The radius of a single atom of a generic element X is 179 picometers (pm) and a crystal of X has a unit cell that is body-centered cubic. Calculate the volume of the unit cell.
For a bcc crystal, a*(3)^1/2 = 4r. You know r, calculate a, then
(V)^1/3 = a so a^3 = volume.
Well, that's quite a small little atom! Just be careful not to drop it. Now, onto the calculation!
To find the volume of a unit cell in a body-centered cubic (BCC) crystal structure, we need to know the length of one side of the cube. Lucky for us, in a BCC structure, the body diagonal of the cube is equal to four times the atomic radius.
So, the length of the body diagonal (D) can be found using the formula D = 4r, where r is the radius of the atom. Plugging in the given radius of 179 picometers (pm), we get:
D = 4 * 179 pm = 716 pm
Now that we have the length of the body diagonal, we can calculate the length of one side of the cube (a) using the Pythagorean theorem. In a cube, all sides are equal to each other, so a = D / √3:
a = 716 pm / √3 ≈ 413.4 pm
Finally, we can find the volume of the unit cell (V) by cubing the length of one side:
V = a³ ≈ (413.4 pm)³ ≈ 70,124,058.2 pm³
In scientific notation, that would be approximately 7.01 x 10^7 pm³.
So, the volume of the unit cell in a body-centered cubic crystal structure of element X would be around 7.01 x 10^7 picometers cubed.
Hope that helps, and remember to keep an eye on that tiny atom! It might get lost in your carpet!
To calculate the volume of the unit cell in a body-centered cubic crystal, we need to use the formula:
Volume of the unit cell = (4 * Radius^3) / 3
Given that the radius of the atom (Radius) is 179 picometers (pm), we can substitute this value into the formula:
Volume of the unit cell = (4 * 179^3) / 3 = (4 * 5730481) / 3 = 19101627 pm^3
Therefore, the volume of the unit cell is 19101627 cubic picometers (pm^3).
To calculate the volume of the unit cell, we need to find the length of one side of the cubic cell.
In a body-centered cubic (bcc) unit cell, the atoms are arranged such that there is one atom at each of the eight corners of the cube and one atom at the center of the cube.
The body diagonal of a bcc unit cell can be found using the formula:
d = 4r (where d is the length of the body diagonal and r is the radius of an atom)
In this case, the radius of the atom is given as 179 picometers (pm), so we have:
r = 179 pm
Therefore, the length of the body diagonal (d) can be calculated as:
d = 4r = 4 * 179 pm
Now, we need to calculate the length of one side of the cube (a).
For a body-centered cubic lattice, the length of the side of the cube (a) is related to the length of the body diagonal (d) as follows:
a = (sqrt(3)/4) * d
Now, let's calculate the value of a using the given value of r:
a = (sqrt(3)/4) * (4 * 179 pm)
Simplifying, we have:
a = sqrt(3) * 179 pm
Finally, we can calculate the volume of the unit cell (V) by cubing the side length (a):
V = a^3 = (sqrt(3) * 179 pm)^3
Evaluating this expression will give us the volume of the unit cell for the given element X.