What is the theoretical density of tantalum (Ta) with the following characteristics:
Crystal structure: BCC
Atomic radius: 14.6 Angstroms
Atomic weight: 180.9 g/mol
density needs to be in g/cm^3
To calculate the theoretical density of tantalum (Ta), we need to use the given information. The theoretical density can be calculated using the following formula:
Theoretical density = (Atomic weight) / (Volume of the unit cell)
First, let's determine the volume of the unit cell. For a body-centered cubic (BCC) crystal structure, the volume of the unit cell can be calculated using the formula:
Volume of BCC unit cell = (4 * π * (Atomic radius)^3) / 3
Now, let's insert the given values into the equations:
Atomic radius = 14.6 Angstroms = 14.6 * 10^(-10) meters
Volume of BCC unit cell = (4 * π * (14.6 * 10^(-10))^3) / 3
Next, simplify the equation further:
Volume of BCC unit cell ≈ 9.5 * 10^(-30) m^3
Now, we need to convert the atomic weight from grams/mol to kilograms/mol:
Atomic weight = 180.9 g/mol = 180.9 * 10^(-3) kg/mol
Finally, we can calculate the theoretical density:
Theoretical density = (180.9 * 10^(-3)) / (9.5 * 10^(-30))
To convert the density to g/cm^3, we need to divide by another factor of 10^6 to convert m^3 to cm^3 and multiply by 10^3 to convert kg to grams:
Theoretical density ≈ (180.9 * 10^(-3)) / (9.5 * 10^(-30)) * (10^3 / 10^6)
Simplifying further, we get:
Theoretical density ≈ 190 g/cm^3
Therefore, the theoretical density of tantalum (Ta) with the given characteristics is approximately 190 g/cm^3.