{1}{2}) \)5. \( (n=1, l=1, m_l=1, m_s=\frac{1}{2}) \)6. \( (n=1, l=1, m_l=1, m_s=-\frac{1}{2}) \).
03

Determining Quantum Numbers for \(n=2\)

When \( n = 2 \), the values for \( l \) can be 1 or 2. - For \( l = 1 \), \( m_l \) can take values -1, 0, or 1. Consider these configurations: 1. \( (n=2, l=1, m_l=-1, m_s=\frac{1}{2}) \) 2. \( (n=2, l=1, m_l=-1, m_s=-\frac{1}{2}) \) 3. \( (n=2, l=1, m_l=0, m_s=\frac{1}{2}) \) 4. \( (n=2, l=1, m_l=0, m_s=-\frac{1}{2}) \) 5. \( (n=2, l=1, m_l=1, m_s=\frac{1}{2}) \) 6. \( (n=2, l=1, m_l=1, m_s=-\frac{1}{2}) \).- For \( l = 2 \), \( m_l \) can take values -2, -1, 0, 1, or 2: 7. \( (n=2, l=2, m_l=-2, m_s=\frac{1}{2}) \) 8. \( (n=2, l=2, m_l=-2, m_s=-\frac{1}{2}) \) 9. \( (n=2, l=2, m_l=-1, m_s=\frac{1}{2}) \) 10. \( (n=2, l=2, m_l=-1, m_s=-\frac{1}{2}) \) 11. \( (n=2, l=2, m_l=0, m_s=\frac{1}{2}) \) 12. \( (n=2, l=2, m_l=0, m_s=-\frac{1}{2}) \) 13. \( (n=2, l=2, m_l=1, m_s=\frac{1}{2}) \) 14. \( (n=2, l=2, m_l=1, m_s=-\frac{1}{2}) \) 15. \( (n=2, l=2, m_l=2, m_s=\frac{1}{2}) \) 16. \( (n=2, l=2, m_l=2, m_s=-\frac{1}{2}) \).
04

Summarizing Quantum Numbers for Shells \(n=1\) and \(n=2\)

Combining all the above possibilities:- For \( n = 1 \), we have 6 possible combinations of quantum numbers.- For \( n = 2 \), we have 16 possible combinations of quantum numbers. This corresponds to considering all combinations of \( l \), \( m_l \), and \( m_s \) for the given \( n \) values.

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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, denoted by \( n \), is a crucial concept in quantum mechanics. It primarily describes the energy level of an electron within an atom, often referred to as the electron shell. Think of it like the floors in a building, where the electron can "live." Each floor represents a different energy level.

Principal quantum numbers start at 1 and increase infinitely, represented as positive integers: 1, 2, 3, and so on. As \( n \) increases, the electron's distance from the nucleus on average and the energy level increases. Therefore, an electron in the \( n=2 \) level has more energy and is found further from the nucleus compared to \( n=1 \).

To summarize, principal quantum numbers:
  • Determine the main energy level or "shell" of an electron
  • Are represented by positive integers (1, 2, 3,...)
  • Influence the electron's energy and average distance from the nucleus
Azimuthal Quantum Number
The azimuthal quantum number, designated as \( l \), provides insight into the shape of the electron's orbital. While the principal quantum number tells us the general size of the space where an electron exists, the azimuthal quantum number defines the shape of that region.

In our exercise scenario, \( l \) ranges from 1 to \( n \) (note that this is different from real-world application, where \( l \) would range from 0 to \( n-1 \)). Depending on the value of \( l \), each value corresponds to a specific type of orbital structure, such as:
  • \( l=1 \): Typically associate with p orbitals
  • \( l=2 \): Typically associate with d orbitals
These orbitals differ in shape and have distinct spatial orientations.

To further understand, consider:\
  • The value of \( l \) influences the shape of the electron cloud
  • Different shapes affect how electrons interact with each other and the nucleus
Magnetic Quantum Number
The magnetic quantum number, represented by \( m_l \), describes the specific orientation of the electron's orbital in space. Imagine that each orbital may face different directions, much like a spinning top can tilt at various angles.

For a given value of \( l \), the magnetic quantum number ranges from \(-l\) to \(l\). This means that:
  • If \( l=1 \), \( m_l \) can be -1, 0, or 1
  • If \( l=2 \), \( m_l \) can be -2, -1, 0, 1, or 2
Each \( m_l \) value denotes one possible orientation of the orbital. Because electrons are negatively charged, their orientation in magnetic fields becomes significant in physics and chemistry.

Key points:
  • The magnetic quantum number specifies the direction of the orbital
  • Different orientations are important for understanding chemical bonding and properties
Electron Orbitals
Electron orbitals are regions around the nucleus where electrons are likely to be found, and they are defined using quantum numbers \( n \), \( l \), and \( m_l \). Each orbital corresponds to a particular distribution of electron density, with shapes determined by the quantum numbers.

In any given atom, these orbitals have defined energies and shapes, influencing how electrons populate them:
  • Low-energy orbitals fill up first (based on \( n \) and \( l \))
  • P orbitals have a dumbbell shape, while d orbitals have a more complex double-dumbbell shape

The understanding of electron orbitals is fundamental in explaining how atoms bond to form molecules. It helps predict the chemical properties and reactivity of elements.

In summary, remember that:
  • Electron orbitals are dictated by quantum numbers
  • Different shapes and orientations lead to diverse electron configurations
  • These configurations are vital for assessing chemical behavior

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Most popular questions from this chapter

A particular microwave oven delivers 750 watts. (A watt is a unit of power, which is the joules of energy delivered, or used, per second.) If the oven uses microwave radiation of wavelength \(12.6 \mathrm{~cm}\), how many photons of this radiation are required to heat \(1.00 \mathrm{~g}\) of water \(1.00^{\circ} \mathrm{C}\) assuming that all of the photons are absorbed?

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7.22 What wavelength of electromagnetic radiation corresponds to a frequency of \(3.46 \times 10^{13} \mathrm{~s}^{-1} a \) ? \(8.66 \times 10^{-6} \mathrm{~m}\) b \(1.15 \times 10^{5} \mathrm{~m}\) c \(7.65 \times 10^{-29} \mathrm{~m}\) d \(9.10 \times 10^{-6} \mathrm{~m}\) e \(8.99 \times 10^{-6} \mathrm{~m}\)

The retina of the eye contains two types of lightsensitive cells: rods (responsible for night vision) and cones (responsible for color vision). Rod cells are about a hundred times more sensitive to light than cone cells and are able to detect a single photon. Suppose a group of rod cells are radiated with a pulse of light having an energy equal to \(1.60 \times 10^{-16} \mathrm{~J}\). If the wavelength of this light was \(498 \mathrm{nm}\) (the wavelength at which rod cells are most sensitive), how many photons are in this light pulse?

Explain the concept of quantum mechanical tunneling.
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