Suppose you are given a cube made of magnesium (Mg) metal of...

Suppose you are given a cube made of magnesium (Mg) metal of edge length $1.0 \mathrm{~cm} .$ (a) Calculate the number of $\mathrm{Mg}$ atoms in the cube. (b) Atoms are spherical in shape. Therefore, the $\mathrm{Mg}$ atoms in the cube cannot fill all of the available space. If only 74 percent of the space inside the cube is taken up by Mg atoms, calculate the radius in picometers of a Mg atom. (The density of $\mathrm{Mg}$ is $1.74 \mathrm{~g} / \mathrm{cm}^{3}$ and the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3} .$

Suppose you are given a cube made of magnesium (Mg) metal of edge length $1.0 \mathrm{~cm} .$
(a) Calculate the number of $\mathrm{Mg}$ atoms in the cube.
(b) Atoms are spherical in shape. Therefore, the $\mathrm{Mg}$ atoms in the cube cannot fill all of the available space. If only 74 percent of the space inside the cube is taken up by Mg atoms, calculate the radius in picometers of a Mg atom. (The density of $\mathrm{Mg}$ is $1.74 \mathrm{~g} / \mathrm{cm}^{3}$ and the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3} .$

Key Concepts

Volume of a Sphere

The volume of a sphere is given by the formula (4/3)?r³, where r is the radius. This geometric relationship is essential when determining the size of an individual atom, once the total volume occupied by all atoms is established from macroscopic measurements.

Unit Conversions

Unit conversions are critical for ensuring that all quantities in a calculation are expressed in compatible units. In particular, converting lengths (such as from centimeters to picometers) is necessary when working between macroscopic measurements and microscopic properties, thereby ensuring accuracy in the final result.

Moles and Avogadro's Number

The mole is a unit that relates a quantity of substance to the number of constituent particles, with Avogadro's number defining the number of atoms or molecules in one mole. This concept is crucial for converting between the mass of a substance and the number of atoms it contains, linking macroscopic mass measurements to atomic-scale counts.

Density

Density is a measure of mass per unit volume, used to relate the mass of a material to the space it occupies. In problems involving materials, knowing the density allows you to calculate the mass of a known volume, or vice versa, which is essential when connecting macroscopic measurements to microscopic properties.

Atomic Packing Factor

The atomic packing factor refers to the fraction of space occupied by atoms in a given structure, taking into account that atoms are not perfect cubes but spheres and do not pack perfectly. It quantifies the efficiency of space usage in a material, which is key when translating overall volume into the actual volume occupied by atoms.

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