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Chapter 9: Problem 68

Plot a graph of the square roots of the ionization energies versus the nuclear charge for the two series \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+},\) and \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+} .\) Explain the observed relationship with the aid of Bohr's expression for the binding energy of an electron in a one electron atom.

Short Answer

Expert verified
First identify the ionization energies for each ion in both series and calculate their square roots. Identify the nuclear charge for each ion. Using this information, plot square root of ionization energy against nuclear charge for both series. The trend in this plot can be explained by Bohr's expression for the binding energy in a hydrogen-like atom.

Step by step solution

01

Identify Ionization Energies

The ionization energies for ions in the series \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+}\) and \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+}\) are identified. These would need to be researched or provided by a periodic table or a reliable database.
02

Calculate Square Roots of Ionization Energies

Calculate the square root of the ionization energy for each ion in the series. Remember that the unit of ionization energy is typically electronvolts (\(eV\)) or joules (\(J\)). Make sure to keep the units consistent!
03

Identify Nuclear Charges

Identify the nuclear charge for each ion in the series. The nuclear charge corresponds to the atomic number, which is the number of protons in the nucleus of an atom. Recall that for a net positive ion, the nuclear charge is higher than the number of electrons. For instance, the \(Be^{+}\) ion has a nuclear charge of 4.
04

Plot the Graph

With the above-collected information, plot the graph with the square root of the ionization energy on the y-axis and the nuclear charge on the x-axis. Two separate series are to be plotted: one for ions \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+}\), and one for \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+}\).
05

Analyze and Explain the Graph Using Bohr’s Theory

Analyze the obtained plots and use Bohr's expression for the binding energy of an electron to explain the observed relationship. According to Bohr’s model, the binding energy of an electron in a one-electron atom is given by \(E = -\frac{13.6eV}{n^2} \times Z^2\), where \(E\) is the energy, \(n\) is the quantum number (principle shell), and \(Z\) is the atomic (nuclear) number. This equation suggests that the energy required to remove an electron (ionization energy) from an atom is directly proportional to the square of the atomic number, which can explain the trend observed in the graph.

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Most popular questions from this chapter

The fourth-period element with the largest atom is (a) \(\mathrm{K} ;\) (b) \(\mathrm{Br} ;\) (c) \(\mathrm{Pb} ;\) (d) \(\mathrm{Kr}\).

In multielectron atoms many of the periodic trends can be explained in terms of \(Z_{\text {eff }}\) Consider the following statements and discuss whether or not the statement is true or false. (a) Electrons in a \(p\) orbital are more effective than electrons in the \(s\) orbitals in shielding other electrons from the nuclear charge. (b) \(\mathrm{Z}_{\text {eff }}\) for an electron in an \(s\) orbital is lower than that for an electron in a \(p\) orbital in the same shell. (c) \(Z_{\text {eff }}\) is usually less than \(Z.\) (d) Electrons in orbitals having \(\ell=1\) penetrate better than those with \(\ell=2.\) (e) \(\mathrm{Z}_{\text {eff }}\) for the orbitals of the elements \(\mathrm{Na}(3 s)\) \(\mathrm{Mg}(3 s), \mathrm{Al}(3 p), \mathrm{P}(3 p),\) and \(\mathrm{S}(3 p)\) are in the order \(Z_{\text {eff }}(\mathrm{Na})<\mathrm{Z}_{\text {eff }}(\mathrm{Mg})>\mathrm{Z}_{\text {eff }}(\mathrm{Al})<\mathrm{Z}_{\text {eff }}(\mathrm{P})>\mathrm{Z}_{\text {eff }}(\mathrm{S}).\)

Compare the elements \(\mathrm{Al}, \mathrm{Si}, \mathrm{S},\) and \(\mathrm{Cl}.\) (a) Place the elements in order of increasing ionization energy. (b) Place the elements in order of increasing electron affinity.

For the atom \(^{119} \mathrm{Sn}\), indicate the number of (a) protons in the nucleus; (b) neutrons in the nucleus; (c) \(4 d\) electrons; (d) 3s electrons; (e) 5 \(p\) electrons; (f) electrons in the valence shell.

Indicate the smallest and the largest species (atom or ion) in the following group: Al atom, F atom, As atom, \(\mathrm{Cs}^{+}\) ion, \(\mathrm{I}^{-}\) ion, \(\mathrm{N}\) atom.
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