Calculate the radius of a palladium (Pd) atom, given that Pd has an FCC crystal structure, a density of 12.0 g/cm3 , and an atomic weight of 106.4 g/mol.
The FCC means 4 atoms/unit cell.
106.4 g/mol x (1/6.022E23) = 1.77E-22 g
1.77E-22 x 4 atoms/unit cell = 7.07E-22 grams/unit cell
density of unit cell = 12.0 g/cc = mass/volume so
volume = 7.07 E-22 g/12.0 g/cc = 5.89E-23 cc. Since volume a^3 so a^1/3 so (5.89E-23)^1/3 = a = 3.92E-8 cm
r = 1.414*a/4 = ?
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To calculate the radius of a palladium (Pd) atom, we can use the formula:
radius = (3 x atomic weight) / (4 x density x Avogadro's number),
where:
- Atomic weight is the molar mass of palladium (Pd), which is 106.4 g/mol.
- Density is the density of palladium (Pd), which is 12.0 g/cm3.
- Avogadro's number is 6.022 x 10^23 atoms/mol.
Let's plug in the values into the formula and calculate the radius:
radius = (3 x 106.4 g/mol) / (4 x 12.0 g/cm3 x 6.022 x 10^23 atoms/mol)
Now, we need to convert units to ensure they are consistent. The given density is in g/cm3, and Avogadro's number is in atoms/mol. To cancel out the units, we can convert grams (g) to kilograms (kg) and cubic centimeters (cm3) to cubic meters (m3).
1 g/cm3 = 1000 kg/m3 (since 1 kg = 1000 g)
1 cm3 = (1/100)^3 m3 = 1 x 10^-6 m3 (since 1 m = 100 cm)
Substituting the values and simplifying, we get:
radius = (3 x 106.4 g/mol) / (4 x 12.0 g/cm3 x 6.022 x 10^23 atoms/mol)
= (3 x 106.4 x 1000 kg/mol) / (4 x 12.0 x 1 x 10^-6 m3 x 6.022 x 10^23 atoms/mol)
Now, we can cancel out the units:
radius = (3 x 106.4 x 1000) / (4 x 12.0 x 1 x 10^-6 x 6.022 x 10^23) m
Simplifying further, we get:
radius = (3 x 106.4 x 1000) / (4 x 12.0 x 6.022 x 10^17) m
Calculating the numerator and denominator, we get:
radius ≈ 1.697 x 10^-10 m
Therefore, the radius of a palladium (Pd) atom is approximately 1.697 x 10^-10 meters.