QuestionJune 4, 2025

Atoms are spherical in shape. Therefore, the Mg atoms in the cube cannot fill all the available space. If only 74.0 percent of the space inside the cube is taken up by Mg atoms, calculate the radius in picometers of an Mg atom. square pm

Atoms are spherical in shape. Therefore, the Mg atoms in the cube cannot fill all the available space. If only 74.0 percent of the space inside the cube is taken up by Mg atoms, calculate the radius in picometers of an Mg atom. square pm
Atoms are spherical in shape. Therefore, the Mg atoms in the cube cannot fill all the available space. If only 74.0 percent of the space inside the cube is taken up by Mg atoms, calculate the radius in picometers of an Mg atom. square pm

Solution
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Answer

160.9 pm Explanation 1. Calculate the volume occupied by Mg atoms Let the side length of the cube be a. The volume of the cube is a^3. Since 74.0% of the space is occupied, the volume occupied by Mg atoms is 0.74 \times a^3. 2. Relate volume to atomic radius Assume Mg atoms form a simple cubic structure. In this case, each atom occupies a sphere with volume \frac{4}{3}\pi r^3, where r is the radius of an Mg atom. For one atom per unit cell, a = 2r. 3. Solve for radius Equating the occupied volume to the volume of the sphere: 0.74 \times (2r)^3 = \frac{4}{3}\pi r^3. Simplify and solve for r: 0.74 \times 8r^3 = \frac{4}{3}\pi r^3. Thus, r^3 = \frac{0.74 \times 8}{\frac{4}{3}\pi}. Calculate r.

Explanation

1. Calculate the volume occupied by Mg atoms<br /> Let the side length of the cube be $a$. The volume of the cube is $a^3$. Since 74.0% of the space is occupied, the volume occupied by Mg atoms is $0.74 \times a^3$.<br /><br />2. Relate volume to atomic radius<br /> Assume Mg atoms form a simple cubic structure. In this case, each atom occupies a sphere with volume $\frac{4}{3}\pi r^3$, where $r$ is the radius of an Mg atom. For one atom per unit cell, $a = 2r$.<br /><br />3. Solve for radius<br /> Equating the occupied volume to the volume of the sphere: $0.74 \times (2r)^3 = \frac{4}{3}\pi r^3$. Simplify and solve for $r$: $0.74 \times 8r^3 = \frac{4}{3}\pi r^3$. Thus, $r^3 = \frac{0.74 \times 8}{\frac{4}{3}\pi}$. Calculate $r$.
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