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Chapter 30: Problem 53

Write down the fourteen sets of the four quantum numbers that correspond to the electrons in a completely filled \(4 \mathrm{f}\) subshell.

Short Answer

Expert verified
The fourteen sets of quantum numbers are: (4, 3, -3, 卤1/2), (4, 3, -2, 卤1/2), (4, 3, -1, 卤1/2), (4, 3, 0, 卤1/2), (4, 3, 1, 卤1/2), (4, 3, 2, 卤1/2), (4, 3, 3, 卤1/2).

Step by step solution

01

Understanding Quantum Numbers

Quantum numbers are used to describe the properties and position of electrons in atoms. They include the principal quantum number ( ), azimuthal or angular momentum quantum number ( ), magnetic quantum number ( ), and spin quantum number ( ). The principal quantum number ( ) for a 4f subshell is 4, and the azimuthal quantum number ( ) is 3.
02

Identify Magnetic Quantum Numbers

For the 4f subshell, where the azimuthal quantum number ( ) is 3, the possible magnetic quantum numbers ( ) range from -3 to +3. This includes the values: -3, -2, -1, 0, 1, 2, 3.
03

Assigning Spin Quantum Numbers

Each magnetic quantum number ( ) can hold two electrons, each with a different spin quantum number ( ), which can be +1/2 or -1/2. This gives each magnetic quantum number two possible sets of quantum numbers.
04

List the Fourteen Sets of Quantum Numbers

Combine all the possible values: 1. (4, 3, -3, +1/2) 2. (4, 3, -3, -1/2) 3. (4, 3, -2, +1/2) 4. (4, 3, -2, -1/2) 5. (4, 3, -1, +1/2) 6. (4, 3, -1, -1/2) 7. (4, 3, 0, +1/2) 8. (4, 3, 0, -1/2) 9. (4, 3, 1, +1/2) 10. (4, 3, 1, -1/2) 11. (4, 3, 2, +1/2) 12. (4, 3, 2, -1/2) 13. (4, 3, 3, +1/2) 14. (4, 3, 3, -1/2)

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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, denoted by \( n \), plays a fundamental role in defining the energy level of an electron within an atom. This number is always a positive integer: 1, 2, 3, and so on. Each value of \( n \) corresponds to a specific electron shell or energy level surrounding the nucleus.
  • The higher the value of \( n \), the higher the energy and the larger the size of the electron cloud.
  • The principal quantum number essentially determines the maximum number of electrons that can be accommodated in a specific shell, given by the formula \( 2n^2 \).
For example, if we have \( n = 4 \), this indicates that the electron is in the fourth energy level, as in the 4f subshell we are discussing. This level is where electrons have enough energy to occupy the 4f orbitals as mentioned.
Azimuthal Quantum Number
The azimuthal quantum number, often represented by \( l \), is instrumental in defining the shape of the electron's orbital. It is also known as the angular momentum quantum number. This number is an integer that ranges from 0 to \( n-1 \). Each value of \( l \) corresponds to a particular subshell or orbital type.
  • For \( l = 0 \), we have an s orbital, which is spherical.
  • For \( l = 1 \), the p orbital, which is dumbbell-shaped.
  • For \( l = 2 \), the d orbital, which can have more complex shapes.
  • For \( l = 3 \), the f orbital, which is even more complex and pertains directly to our 4f subshell.
Since the 4f subshell is being filled, the azimuthal quantum number \( l \) is specifically set to be 3 for the f orbitals. This specifies the particular shape and complexity associated with this type of subshell.
Magnetic Quantum Number
Each orbital within a subshell can hold a different orientation in space, defined by the magnetic quantum number, \( m_l \). This number indicates the orientation of the orbital relative to an external magnetic field and ranges from \( -l \) to \( +l \). Therefore, for a given \( l = 3 \), \( m_l \) can have values of -3, -2, -1, 0, 1, 2, and 3.
  • This results in a total of seven orientations available for any energy level's f orbitals.
  • These orientations combine to form the 4f subshell's capacity for holding 14 electrons total, as each orbital orientation can accommodate two electrons with opposite spins.
The variety of \( m_l \) values allows for a precise description of electron distributions and helps to explain the atom's chemical properties based on its electron configuration.
Spin Quantum Number
While the other quantum numbers describe the electron's energy, position, and orbital, the spin quantum number, \( m_s \), addresses the fundamental property known as electron spin. This is analogous to the electron's intrinsic angular momentum.
  • Electron spin can have one of two values: +1/2 or -1/2.
  • These spins indicate the two possible spin directions, often thought of as "spin up" and "spin down".
Because of the Pauli exclusion principle, no two electrons can have the same set of all four quantum numbers, ensuring that two electrons in an identical orbital have opposite spins. Thus, every \( m_l \) can pair with two values of \( m_s \), allowing electrons to fully populate the 4f subshell properly, with a distinct set of quantum numbers for each electron.

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Most popular questions from this chapter

The nucleus of a copper atom contains 29 protons and has a radius of \(4.8 \times 10^{-15} \mathrm{m} .\) How much work (in electron volts) is done by the electric force as a proton is brought from infinity, where it is at rest, to the "surface" of a copper nucleus?

A sodium atom \((Z=11)\) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value \(Z_{\text {effective }}\) rather than 11 for the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is \(5.1 \mathrm{eV}\). Use the Bohr model with \(Z=Z_{\text {effective }}\) to calculate a value for \(Z_{\text {effective }}\) (b) Using \(Z=11\) and \(Z=Z_{\text {effective }},\) determine the corresponding two values for the radius of the outermost Bohr orbit.

For a doubly ionized lithium atom \(\mathrm{Li}^{2+}(Z=3)\), what is the principal quantum number of the state in which the electron has the same total energy as a ground-state electron has in the hydrogen atom?

A laser is used in eye surgery to weld a detached retina back into place. The wavelength of the laser beam is \(514 \mathrm{nm}\) and the power is 1.5 W. During surgery, the laser beam is turned on for 0.050 s. During this time, how many photons are emitted by the laser?

The K-shell and L-shell ionization energies of a metal are \(8979 \mathrm{eV}\) and \(951 \mathrm{eV},\) respectively. Concepts: (i) How is a \(K_{\alpha}\) photon produced, and how much energy does it have? (ii) What must be the minimum voltage across the X-ray tube to produce a \(K_{\alpha}\) photon? (iii) What is meant by the phrases "K-shell ionization energy" and "L-shell ionization energy"? (iv) What does the difference between the K-shell and L-shell ionization energies represent? Calculations: (a) Assuming that there is a vacancy in the \(\mathrm{L}\) shell, what must be the minimum voltage across an X-ray tube with a target made from this metal to produce \(K_{\alpha}\) X-ray photons? (b) Determine the wavelength of a \(K_{\alpha}\) photon.
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