1. The eleent gold, Au, has a face-centered cubic structure. (Density - 19.3 g/cm^3)
(a) What are the # of gold atoms in 1 unit cell?
(b) What are the # of unit cells in 1 mol?
(c) What is the volume of 1 mol gold unit cells?
(d) What is the volume of 1 unit cell?
(e) what is the length of 1 side of a unit cell?
(f) What is the metallic diameter of 1 atom?
a. 4
b. There are 6.022E23 unit cells in 1 mol units cells.
c. volume = mass/density = 4*196.67/19.3 = ?
d. c/6.022E23 = ?
e. volume = a^3. Solve for a.
f. 4*radius = a*(2)^1/2. You want the diameter so that will be 2*r.
My answer is about 1.44E-8 cm for r or about 2.88E-8 cm for diameter but check my work.
So what would the units for (c) be?
And, what's the radius to work out (f)?
To answer these questions, we need to understand the structure of a face-centered cubic (FCC) unit cell and use some formulas.
(a) The face-centered cubic (FCC) structure consists of atoms at the corners of the unit cell and one atom at the center of each face. However, the atom shared by two adjacent unit cells counts as 1/2 in each unit cell. Therefore, there are 4 atoms at the corners (each corner is shared by 8 unit cells) and 6 atoms (1/2 atom in each unit cell) at the faces (each face is shared by 2 unit cells). So, the number of gold atoms in one unit cell is calculated as:
4 atoms x 1/8 + 6 atoms x 1/2 = 4(1/8) + 6(1/2) = 1 + 3 = 4 gold atoms.
(b) The molar mass of gold is 197.0 g/mol. Since 1 mol of a substance contains Avogadro's number (6.022 x 10^23) of particles, the number of unit cells in 1 mol can be determined by dividing the total number of atoms in 1 mol by the number of atoms in one unit cell.
Number of unit cells in 1 mol = Avogadro's number / number of atoms in one unit cell
= 6.022 x 10^23 / 4
= 1.5055 x 10^23 unit cells.
(c) To calculate the volume of 1 mol of gold unit cells, we need to multiply the number of unit cells in 1 mol by the volume of one unit cell. The volume of one unit cell in an FCC structure can be determined using the formula:
Volume of 1 unit cell = (side length)^3.
(d) To determine the volume of one unit cell, we first need to find the side length of the unit cell. The density of gold is given as 19.3 g/cm^3. Knowing the molar mass (197.0 g/mol) and density of gold, we can calculate the side length using the following relationship:
Density = (molar mass)/(number of particles in one unit cell x volume of one unit cell).
(e) Once we know the side length of the unit cell (which is the length of one side), we can directly use it to answer this question.
(f) The metallic diameter of one atom can be calculated using the side length of the unit cell and the relationship between the side length and the diameter of an atom in an FCC structure, which is given by the formula:
Metallic diameter = (side length) x (√2).
By following these steps, we can find the answers to each of the above questions using the given information.