Question

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 the available space. If only 74 percent of the space inside the cube is taken up by \(\mathrm{Mg}\) atoms, calculate the radius in picometers of an \(\mathrm{Mg}\) atom. (The density of \(\mathrm{Mg}\) is \(1.74 \mathrm{~g} / \mathrm{cm}^{3},\) and the volume of a sphere of radius \(r\) is \(\left.\frac{4}{3} \pi r^{3} .\right)\)

Short answer

(a) \( 4.31 \times 10^{22} \) atoms; (b) Radius \( \approx 160 \) pm.

Step-by-step solution

Calculate the Mass of the Magnesium Cube

The volume of the cube is calculated using the formula for the volume of a cube, which is the cube of the edge length. Given the edge length of 1.0 cm, the volume is \( 1.0^3 = 1.0 \text{ cm}^3 \). Given the density of magnesium, \( 1.74 \text{ g/cm}^3 \), use the formula \( \text{Mass} = \text{Density} \times \text{Volume} \) to find that the mass of the cube is \( 1.74 \text{ g} \).

Calculate the Number of Moles of Magnesium in the Cube

To determine the number of moles, use the equation \( \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \). The molar mass of magnesium is approximately \( 24.305 \text{ g/mol} \). Therefore, \( \text{Moles of Mg} = \frac{1.74 \text{ g}}{24.305 \text{ g/mol}} \approx 0.0716 \text{ mol} \).

Calculate the Number of Magnesium Atoms in the Cube

Avogadro's number (approximately \( 6.022 \times 10^{23} \text{ atoms/mol} \)) gives the number of atoms per mole. Multiply Avogadro's number by the number of moles calculated in the previous step: \( 0.0716 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 4.31 \times 10^{22} \text{ atoms} \).

Determine the Total Volume Occupied by Magnesium Atoms

Since only 74% of the cube's volume is occupied by the magnesium atoms, calculate this occupied volume as \( 0.74 \times 1.0 \text{ cm}^3 = 0.74 \text{ cm}^3 \).

Calculate the Volume of a Single Magnesium Atom

Divide the total occupied volume by the total number of magnesium atoms to find the volume of a single magnesium atom: \( \frac{0.74 \text{ cm}^3}{4.31 \times 10^{22}} \approx 1.72 \times 10^{-23} \text{ cm}^3 \) per atom.

Calculate the Radius of a Magnesium Atom in Picometers

The volume of a sphere is given by \( \frac{4}{3}\pi r^3 \). Set \( \frac{4}{3}\pi r^3 \) equal to the volume of a single atom found in the previous step. Solving for \( r \), \( r = \left(\frac{3 \times 1.72 \times 10^{-23} \text{ cm}^3}{4\pi}\right)^{1/3} \approx 1.60 \times 10^{-8} \text{ cm} \). Converting to picometers (1 cm = \( 10^{10} \) pm), \( r \approx 160 \text{ pm} \).

Key concepts

Cube Volume Calculation

Understanding how to calculate the volume of a cube is essential in this exercise. A cube is a three-dimensional shape with equal edge lengths. The formula to find its volume is relatively simple. You raise the edge length to the third power. In this case, the edge is given as 1.0 cm. So, the volume is calculated as \(1.0^3 = 1.0 \, \text{cm}^3\). This calculation is the first step in determining other properties of the cube filled with magnesium.Knowing this volume allows you to further calculate other needed quantities, such as the mass of the magnesium cube.

Density of Magnesium

Density is a key concept in physics and chemistry. It indicates how much mass is contained in a unit volume. It's given in units of grams per cubic centimeter (g/cm³). For magnesium, it is provided as 1.74 g/cm³.To find the mass of the cube, use the formula:\[ \text{Mass} = \text{Density} \times \text{Volume} \]Since we know the volume is 1.0 cm³, the mass is simply:\(1.74 \, \text{g/cm}^3 \times 1.0 \, \text{cm}^3 = 1.74 \, \text{g}\).Density helps us connect the physical size of an object to its mass, facilitating further calculations like determining the number of moles.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry. It provides a bridge between the macroscopic scale and the atomic scale by telling us the number of atoms or molecules in a mole, approximately \(6.022 \times 10^{23}\) units.To find how many atoms of magnesium are in our cube, we need to first determine how many moles are present. Using the formula:\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \]With a mass of 1.74 g and molar mass of magnesium as 24.305 g/mol, the moles are about 0.0716. Then multiply by Avogadro’s number to find the atoms:\(0.0716 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 4.31 \times 10^{22} \, \text{atoms}\).

Sphere Volume Formula

Atoms are often modeled as spheres. To understand how much space an atom occupies, use the sphere volume formula:\[ \text{Volume} = \frac{4}{3}\pi r^3 \]In the problem, we need to determine the radius of a magnesium atom. Since only 74% of the cube's volume is filled by the atoms, calculate the occupied volume:\(0.74 \, \text{cm}^3\).The volume of a single atom then becomes \(1.72 \times 10^{-23} \, \text{cm}^3\).Using the sphere volume formula and solving for the radius \(r\), we transform the volume equation:\[ r = \left(\frac{3 \times 1.72 \times 10^{-23} \, \text{cm}^3}{4\pi}\right)^{1/3} \approx 1.60 \times 10^{-8} \, \text{cm} \]Convert to picometers for clarity:\(1.60 \times 10^{-8} \, \text{cm} = 160 \, \text{pm}\). This conversion is vital, as chemistry often uses picometers for atomic scale measurements.