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Chapter 10: Problem 139

For a simple cubic array, solve for the volume of an interior sphere (cubic hole) in terms of the radius of a sphere in the array.

Short Answer

Expert verified
The volume of the interior sphere (cubic hole) in terms of the radius of a sphere in the array is given by: \[V_{interior} = \frac{4}{3}\pi \left(\frac{\sqrt{3}(4r) - 4r}{2}\right)^3\]

Step by step solution

01

Understanding the Simple Cubic Array Arrangement

In a simple cubic array, sphere particles are arranged such that there are equal spaces between each pair of adjacent spheres in all three dimensions of the cube. In this scenario, we will consider that each sphere is touching one another creating a hole between them.
02

Calculate the Length of the Cubic Edge

Since the spheres are touching each other, the length of the cubic edge would be equal to the sum of their diameters. Let's denote the radius of the sphere as r. The diameter of one sphere is 2r. As there are two diameters along the edge of the cubic array, the length of the cubic edge(L) would be: \[L = 2(2r) = 4r\]
03

Determine the Interior Sphere Diameter

Since our goal is to find the sphere in the cubic hole's volume, we need to find the diameter(d) of the interior sphere. To achieve this, we can use the Pythagorean theorem on the diagonal of the cubic array. The diagonal length is the diameter of the interior sphere plus the diameters of two spheres located on opposite corners of the diagonal: \[D = d + 2(2r)\] \[D^2 = L^2 + L^2 + L^2 \] \[D = \sqrt{3}L\] Now, plug the relationship between L and r: \[D = \sqrt{3}(4r)\] Now, we can derive the diameter of the interior sphere (d) in terms of the radius r: \[d = D - 4r\] \[d = \sqrt{3}(4r) - 4r\]
04

Calculate the Volume of the Interior Sphere

To determine the volume, we need to find the radius of the interior sphere first. Divide the diameter d by 2: \[r_{interior} = \frac{d}{2} = \frac{\sqrt{3}(4r) - 4r}{2}\] Now we can find the volume of the interior sphere using the volume formula for a sphere: \[V_{interior} = \frac{4}{3}\pi r_{interior}^3\] \[V_{interior} = \frac{4}{3}\pi \left(\frac{\sqrt{3}(4r) - 4r}{2}\right)^3\] So, the volume of the interior sphere (cubic hole) in terms of the radius of a sphere in the array is: \[V_{interior} = \frac{4}{3}\pi \left(\frac{\sqrt{3}(4r) - 4r}{2}\right)^3\]

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Most popular questions from this chapter

General Zod has sold Lex Luthor what Zod claims to be a new copper-colored form of kryptonite, the only substance that can harm Superman. Lex, not believing in honor among thieves, decided to carry out some tests on the supposed kryptonite. From previous tests, Lex knew that kryptonite is a metal having a specific heat capacity of \(0.082 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\), and a density of \(9.2 \mathrm{~g} / \mathrm{cm}^{3}\). Lex Luthor's first experiment was an attempt to find the specific heat capacity of kryptonite. He dropped a \(10 \mathrm{~g} \pm 3 \mathrm{~g}\) sample of the metal into a boiling water bath at a temperature of \(100.0^{\circ} \mathrm{C} \pm 0.2^{\circ} \mathrm{C} .\) He waited until the metal had reached the bath temperature and then quickly transferred it to \(100 \mathrm{~g} \pm 3 \mathrm{~g}\) of water that was contained in a calorimeter at an initial temperature of \(25.0^{\circ} \mathrm{C} \pm 0.2^{\circ} \mathrm{C}\). The final temperature of the metal and water was \(25.2^{\circ} \mathrm{C}\). Based on these results, is it possible to distinguish between copper and kryptonite? Explain. When Lex found that his results from the first experiment were inconclusive, he decided to determine the density of the sample. He managed to steal a better balance and determined the mass of another portion of the purported kryptonite to be \(4 \mathrm{~g} \pm 1 \mathrm{~g} .\) He dropped this sample into water contained in a 25-mL graduated cylinder and found that it displaced a volume of \(0.42 \mathrm{~mL} \pm 0.02 \mathrm{~mL}\). Is the metal copper or kryptonite? Explain. Lex was finally forced to determine the crystal structure of the metal General Zod had given him. He found that the cubic unit cell contained 4 atoms and had an edge length of 600 . pm. Explain how this information enabled Lex to identify the metal as copper or kryptonite. Will Lex be going after Superman with the kryptonite or seeking revenge on General Zod? What improvements could he have made in his experimental techniques to avoid performing the crystal structure determination?

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MnO has either the \(\mathrm{NaCl}\) type structure or the \(\mathrm{CsCl}\) type structure (see Exercise 67). The edge length of the \(\mathrm{MnO}\) unit cell is \(4.47 \times 10^{-8} \mathrm{~cm}\) and the density of \(\mathrm{MnO}\) is \(5.28 \mathrm{~g} / \mathrm{cm}^{3}\) a. Does \(\mathrm{Mn} \mathrm{O}\) crystallize in the \(\mathrm{NaCl}\) or the \(\mathrm{CsCl}\) type structure? b. Assuming that the ionic radius of oxygen is \(140 . \mathrm{pm}\), estimate the ionic radius of manganese.

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