When spheres of radius r are packed in a simple cubic arrangement, they occupy 52.0 %...

When spheres of radius r are packed in a simple cubic arrangement, they occupy 52.0 % of the available volume. Use the fraction of occupied volume to calculate the value of a, the length of the edge of the cube in terms of r.

Expert Solution

In a simple cubic arrangement, the vertices of the cube coincides with the center of the spheres.

There are 8 spheres on the 8 vertices of the cube. Hence, only 1/8 th of each sphere lies inside the cube.

Volume of each sphere of radius r is .

Hence, 1/8 the of the volume of 8 spheres that lies inside the cube must equal the volume of 1 complete sphere.

Hence, the volume of all the spheres that lies inside the cube must be .

Now, the volume of the cube with edge length a is .

It is given that only 52% of the cube is occupied by the spheres.

Hence,

We can easily solve for the edge length a in terms of r from the above relation as shown below.

Hence, the relation between the edge length a and radius of sphere r is .

queen_honey_blossom answered 2 years ago

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