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Chapter 41: Problem 30

For germanium \((\mathrm{Ge}, Z=32),\) make a list of the number of electrons in each subshell \((1 s, 2 s, 2 p, \ldots) .\) Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3 .

Short Answer

Expert verified
The Germanium atomhas the following electron configuration: \(1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^{10}, 4p^2\)

Step by step solution

01

Determine the Subshells and their Maximum Capacities

Start by listing down all the subshells and their maximum electron capacities according to the four quantum numbers. Every subshell can fit a maximum of \(4l+2\) electrons where l is the azimuthal quantum number. In order: \(1s\) can hold a maximum of 2 electrons, \(2s\) can hold max 2 electrons, \(2p\) max 6 electrons, \(3s\) max 2 electrons, \(3p\) max 6 electrons, \(4s\) max 2 electrons, \(3d\) max 10 electrons, \(4p\) max 6 electrons, \(5s\) max 2 electrons and so forth.
02

Fill up the Subshells

Starting from the lower energy levels to the higher ones, begin filling up the subshells with the 32 electrons of Germanium. First subshell is \(1s\) which gets 2 electrons. The second subshell \(2s\) also gets 2 electrons. The third subshell \(2p\) gets 6. Continue this way until all 32 electrons are used. Always remember that a subshell cannot accommodate more than \( (4l+2) \) electrons.
03

Confirm the Electron Configuration

After distributing the electrons, the configuration would be \(1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^{10}, 4p^2\). Confirming this configuration and doing a quick count indicates that indeed all 32 electrons of the Germanium atom have been placed into the various subshells.

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Key Concepts

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

Quantum Numbers
Quantum numbers are essential in understanding atoms as they describe the distribution of electrons around a nucleus. For each electron in an atom, four quantum numbers are needed to specify its state precisely. These are:
  • The principal quantum number, \( n \), indicates the energy level of the electron and is always a positive integer (1, 2, 3, ...). Higher values of \( n \) correspond to electrons further from the nucleus.
  • The azimuthal quantum number, \( l \), defines the shape of the orbital and depends on \( n \). It can be any integer from 0 to \( n-1 \). Each value of \( l \) corresponds to a different subshell type: \( s \) (sphere, \( l=0 \)), \( p \) (dumbbell, \( l=1 \)), \( d \) (cloverleaf, \( l=2 \)), and \( f \) (complex shapes, \( l=3 \)).
  • The magnetic quantum number, \( m_l \), determines the orientation of the orbital in space. It ranges from \( -l \) to \( +l \), including zero.
  • The spin quantum number, \( m_s \), describes the spin direction of the electron. It can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \), representing the two possible spin states.
These quantum numbers together help define the unique position and properties of each electron in an atom's electron cloud.
Pauli Exclusion Principle
The Pauli Exclusion Principle is a fundamental rule in quantum mechanics and states that no two electrons in an atom can have identical quantum numbers. This principle is critical in determining how electrons are arranged within atoms.

Since there are four quantum numbers, each electron must differ by at least one when in the same atom. This is particularly important in ensuring the diverse properties of the elements in the periodic table.

When filling electrons into subshells, this principle implies:
  • Each orbital can hold a maximum of two electrons, provided they have opposite spins (\( +\frac{1}{2} \) and \( -\frac{1}{2} \)).
  • In practice, this means subshells fill systematically: electrons populate the lower-energy orbitals first and pair up in individual orbitals when necessary.
This principle explains why atoms have specific electron configurations, directly influencing their chemical behavior and interactions.
Subshell Electron Capacity
Subshell electron capacity describes the maximum number of electrons each type of subshell can contain, determined by the azimuthal quantum number \( l \).

Each subshell's capacity follows the rule \( 2(2l+1) \), often simplified to \( 4l+2 \). Here鈥檚 a practical example:
  • The \( s \) subshell (\( l=0 \)) can hold up to 2 electrons. This includes subshells like \( 1s \), \( 2s \), and \( 3s \).
  • The \( p \) subshell (\( l=1 \)) can manage up to 6 electrons, explaining configurations in \( 2p \), \( 3p \), and \( 4p \).
  • The \( d \) subshell (\( l=2 \)) can accommodate a maximum of 10 electrons, as seen in \( 3d \) and \( 4d \).
  • The \( f \) subshell (\( l=3 \)) possesses a larger capacity and holds up to 14 electrons, such as in \( 4f \) and \( 5f \).
Understanding how many electrons fit in each subshell is fundamental to accurately determining an atom's electron configuration and ensuring compliance with quantum rules.

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Most popular questions from this chapter

Effective Magnetic Field. An clectron in a hydrogen atom is in the \(2 p\) state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius \(r\) equal to the Bohr-model radius for \(n=2 .\) Assume that the speed \(v\) of the orbiting electron can be calculated by sctting \(L=m v r\) and taking \(L\) to have the quantum-mechanical value for a \(2 p\) state. In the frame of the electron, the proton orbits with radius \(r\) and speed \(v .\) Model the orbiting proton as a circular current loop. and calculate the magnetic field it produces at the location of the clectron.

Consider the seventh excited level of the hydrogen atom. (a) What is the energy of this level? (b) What is the largest magnitude of the orbital angular momentum? (c) What is the largest angle between the orbital angular momentum and the z-axis?

CP Classical Electron Spin. (a) If you treat an electron as a classical spherical object with a radius of \(1.0 \times 10^{-17} \mathrm{~m}\), what angular speed is necessary to produce a spin angular momentum of magnitude \(\sqrt{\frac{3}{4}} h ?\) (b) Use \(v=r \omega\) and the result of part (a) to calculate the speed \(v\) of a point at the electron's cquator. What does your result suggest about the validity of this model?

(a) The doubly charged ion \(\mathrm{N}^{2+}\) is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the \(\mathrm{N}^{2+}\) ion? (b) Estimate the energy of the least strongly bound level in the \(L\) shell of \(\mathrm{N}^{2+}\). (c) The doubly charged ion \(\mathrm{P}^{2+}\) is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the \(\mathrm{P}^{2+}\) ion? (d) Estimate the energy of the least strongly bound level in the \(M\) shell of \(\mathrm{P}^{2+}\)

Assume that the researchers place an atom in a state with \(n=100, l=2 .\) What is the magnitude of the orbital angular momentum \(\overrightarrow{\boldsymbol{L}}\) associated with this state? (a) \(\sqrt{2} h ;\) (b) \(\sqrt{6} h ;\) (c) \(\sqrt{200} h\) (d) \(\sqrt{10.100} h\)
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