An element crystallizes in a body centered cubic unit cell. The radius of this element is 2.20A , and the density is .991 g/cm^3. how many moles of atoms are within one unit cell?
There are two atoms/unit cell in a bcc and a mole of atoms is 6.02E32 atoms. You make the conversion from atoms to mols.
To calculate the number of moles of atoms within one unit cell, we need to find the number of atoms present in the unit cell.
In a body-centered cubic (BCC) unit cell, there is one atom at each corner and one atom at the center of the unit cell.
First, let's calculate the volume of the unit cell:
The BCC unit cell has 8 corners, where each corner atom contributes 1/8th to the volume, so 8 x 1/8 = 1 atom contributes to the volume.
Additionally, there is 1 whole atom at the center, so in total, there are 2 atoms contributing to the volume of the unit cell.
The volume of the unit cell can be calculated as:
Volume = (4/3) * π * r^3
= (4/3) * π * (2.20A)^3
Now, let's convert the volume from cubic Ångstroms to cubic centimeters, as the density is given in g/cm^3.
1 Ångstrom (A) = 1 x 10^-8 cm
1 Ångstrom^3 = (1 x 10^-8 cm)^3
= 1 x 10^-24 cm^3
Therefore, the volume of the unit cell in cm^3 can be calculated as:
Volume_cm^3 = Volume_A^3 * (1 x 10^-24 cm^3)
Next, divide the density (0.991 g/cm^3) by the molar mass (in g/mole) to find the number of moles per cm^3.
Finally, multiply the number of moles per cm^3 by the volume of the unit cell in cm^3 to find the moles of atoms within one unit cell.
To determine the number of moles of atoms in one unit cell of a body-centered cubic lattice, we need to follow these steps:
Step 1: Calculate the volume of the unit cell.
The body-centered cubic unit cell consists of one full atom at each of the eight corners of the cube and an additional atom at the center of the cube. The volume of the unit cell can be expressed as:
Volume = ((side length)^3) * (number of atoms in the unit cell)
= (4 * r^3) * (2)
Given that the radius (r) of the element is 2.20 Å, we convert it to centimeters:
1 Å = 1 × 10^(-8) cm
2.20 Å = 2.20 × 10^(-8) cm
Now, plug in the values to calculate the volume.
Volume = (4 * (2.20 × 10^(-8) cm)^3) * (2)
Step 2: Convert the density to grams per cubic centimeter.
The given density is 0.991 g/cm^3, so we already have the density in the desired units.
Step 3: Calculate the molar mass of the element.
This step requires you to know the identity of the element. Look up the molar mass or atomic mass of the element of interest.
Step 4: Calculate the number of moles of atoms.
The number of moles can be determined using the formula:
Number of moles = (Density * Volume) / Molar mass
Plugging in the respective values, calculate the number of moles of atoms in one unit cell.
It is worth noting that the body-centered cubic lattice assumes that the atoms are perfectly aligned and in a regular arrangement. In reality, the arrangement of atoms in a crystal may deviate from this idealized structure.