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Chapter 10: Problem 15

Atoms are assumed to touch in closest packed structures, yet every closest packed unit cell contains a significant amount of empty space. Why?

Short Answer

Expert verified
In closest-packed structures, such as hexagonal closed-packed (hcp) and face-centered cubic (fcc), atoms are arranged in a way that maximizes density and minimizes empty space. However, due to the spherical shape of atoms, it is geometrically inevitable that there will still be voids or interstitial sites within the unit cell. As a result, even in these optimal arrangements, the packing efficiency is only around 74.05%, meaning that approximately 25.95% of the volume remains empty.

Step by step solution

01

Understanding Closest-packed Structures

Closest-packed structures are arrangements of atoms that result in the highest possible density. In these structures, atoms are packed close to each other in a manner that minimizes empty space. There are two main types of closest-packed structures: hexagonal closed-packed (hcp) and face-centered cubic (fcc). Note that despite packing atoms close to each other, some empty space will always be present, as we will see in the next steps.
02

Estimating the Packing Efficiency

The packing efficiency is a measure of how much space is filled by atoms in a given structure. In the closest-packed structures, the highest possible packing efficiency is approximately 74.05%. This means that out of the total volume of the unit cell, about 74.05% is filled by atoms, with the remaining 25.95% being empty space. To understand why there is empty space, we should consider the nature of atoms and their arrangements.
03

Atoms and their Spherical Shape

Atoms can be modeled as hard spheres with a fixed diameter; this model helps us understand their packing behavior in solid structures. When trying to fill a space with spheres, there will always be empty space left, regardless of their arrangement. Imagine trying to fill a box with oranges; no matter how you place them, there will still be gaps between the oranges.
04

Atomic Arrangement in Closest-packed Structures

In closest-packed structures, atoms occupy positions that minimize empty space. For example, in both hcp and fcc structures, each atom has 12 neighboring atoms that touch it. However, even in these most optimized arrangements, there are unoccupied spaces within the unit cell. These spaces, called voids or interstitial sites, are geometrically inevitable, as multiple spheres can never be placed perfectly together with no space in between.
05

Conclusion

To answer the given exercise, even though atoms touch in closest-packed structures (hcp and fcc), there is still a significant amount of empty space within the unit cell. This is due to the spherical shape of atoms and the geometric imperfection when trying to fill a space with spheres. Even when atoms are optimally arranged in a closest-packed structure, there are still voids or interstitial sites that cause around 25.95% of the volume to be empty.

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Most popular questions from this chapter

Which are stronger, intermolecular or intramolecular forces for a given molecule? What observation(s) have you made that support this? Explain.

Nickel has a face-centered cubic unit cell. The density of nickel is \(6.84 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate a value for the atomic radius of nickel.

Iodine, like most substances, exhibits only three phases: solid. liquid, and vapor. The triple point of iodine is at 90 torr and \(115^{\circ} \mathrm{C}\). Which of the following statements concerning liquid \(\mathrm{I}_{2}\) must be true? Explain your answer. a. \(\mathrm{I}_{2}(l)\) is more dense than \(\mathrm{I}_{2}(g)\). b. \(\mathrm{I}_{2}(l)\) cannot exist above \(115^{\circ} \mathrm{C}\). c. \(\mathrm{I}_{2}(l)\) cannot exist at 1 atmosphere pressure. d. \(\mathrm{I}_{2}(l)\) cannot have a vapor pressure greater than 90 torr. e. \(\mathrm{I}_{2}(l)\) cannot exist at a pressure of 10 torr.

The molar enthalpy of vaporization of water at \(373 \mathrm{~K}\) and \(1.00\) atm is \(40.7 \mathrm{~kJ} / \mathrm{mol}\). What fraction of this energy is used to change the internal energy of the water, and what fraction is used to do work against the atmosphere? (Hint: Assume that water vapor is an ideal gas.)

What quantity of energy does it take to convert \(0.500 \mathrm{~kg}\) ice at \(-20 .{ }^{\circ} \mathrm{C}\) to steam at \(250 .{ }^{\circ} \mathrm{C}\) ? Specific heat capacities: ice, \(2.03 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ;\) liquid, \(4.2 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ;\) steam, \(2.0 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ; \Delta H_{\text {vep }}=\) 40.7 \(\mathrm{kJ} / \mathrm{mol} ; \Delta H_{\mathrm{fec}}=6.02 \mathrm{~kJ} / \mathrm{mol}\).
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