Calculate the radius of a tantalum (Ta) atom, given that Ta has a BBC crystal structure, a density of 16.6g/cm³, and an atomic weight of 180.9 g/mol.
Please some one help me on this, I don't have the answer myself
16.6 g/cm^3 * 1mole/180.9g * 6.023*10^23 atoms/mole = 5.535*10^22 atoms/cm^3
The article below shows how to relate the atomic radius to the BCC packing structure.
msestudent.com/body-centered-cubic-bcc-unit-cell/
To calculate the radius of a tantalum (Ta) atom, we need to make use of the formula for the volume of a face-centered cubic (FCC) unit cell and the atomic weight of tantalum.
Step 1: Calculate the volume of the unit cell.
For an FCC crystal structure, the volume of the unit cell can be determined using the formula:
Volume = (4/3) * pi * r^3,
where r is the radius of the atom.
Step 2: Calculate the edge length of the unit cell.
For an FCC crystal structure, the edge length (a) of the unit cell is related to the radius (r) by the equation:
a = 4 * r / sqrt(2).
Step 3: Calculate the mass of the unit cell.
The mass of the unit cell can be calculated using the atomic weight (180.9 g/mol) and density (16.6 g/cm³):
Mass = (atomic weight / Avogadro's number) * volume = density * volume.
Step 4: Solve for the radius.
Substitute the equations from step 1 and step 3 into the equation from step 2. Simplify the equation and solve for r.
Let's calculate the radius step by step:
Step 1: Calculate the volume of the unit cell.
Volume = (4/3) * pi * r^3
Step 2: Calculate the edge length of the unit cell.
a = 4 * r / sqrt(2)
Step 3: Calculate the mass of the unit cell.
Mass = density * volume
Step 4: Solve for the radius.
Substitute the equations from step 1 and step 3 into the equation from step 2. Simplify the equation and solve for r.
To calculate the radius of a tantalum (Ta) atom, we need to use the information provided about its crystal structure, density, and atomic weight.
The crystal structure of a BBC (Body-Centered Cubic) unit cell consists of one atom at each corner of the cube and one atom at the center of the cube. This means that in one unit cell, there are 8 corner atoms and 1 center atom.
First, we need to find the volume of the unit cell. In a BBC structure, the length of the unit cell (a) can be determined using the formula:
a = (4 * r) / sqrt(3)
Where r is the radius of the atom.
Now, let's calculate the volume of the unit cell. The volume (V) can be calculated by multiplying the length of the unit cell (a) by itself three times:
V = a^3
Since there is one atom per unit cell, the volume occupied by each atom can be calculated by dividing the total volume of the unit cell by the number of atoms:
Volume of each atom = V / 9
Finally, we can calculate the radius (r) of the tantalum atom by rearranging the formula for volume of a sphere:
radius = (3 * Volume of each atom) / (4 * pi)^(1/3)
Let's plug in the values and calculate the radius:
Given:
Density (ρ) = 16.6 g/cm³
Atomic weight (M) = 180.9 g/mol
First, we need to convert the density into kg/m³:
Density (ρ) = 16.6 * 1000 = 16600 kg/m³
Next, we calculate the molar volume (V_m) using the formula:
V_m = M / ρ
V_m = (180.9 g/mol) / (16600 kg/m³)
Now, we need to convert grams to kilograms and cubic centimeters to cubic meters in order to maintain consistent units:
V_m = (180.9 / 1000 kg/mol) / (16600 / 1000000 m³/kg)
Simplifying the units and the values:
V_m = 0.0109 m³/mol
Now, we can obtain the volume of each atom:
Volume of each atom = V_m / Avogadro's number
Avogadro's number (N_A) = 6.022 x 10^23 mol⁻¹
Volume of each atom = 0.0109 m³/mol / (6.022 x 10^23 mol⁻¹)
Finally, we can calculate the radius of the tantalum atom:
radius = (3 * Volume of each atom) / (4 * pi)^(1/3)
Now, you can calculate the radius of the tantalum atom using the given values and the formulas explained above.