The edge of a body-centered -cubic unit cell (which contains two atoms per unit cell) of an element Y was found to be 3.16x10^-8cm.The density of the metalis 19.35g*cm^-3. What is the approximate molar mass of Y?
first you take (3.16x10^-8)^3)=3.15^-23
thenyou take the 2 atoms and divide it by (3.15x10^-23)= 6.34x10^22
then:you take(6.34x10^22)and divide it by( 6.022x10^23)= 0.105
last you take 19.35 divid by 0.105= and your answer is 184g*mol
To find the approximate molar mass of element Y, we need to use the given information regarding the unit cell and the density of the metal.
Step 1: Calculate the volume of the unit cell
The body-centered cubic (bcc) unit cell has two atoms per unit cell. Since each atom is located at the center of eight adjacent unit cells, the effective contribution of each atom is 1/2. Therefore, the volume of the unit cell can be calculated using the formula:
Volume = (Edge length)^3 * (1/2)
Given:
Edge length = 3.16 x 10^-8 cm
Calculating the volume:
Volume = (3.16 x 10^-8 cm)^3 * (1/2)
Volume = 3.16 x 10^-8 cm * 3.16 x 10^-8 cm * 3.16 x 10^-8 cm * (1/2)
Volume = 3.16^3 x 10^(-8 + -8 + -8) * (1/2)
Volume = 3.16^3 x 10^-24 cm^3 * (1/2)
Volume = 3.16^3 x 10^-24 cm^3 * 0.5
Volume ≈ 1.260 x 10^-24 cm^3
Step 2: Calculate the molar mass of element Y
The molar mass of an element is the mass of one mole of that element. We can calculate it using the following formula:
Molar mass = (Mass of one unit cell) / (Number of unit cells in one mole)
Given:
Density = 19.35 g/cm^3
To determine the mass of one unit cell, we need to convert the volume to cm^3 and multiply it by the density:
Mass of one unit cell = Volume (cm^3) * Density (g/cm^3)
Converting volume to cm^3:
Volume ≈ 1.260 x 10^-24 cm^3
Calculating the mass of one unit cell:
Mass of one unit cell = 1.260 x 10^-24 cm^3 * 19.35 g/cm^3
Mass of one unit cell ≈ 2.438 x 10^-23 g
Now, we need to find the number of unit cells in one mole. One mole is equal to Avogadro's number (6.022 x 10^23) of unit cells:
Number of unit cells in one mole = Avogadro's number
Molar mass = Mass of one unit cell / Number of unit cells in one mole
Molar mass ≈ (2.438 x 10^-23 g) / (6.022 x 10^23)
Molar mass ≈ 4.05 x 10^-47 g * (1/6.022 x 10^23)
Molar mass ≈ 4.05 x 10^-47 g * 1.66 x 10^(-23 - 47)
Molar mass ≈ 6.723 x 10^-70 g
Therefore, the approximate molar mass of element Y is approximately 6.723 x 10^-70 g.