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Chapter 7: Problem 5

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 X₂ molecule is equal to d₂.

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 A₂X₄.
02

Find the bonding atomic radii of A and X in terms of d₁ and d₂

According to the given information, the A-A bond length is d₁, and the A-X bond length is d₂. 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) = d₁ / 2 Bonding atomic radius of X (r_X) = d₂ / 2
03

Predict the X-X bond length for an X₂ molecule

When two X atoms form a bond in an X₂ molecule, the bonding atomic radii of both X atoms contribute to the bond length. Hence, the X-X bond length (d₃) can be calculated as: d₃ = Bonding atomic radius of X (r_X) + Bonding atomic radius of X (r_X) d₃ = r_X + r_X As we already defined r_X in step 2, we can substitute its value to calculate d₃: d₃ = (d₂ / 2) + (d₂ / 2) d₃ = d₂ Therefore, the X-X bond length for an X₂ molecule is predicted to be equal to d₂.

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Most popular questions from this chapter

The following observations are made about two hypothetical elements \(\mathrm{A}\) and \(\mathrm{B}:\) The \(\mathrm{A}-\mathrm{A}\) and \(\mathrm{B}-\mathrm{B}\) bond lengths in the elemental forms of \(\mathrm{A}\) and \(\mathrm{B}\) are 236 and \(194 \mathrm{pm}\), respectively. A and \(B\) react to form the binary compound \(A B_{2}\), which has a linear structure (that is \(\left.\angle \mathrm{B}-\mathrm{A}-\mathrm{B}=180^{\circ}\right) .\) Based on these statements, predict the separation between the two B nuclei in a molecule of \(\mathrm{AB}_{2}\).

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