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Chapter 18: Problem 21

Which unit cell has the greater packing efficiency, simple cubic or body- centered cubic?

Short Answer

Expert verified
Answer: The body-centered cubic (BCC) unit cell has a greater packing efficiency than the simple cubic (SC) unit cell.

Step by step solution

01

Determine the number of spheres in the unit cells

For a simple cubic unit cell, there is only one sphere located at each corner of the cube. Since there are 8 corners, and each corner is shared by 8 adjacent cells, there is a total of (8 corners 脳 1/8) = 1 sphere in the unit cell. For a body-centered cubic unit cell, there is one sphere at each corner of the cube (8 corners 脳 1/8) and one sphere at the center of the cube, giving a total of (8 corners 脳 1/8) + 1 = 2 spheres in the unit cell.
02

Calculate the volume of a sphere

The volume of a sphere can be calculated using the formula: Volume = (4/3) 脳 蟺 脳 r鲁, where r is the radius of a sphere (or atom). Since both the SC and BCC unit cells have spheres (atoms) of equal size, we can simplify our calculations by comparing their packing efficiency with respect to their respective volume ratios.
03

Determine the edge length and total volume of the unit cells

For a simple cubic unit cell, the edge length (a) is equal to 2r, where r is the radius of the sphere (or atom). The total volume of the unit cell can be calculated as: Total volume (SC) = a鲁 = (2r)鲁 = 8r鲁 For a body-centered cubic unit cell, the edge length (a) is related to the radius by the equation: a = 4r/鈭�3 The total volume of the unit cell can be calculated as: Total volume (BCC) = a鲁 = (4r/鈭�3)鲁 = 64r鲁/3鈭�3
04

Calculate packing efficiency

Now we can calculate the packing efficiency for both unit cells: Packing Efficiency (SC) = (Volume of spheres in SC / Total volume of SC) 脳 100 = ((4/3)蟺r鲁 / 8r鲁) 脳 100 鈮� 52.4% Packing Efficiency (BCC) = (Volume of spheres in BCC / Total volume of BCC) 脳 100 = (2 脳 (4/3)蟺r鲁 / (64r鲁/3鈭�3)) 脳 100 鈮� 68.0%
05

Compare packing efficiencies

Comparing the packing efficiencies of the simple cubic (52.4%) and body-centered cubic (68.0%) unit cells, we can conclude that the body-centered cubic unit cell has a greater packing efficiency than the simple cubic unit cell.

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Most popular questions from this chapter

In the fullerene known as buckminsterfullerene, molecules of \(\mathrm{C}_{60}\) form a cubic closest-packed array of spheres with a unit cell edge length of \(1410 \mathrm{pm}\). a. What is the density of crystalline \(C_{60} ?\) b. If we treat each \(\mathrm{C}_{60}\) molecule as a sphere of 60 carbon atoms, what is the radius of the \(C_{60}\) molecule? c. \(C_{60}\) reacts with alkali metals to form \(\mathrm{M}_{3} \mathrm{C}_{60}\) (where \(\mathrm{M}=\mathrm{Na} \text { or } \mathrm{K}) .\) The crystal structure of \(\mathrm{M}_{3} \mathrm{C}_{60}\) contains cubic closest-packed spheres of \(C_{60}\) with metal ions in holes. If the radius of a \(\mathrm{K}^{+}\) ion is \(138 \mathrm{pm},\) which type of hole is a \(\mathrm{K}^{+}\) ion likely to occupy? What fraction of the holes will be occupied? d. Under certain conditions, a different substance, \(\mathrm{K}_{6} \mathrm{C}_{60}\) can be formed in which the \(\mathrm{C}_{60}\) molecules have a bcc unit cell. Calculate the density of a crystal of \(\mathrm{K}_{6} \mathrm{C}_{60}\)

A number of crystalline transition metals (including titanium, zirconium, and hafnium) can store hydrogen as metal hydrides for use as fuel in a hydrogen- powered vehicle. Are the H atoms (radius \(37 \mathrm{pm}\) ) more likely to be in the tetrahedral or octahedral holes of these three metals whose atomic radii are \(147,160,\) and \(159 \mathrm{pm},\) respectively?

As the cation-anion radius ratio increases for an ionic compound with the rock salt crystal structure, is the calculated density more likely to be greater than or less than the measured value?

Kaolinite \(\left[\mathrm{Al}_{2}\left(\mathrm{Si}_{2} \mathrm{O}_{5}\right)(\mathrm{OH})_{4}\right]\) is formed by weathering of the mineral \(\mathrm{KAlSi}_{3} \mathrm{O}_{8}\) in the presence of carbon dioxide and water, as described by the following unbalanced reaction: \(\mathrm{KAISi}_{3} \mathrm{O}_{8}(s)+\mathrm{H}_{2} \mathrm{O}(\ell)+\mathrm{CO}_{2}(g) \rightarrow\) $$ \mathrm{Al}_{2}\left(\mathrm{Si}_{2} \mathrm{O}_{5}\right)(\mathrm{OH})_{4}(s)+\mathrm{SiO}_{2}(s)+\mathrm{K}_{2} \mathrm{CO}_{3}(a q) $$ Balance the reaction and determine whether or not it is a redox reaction.

A chemical reaction between \(\mathrm{H}_{2} \mathrm{S}_{4}\) and \(\mathrm{S}_{2} \mathrm{Cl}_{2}\) produces cyclic \(S_{6} .\) What are the bond angles in \(S_{6} ?\)
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