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Chapter 30: Problem 60

(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. Verify that your answers are consistent with your answers to the Concept Questions.

Short Answer

Expert verified
(a) \( Z_{\text{eff}} \approx 1.84 \) (b) Radius with \( Z = 11 \) is \( 0.43 \text{ 脜} \); with \( Z_{\text{eff}} = 1.84 \) is \( 2.60 \text{ 脜} \).

Step by step solution

01

Understanding the Bohr Model Energy Formula

The energy of an electron in a Bohr model atom is given by the formula: \( E_n = -\frac{13.6Z_{ ext{eff}}^2}{n^2} \text{ eV} \),where \( n \) is the principal quantum number and \( Z_{\text{eff}} \) is the effective nuclear charge. For the outermost electron of sodium in the ground state, \( n = 3 \) and \( E_n = -5.1 \text{ eV} \).
02

Solve for Effective Nuclear Charge

Using the energy formula for the ground state, substitute \( E_n = -5.1 \text{ eV} \) and \( n = 3 \) into the equation to find \( Z_{\text{eff}} \):\[ -\frac{13.6Z_{\text{eff}}^2}{3^2} = -5.1 \]Simplifying, we get:\[ 13.6Z_{\text{eff}}^2 = 5.1 \times 9 \]\[ Z_{\text{eff}}^2 = \frac{45.9}{13.6} \]\[ Z_{\text{eff}} = \sqrt{\frac{45.9}{13.6}} \approx 1.84 \].
03

Calculating Radius with Full Nuclear Charge

The radius of the Bohr orbit is given by:\[ r_n = \frac{n^2a_0}{Z} \], where \( a_0 = 0.529 \text{ 脜} \). For \( Z = 11 \) (full nuclear charge for sodium):\[ r_3 = \frac{3^2 \times 0.529}{11} \approx 0.43 \text{ 脜} \].
04

Calculating Radius with Effective Nuclear Charge

For \( Z_{\text{eff}} = 1.84 \), the radius is:\[ r_3 = \frac{3^2 \times 0.529}{1.84} \approx 2.60 \text{ 脜} \].
05

Conclusion and Verification

The different radii calculated reflect the concepts of effective nuclear charge in atomic models. Using \( Z = 11 \) gives a smaller radius because the electron experiences the full nuclear attraction, while with \( Z_{\text{eff}} = 1.84 \), the electron only effectively experiences that smaller charge, resulting in a larger radius. This is consistent with the Concept Questions results.

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Key Concepts

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

Ionization Energy
Ionization energy is a fundamental concept in chemistry and physics that represents the amount of energy required to remove the outermost electron from an atom, turning it into a positively charged ion. In the context of the Bohr model, we can relate this energy expenditure to the difference in energy levels that the electron occupies before and after the ionization process.
The ionization energy for sodium's outermost electron is specified in the problem as 5.1 eV. This value indicates the energy needed to remove an electron from the third quantum level (n = 3) in a sodium atom.
Understanding ionization energy helps us comprehend atomic stability and reactivity. Atoms with high ionization energies hold onto their electrons more tightly, making them less reactive, while those with low ionization energies lose electrons more readily and are more reactive.
Effective Nuclear Charge
Effective nuclear charge ( Z_{ ext{eff}} ) is an important factor in understanding how electrons are affected by the nucleus in multi-electron atoms. We can think of it as the net charge an electron experiences after accounting for the electron shielding effect caused by other electrons present in the atom.
In this exercise, we used the value Z_{ ext{eff}} = 1.84. This effective charge calculation is essential because it acknowledges that outer electrons are less attracted to the nucleus due to interference by inner-shell electrons.
  • The effective nuclear charge is calculated by considering both the total number of protons and the repelling influence of other electrons.
  • In sodium, the significant difference between the full nuclear charge (Z = 11) and Z_{ ext{eff}} = 1.84 reflects the substantial shielding effect of the ten inner electrons.
Effective nuclear charge accentuates the differences in chemical properties across period and group elements.
Bohr Radius
The Bohr radius is a critical concept in the Bohr model of the atom, referring to the radius of the simplest possible orbit of an electron around the nucleus in a hydrogen atom. The value for a Bohr radius is commonly denoted as a_0 = 0.529 ext {脜} . For other atoms and different energy levels, this value gets adjusted based on the nuclear charge.
In this exercise, we calculated the radii for sodium's outermost electron using both the full nuclear charge and the effective nuclear charge.
  • Using the full nuclear charge ( Z = 11 ), the radius was found to be 0.43 脜. This shows a more compact orbit since the electron is subjected to the full nuclear pull.
  • With Z_{ ext{eff}} = 1.84 , the calculated radius was 2.60 脜, illustrating a more spread-out electron path due to reduced effective nuclear attraction.

The concept of the Bohr radius enhances our understanding of atomic dimensions and their dependence on nuclear charge.

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Most popular questions from this chapter

The nucleus of a hydrogen atom is a single proton, which has a radius of about \(1.0 \times 10^{-15} \mathrm{~m} .\) The single electron in a hydrogen atom normally orbits the nucleus at a distance of \(5.3 \times 10^{-11} \mathrm{~m} .\) What is the ratio of the density of the hydrogen nucleus to the density of the complete hydrogen atom?

When an electron makes a transition between energy levels of an atom, there are no restrictions on the initial and final values of the principal quantum number \(n\). According to quantum mechanics, however, there is a rule that restricts the initial and final values of the orbital quantum number \(\ell\). This rule is called a selection rule and states that \(\Delta \ell=\pm 1 .\) In other words, when an electron makes a transition between energy levels, the value of \(\ell\) can only increase or decrease by one. The value of \(\ell\) may not remain the same nor increase or decrease by more than one. According to this rule, which of the following energy level transitions are allowed: (a) \(2 \mathrm{~s} \rightarrow 1 \mathrm{~s}\), (b) \(2 \mathrm{p} \rightarrow 1 \mathrm{~s}\), (c) \(4 \mathrm{p} \rightarrow 2 \mathrm{p}\), (d) \(4 \mathrm{~s} \rightarrow 2 \mathrm{p}\), and (e) \(3 \mathrm{~d} \rightarrow 3 \mathrm{~s} ?\)

Answer the following questions using the quantum mechanical model of the atom. (a) For a given principal quantum number \(n\), what values of the angular momentum quantum number \(\ell\) are possible? (b) For a given angular momentum, what values of the magnetic quantum number \(m_{\ell}\) are possible? Which of the following subshell configurations are not allowed? For those that are not allowed, give the reason why. (a) \(3 \mathrm{~s}^{1}\), (b) \(2 \mathrm{~d}^{2}\) (c) \(3 \mathrm{~s}^{4}\) (d) \(4 \mathrm{p}^{8}\) (e) \(5 \mathrm{f}^{12}\)

The \(K_{\beta}\) characteristic \(\mathrm{X}\) -ray line for tungsten has a wavelength of \(1.84 \times 10^{-11} \mathrm{~m} .\) What is the difference in energy between the two energy levels that give rise to this line? Express the answer in (a) joules and (b) electron volts.

The maximum value for the magnetic quantum number in state \(\mathrm{A}\) is \(m_{\ell}=2,\) while in state \(\mathrm{B}\) it is \(m_{p}=1 .\) What is the ratio \(L_{\mathrm{A}} / L_{\mathrm{B}}\) of the magnitudes of the orbital angular momenta of an electron in these two states?
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