Problem 145 - Chemistry An Atoms First Approach

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.

Step-by-Step Solution

Answer

The volume of an interior sphere (cubic hole) in terms of the radius of a sphere in the simple cubic array is: \[V_{interior} = r^3(8 - \frac{32}{3}\pi)\]

Step 1: Express the Length of the Cube in terms of the Sphere's Radius

The cubic array contains eight identical spheres, which means there are two spheres along each edge of the cube. Since we need to find the volume of the interior space, we need to express the length of the cube in terms of the radius of the spheres. Let the radius of the sphere be r, then the length of the cube is 2r.

Step 2: Calculate the Volume of the Cube

Now that we know the length of the cube, we can calculate its volume. The volume of the cube (V_cube) is given by the formula: \[V_{cube} = l^3\] Where l is the length of the cube. In our case, l = 2r, so: \[V_{cube} = (2r)^3 = 8r^3\]

Step 3: Calculate the Volume of a Single Sphere

The cubic array contains eight identical spheres. The volume (V_sphere) of a single sphere is given by the formula: \[V_{sphere} = \frac{4}{3}\pi r^3\]

Step 4: Calculate the Total Volume of all Spheres in the Array

Now that we have the volume of a single sphere, we can calculate the total volume of all spheres in the array (V_all_spheres). Since there are eight identical spheres in the array: \[V_{all\_spheres} = 8 \cdot V_{sphere} = 8 \cdot \frac{4}{3}\pi r^3 = \frac{32}{3}\pi r^3\]

Step 5: Calculate the Volume of the Interior Space (Cubic Hole)

Finally, we can find the volume of the interior space (cubic hole, V_interior) by subtracting the total volume of all spheres from the volume of the cube: \[V_{interior} = V_{cube} - V_{all\_spheres} = 8r^3 - \frac{32}{3}\pi r^3 = r^3(8 - \frac{32}{3}\pi)\] The volume of an interior sphere (cubic hole) in terms of the radius of a sphere in the simple cubic array is: \[V_{interior} = r^3(8 - \frac{32}{3}\pi)\]