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Consider the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule depicted here, where \(\mathrm{A}\) and \(\mathrm{X}\) are elements. The \(A-A\) bond length in this molecule is \(d_{1}\), and the four \(\mathrm{A}-\mathrm{X}\) bond lengths are each \(d_{2}\). (a) In terms of \(d_{1}\) and \(d_{2},\) how could you define the bonding atomic radii of atoms A and X? (b) In terms of \(d_{1}\) and \(d_{2}\), what would you predict for the \(X-X\) bond length of an \(X_{2}\) molecule? [Section \(\left.7.3\right]\)

Short Answer

Expert verified
The bonding atomic radii of atoms A and X in terms of the given bond lengths are: Bonding atomic radius of A (r_A) = \( \frac{d_1}{2} \) and Bonding atomic radius of X (r_X) = \( \frac{d_2}{2} \). The predicted X-X bond length for an X鈧 molecule is equal to d鈧.

Step by step solution

01

Define bonding atomic radii

Bonding atomic radii are the distance between the nucleus of two bonded atoms, divided by two. In other words, it's the distance from the nucleus to the point where the electron cloud of the bonded atom's outermost shell makes contact with another atom. When two atoms are bonded, the bonding atomic radius is half of the bond length between them. Now let's apply this concept to the given molecule A鈧俋鈧.
02

Find the bonding atomic radii of A and X in terms of d鈧 and d鈧

According to the given information, the A-A bond length is d鈧, and the A-X bond length is d鈧. Using the definition of bonding atomic radii, we can define the bonding atomic radii of atoms A and X as follows: Bonding atomic radius of A (r_A) = d鈧 / 2 Bonding atomic radius of X (r_X) = d鈧 / 2
03

Predict the X-X bond length for an X鈧 molecule

When two X atoms form a bond in an X鈧 molecule, the bonding atomic radii of both X atoms contribute to the bond length. Hence, the X-X bond length (d鈧) can be calculated as: d鈧 = Bonding atomic radius of X (r_X) + Bonding atomic radius of X (r_X) d鈧 = r_X + r_X As we already defined r_X in step 2, we can substitute its value to calculate d鈧: d鈧 = (d鈧 / 2) + (d鈧 / 2) d鈧 = d鈧 Therefore, the X-X bond length for an X鈧 molecule is predicted to be equal to d鈧.

