Question

A certain form of lead has a cubic closest packed structure with an edge length of \(492 \mathrm{pm} .\) Calculate the value of the atomic radius and the density of lead.

Short answer

Step 3: Solve for the atomic radius Divide both sides of the equation by \(4\sqrt{2}\) to isolate r: \(r = \frac{492}{4\sqrt{2}}\) Calculating the value, we get: \(r \approx 173.5 \mathrm{pm}\) So, the atomic radius of the cubic closest packed lead structure is approximately 173.5 pm. #tag_title# Step 4: Calculate the density#tag_content# To calculate the density of this lead structure, we will use the formula: \(Density = \frac{mass}{volume} = \frac{mass}{a^3}\) In a cubic closest packed structure, there are four equivalent atoms per unit cell. The molar mass of lead (Pb) is 207.2 g/mol. We can determine the mass of these four atoms in the unit cell by using Avogadro's number (\(6.022 \times 10^{23} \) atoms/mol): \(mass = 4 \cdot \frac{207.2 \mathrm{g/mol}}{6.022 \times 10^{23}\, \mathrm{atoms/mol}}\) Now, we can calculate the volume of the unit cell by using the given edge length: \(volume = a^3 = (492 \mathrm{pm})^3 \) To convert the volume from cubic picometers to cubic centimeters, we can use the following conversion factor: \(1 \mathrm{cm} = 10^{12} \mathrm{pm}\) \(volume = \frac{(492 \mathrm{pm})^3}{(10^{12}\, \mathrm{pm/cm})^3} = \frac{(492 \times 10^{-12}\, \mathrm{cm})^3}{1^3\, \mathrm{cm^3}}\) Finally, we can plug the mass and volume values into the density formula: \(Density = \frac{4 \cdot \frac{207.2\, \mathrm{g/mol}}{6.022 \times 10^{23} \, \mathrm{atoms/mol}}}{\left(\frac{492 \times 10^{-12}\, \mathrm{cm}}{1\, \mathrm{cm}}\right)^3} \approx 11.34\, \mathrm{g/cm^3}\) So, the density of the cubic closest packed lead structure is approximately 11.34 g/cm³.

Step-by-step solution

Calculate the atomic radius

In a cubic closest packed (ccp) structure, atoms occupy the corners and face centers of a cubic unit cell. In this structure, the atoms are arranged in a way that the distance between them is minimized, and they are packed as closely as possible. In a ccp structure, there's a geometrical relationship between the edge length (a) of the unit cell and the atomic radius (r) of the atom. That relationship can be described by the following equation: \(4r = \sqrt{2} \cdot a\) Now, we are given that the edge length of the ccp structure is 492 pm. We can plug this value into the equation to determine the atomic radius: Step 2: Plug the given values into the equation \(4r = \sqrt{2} \cdot 492\)