The radius of a single atom of a generic element X is 199 picometers (pm) and a crystal of X has a unit cell that is face-centered cubic. Calculate the volume of the unit cell.
If a is the edge length of the cell, then
4r = 1.414*a
Solve for a, then
volume = a^3.
Well, well, well, it seems we have a crystal-clear question here! Let's put on our thinking caps and calculate this.
In a face-centered cubic (FCC) crystal, each corner atom is shared by eight unit cells, while each face-centered atom only belongs to one unit cell. So how many atoms do we have in total? Grab your circus calculator, it's time to do some math!
An FCC unit cell has 8 corners and 6 face-centered atoms. Each of these contributes 1/8 of an atom to the unit cell from the corners and 1/2 from the faces, respectively. Therefore, the total number of atoms per unit cell (n) is:
n = (8 x 1/8) + (6 x 1/2) = 4
Now that we have the number of atoms per unit cell, we can calculate the volume (V). The volume of a unit cell is equivalent to the volume of a single atom multiplied by the number of atoms in the unit cell.
V = (4/3)π(199 pm)^3
Now, just plug and chug that equation into your calculator and you'll find your answer! Voila!
To calculate the volume of the unit cell of a face-centered cubic (FCC) crystal, we need to determine the length of the edge of the unit cell.
In an FCC crystal, each corner atom is shared by eight unit cells, and each face-centered atom is shared by two unit cells. Therefore, the total number of atoms per unit cell is:
number of corner atoms = 8 * 1/8 = 1 atom
number of face-centered atoms = 6 * 1/2 = 3 atoms
Total number of atoms per unit cell = 1 + 3 = 4 atoms
Now, let's calculate the edge length of the unit cell:
Given the radius of a single atom of element X = 199 picometers (pm), the diameter of the atom is 2 * radius = 2 * 199 pm = 398 pm.
The edge length of the unit cell can be calculated using the formula:
edge length = 4 * atom radius
Substituting the value of the atom radius, we have:
edge length = 4 * 398 pm = 1592 pm
Now, let's calculate the volume of the unit cell:
The volume of a cube is given by the formula:
volume = (edge length)^3
Substituting the value of the edge length, we have:
volume = (1592 pm)^3 = 4,038,858,688 pm^3
Therefore, the volume of the unit cell is approximately 4,038,858,688 picometers cubed (pm^3).
To calculate the volume of the unit cell in a face-centered cubic (FCC) structure, we first need to understand the arrangement of atoms in this crystal system. In an FCC structure, there are atoms at the corners of the unit cell and additional atoms located at the center of each face of the unit cell.
Given that the radius of a single atom of element X is 199 picometers (pm), we can determine the length of the side of the unit cell.
In a face-centered cubic structure, the body diagonal (d) of the unit cell is twice the length of the side (a). Using this relationship, we have:
d = 2a.
We know that the body diagonal (d) is equal to four times the radius of the atom (4r):
d = 4r.
Let's substitute the values we have:
4r = 2a.
Since we know the radius (r) is 199 picometers (pm), we can solve for the length of the side (a):
2a = 4r
2a = 4 * 199 pm
2a = 796 pm
a = 796 pm / 2
a = 398 pm.
Now that we have the length of one side of the unit cell (a = 398 pm), we can calculate the volume (V) of the unit cell in cubic picometers (pm³).
The volume of the unit cell in FCC structure is given by:
V = a³.
Substituting the value of a, we have:
V = (398 pm)³
V = 63,032,792 pm³.
Therefore, the volume of the unit cell for element X is 63,032,792 cubic picometers (pm³).