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Key Concepts

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

A2X4 molecule
In chemistry, an \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule refers to a particular molecular structure where two atoms of \(\mathrm{A}\) are connected to four atoms of \(\mathrm{X}\). This molecular arrangement can lead to various geometric shapes depending on the spatial orientation of the bonds. Typically, such molecules are studied to understand how the different types of atoms interact and bond within the structure.
The molecular formula, \(\mathrm{A}_{2} \mathrm{X}_{4}\), indicates that for every two atoms of \(\mathrm{A}\), there are four atoms of \(\mathrm{X}\). Thus, each \(\mathrm{A}\) atom is likely bonded to two \(\mathrm{X}\) atoms. This helps define the internal symmetry and bond distribution within the molecule, which can affect both the physical and chemical properties.
Understanding molecules like \(\mathrm{A}_{2} \mathrm{X}_{4}\) is particularly useful for predicting reactivity and bonding patterns in complex compounds. It demonstrates how variations in molecular composition can result in different chemical behaviors.
  • Visualizing the structure helps in predicting bonding interactions.
  • Each \(\mathrm{A}\) atom connects potentially in a symmetric or asymmetric pattern.
  • Considerations include the angles between bonds and the energies associated with these arrangements.
Bond length
Bond length is a fundamental concept in chemistry that represents the average distance between the nuclei of two bonded atoms. In the context of \(\mathrm{A}_{2} \mathrm{X}_{4}\), there are two types of bond lengths: the \(A-A\) bond length \(d_1\) and the \(A-X\) bond length \(d_2\). These bond lengths can tell us a lot about the strength and the nature of the bonds.
In general, shorter bond lengths suggest stronger bonds due to the nuclei being closer together, which allows the sharing of electrons to be stronger. For instance, the \(A-A\) bond length \(d_1\) in an \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule might indicate a particular type of interaction specific to atoms \(A\).
Similarly, the \(A-X\) bond length \(d_2\) indicates how the atoms of \(A\) and \(X\) are bonded. It suggests the extent of interaction and overlap between the electron clouds of \(A\) and \(X\).
  • Shorter lengths typically indicate stronger and more stable bonds.
  • Longer bonds usually result from weaker interactions.
  • Each type of bond length in a molecule contributes to its overall geometric structure.
Atomic structure
Atomic structure refers to the composition and arrangement of subatomic particles within an atom. Each atom comprises a nucleus made of protons and neutrons, with electrons orbiting this nucleus in various energy levels or shells. The arrangement of these electrons significantly influences how atoms bond and form molecules.
In molecules like \(\mathrm{A}_{2} \mathrm{X}_{4}\), understanding the atomic structure of elements \(A\) and \(X\) is crucial. The number of valence electrons, which are the electrons in the outer shell, determines how an atom can bond with others. For example, if \(A\) has two valence electrons available for bonding, this would affect the type of bonds it can form with \(X\).
This structure leads to specific properties and reactivity patterns. By exploring the atomic structure, we gain insight into why certain atoms form particular bonds and how these bonds dictate the molecule's stability and reactivity.
  • Protons determine the elemental identity and are equal to the atomic number.
  • Neutrons contribute to the atomic mass but not to the charge.
  • Electrons form the outer electron cloud, dictating bonding behaviors.
Chemical bonding
Chemical bonding is the process through which atoms combine to form molecules. It arises due to interactions between the electromagnetic forces of the participating atoms' electrons and nuclei. There are several types of chemical bonds, including covalent, ionic, and metallic bonds. Each type has distinct characteristics and leads to different molecular shapes and properties.
In the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule, chemical bonding involves the formation of bonds between \(A\) atoms and \(X\) atoms. These bonds could be covalent if there's a sharing of valence electrons or ionic if electrons are transferred from one atom to another, creating ions.
The type of bond impacts how easily a molecule will react with other substances, its melting and boiling points, and its physical state at room temperature. Being able to predict these properties based on bonding helps chemists understand how molecules will behave in different environments.
  • Covalent bonds involve sharing of electron pairs between atoms.
  • Ionic bonds occur when electrons are transferred, creating charged ions.
  • Complex molecules can contain a mixture of bond types.

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Most popular questions from this chapter

Write balanced equations for the following reactions: (a) boron trichloride with water, (b) cobalt (II) oxide with nitric acid, (c) phosphorus pentoxide with water, (d) carbon dioxide with aqueous barium hydroxide.

It is possible to define metallic character as we do in this book and base it on the reactivity of the element and the ease with which it loses electrons. Alternatively, one could measure how well electricity is conducted by each of the elements to determine how "metallic" the elements are. On the basis of conductivity, there is not much of a trend in the periodic table: Silver is the most conductive metal, and manganese the least. Look up the first ionization energies of silver and manganese; which of these two elements would you call more metallic based on the way we define it in this book?

(a) What is the trend in first ionization energies as one proceeds down the group 17 elements? Explain how this trend relates to the variation in atomic radii. (b) What is the trend in first ionization energies as one moves across the fourth period from \(\mathrm{K}\) to \(\mathrm{Kr}\) ? How does this trend compare with the trend in atomic radii?

Consider the stable elements through lead \((Z=82) .\) In how many instances are the atomic weights of the elements out of order relative to the atomic numbers of the elements?

Elemental barium reacts more violently with water than does elemental calcium. Which of the following best explains this difference in reactivity? (i) Calcium has greater metallic character than does barium. (ii) The electron affinity of calcium is smaller than that of barium. (iii) The first and second ionization energies of barium are less than those of calcium. (iv) The atomic radius of barium is smaller than that of calcium. (v) The ionic radius of the barium ion is larger than that of the calcium ion.

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