Question

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

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

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.

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: \[L = 2(2r) = 4r\]

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: \[d = D - 4r\] \[d = \sqrt{3}(4r) - 4r\]

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