Log out Go to App
Go to App

Learning Materials

Features

Discover

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: \[V_{interior} = \frac{4}{3}\pi \left(\frac{\sqrt{3}(4r) - 4r}{2}\right)^3\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You have three covalent compounds with three very different boiling points. All of the compounds have similar molar mass and relative shape. Explain how these three compounds could have very different boiling points.

The compounds \(\mathrm{Na}_{2} \mathrm{O}, \mathrm{CdS}\), and \(\mathrm{Zr} \mathrm{I}_{4}\) all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case?

Some ionic compounds contain a mixture of different charged cations. For example, some titanium oxides contain a mixture of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions. Consider a certain oxide of titanium that is \(28.31 \%\) oxygen by mass and contains a mixture of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions. Determine the formula of the compound and the relative numbers of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions.

Why are the dipole-dipole interactions between polar molecules not important in the vapor phase?

When 1 mol benzene is vaporized at a constant pressure of \(1.00\) atm and at its boiling point of \(353.0 \mathrm{~K}, 30.79 \mathrm{~kJ}\) of energy (heat) is absorbed and the volume change is \(+28.90 \mathrm{~L}\). What are \(\Delta E\) and \(\Delta H\) for this process?
See all solutions

Recommended explanations on Chemistry Textbooks

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Kinetics

Read Explanation

Chemistry Branches

Read Explanation

Inorganic Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free