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 Mg atoms in the cube cannot fill all 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 an $\mathrm{Mg}$ atom. (The density of $\mathrm{Mg}$ is $\left.1.74 \mathrm{g} / \mathrm{cm}^{3}, \text { and the volume of a sphere of radius } r \text { is } \frac{4}{3} \pi r^{3} .\right)$

Key Concepts

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.

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.

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.

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.

Transcript

00:01 So for this problem, we have a cube of magnesium, and we want to find for part a the number of magnesium atoms in that cube.

00:11 So we know that the volume of the cube is just going to be the edge length cubed.

00:17 So we take this value and you cube it, and we'll get one cubic centimeter.

00:26 So that's the volume of our cube.

00:28 And now we want to convert the volume to atoms of mg.

00:34 So this is our starting unit.

00:37 Now we want to convert this to magnesium somehow.

00:40 And if you look up the density of magnesium, you'll get that the density is 1 .74 grams of magnesium per 1 centimeter cubed.

00:55 Now that we have grams of magnesium, we can then convert this to moles by dividing by its atomic mass, which is 24 .3 grams.

01:07 And then finally multiplying by abogadro's number to get atoms of mg.

01:20 So when we multiply all this through, you get 4 .3 times 10 to the 22 atoms of mg.

01:35 So that is the final answer to part a.

01:39 Now for part b.

01:40 Part b is a little longer, and we want to find the radius in picometers of a magnesium atom.

01:50 So just before we get into the actual problem, we want to find defined units, which is going to be picometers.

01:57 So a picometer is just one times 10 to the 12th of a meter.

02:06 So that's going to be an important note when we get to our units later.

02:12 So we know that our first conversion is just going to be an atom of mg, one atom of magnesium.

02:23 And then, we know that we can convert this to moles of magnesium by dividing by abeladha's number.

02:36 And now for moles, we want to get grams.

02:45 And we can multiply by the atomic mass, which is 24 .31...

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