Final answer:
The correct answer is option (a) 2.1.38 x 10⁷ pm³.
Step-by-step explanation:
The metal vanadium crystallizes in a body-centered cubic lattice.
The unit cell volume for a body-centered cubic structure can be calculated using the formula:
V = a³
Where V is the unit cell volume and a is the edge length of the unit cell.
Given that the density of vanadium is 6.11 g/cm³, we can convert it to g/pm³ by multiplying by 10^15 since there are 10^15 pm³ in 1 cm³. Thus, the density of vanadium is 6.11 x 10^15 g/pm³.
Let's assume the edge length of the unit cell is 'x' pm, then the unit cell volume would be x³ pm³.
Using the given density and the formula for unit cell volume, we can set up the following equation:
x³ pm³ * (6.11 x 10^15 g/pm³) = 6.11 x 10^15 pm³
Solving for x, we find:
x³ pm³ = 6.11 x 10^15 pm³ / (6.11 x 10^15 g/pm³)
x³ = 1 pm³ / g
x = 1^(1/3) pm / g^(1/3)
x = 1 pm / g^(1/3)
As we can see, the unit of 'x' is pm. Therefore, the unit cell volume is 1 pm³ / g^(1/3) pm = g^(-1/3).
Therefore, the correct answer is option (a) 2.1.38 x 10⁷ pm³.