At room temperature and pressure RbI crystallizes with the NaCl-type structure. (a) Use ionic radii to predict the length of the cubic unit cell edge. (b) Use this value to estimate the density. (c) At high pressure the structure transforms to one with a CsCl-type structure. (c) Use ionic radii to predict the length of the cubic unit cell edge for the high-pressure form of RbI. (d) Use this value to estimate the density. How does this density compare with the density you calculated in part (b)?
NaCl has an fcc structure.
This implies that in an RbI crystal lattice, the distance between cation and anion (ionic radii) is half the diagnol of the unit cell.
Hence, if ionic radii = r
=> √2a = 2r
=> a = √2r
b) d = (Z)(M.M)/(Na)(a^3)
You'll find 'a' in the above part. Note that Z = 4 for an fcc lattice.
c) & d) Repeat the same procedure for CsCl which is a bcc structure.
(a) To predict the length of the cubic unit cell edge for RbI with the NaCl-type structure, we can use the ionic radii of Rb+ and I- ions.
The ionic radii of Rb+ and I- are approximately 1.52 Å and 2.20 Å, respectively. In the NaCl-type structure, the cations and anions are arranged in a face-centered cubic (FCC) lattice.
In an FCC lattice, the length of the cubic unit cell edge (a) is related to the ionic radii (r) by the equation:
a = 4√2 * r
For RbI, the length of the cubic unit cell edge (a) can be calculated as:
a = 4√2 * (ionic radius of Rb+ + ionic radius of I-)
Plugging in the values, we get:
a = 4√2 * (1.52 Å + 2.20 Å)
a ≈ 4√2 * 3.72 Å
a ≈ 4 * 1.414 * 3.72 Å
a ≈ 5.30 Å
Therefore, the length of the cubic unit cell edge for RbI with the NaCl-type structure is approximately 5.30 Å.
(b) To estimate the density, we can use the formula:
density = (mass of unit cell) / (volume of unit cell)
For a cubic unit cell with edge length (a), the volume (V) is given by:
V = a^3
Since RbI has the NaCl-type structure, the unit cell contains 4 formula units of RbI. So, the mass of the unit cell is the sum of the masses of 4 Rb ions and 4 I ions.
The molar mass of Rb is approximately 85.47 g/mol, and that of I is approximately 126.90 g/mol.
Hence, the mass of the unit cell (m) is:
m = 4 * (molar mass of Rb) + 4 * (molar mass of I)
Plugging in the values, we get:
m = 4 * (85.47 g/mol) + 4 * (126.90 g/mol)
m ≈ 341.88 g + 507.60 g
m ≈ 849.48 g
Now, we can calculate the density:
density = mass of unit cell / volume of unit cell
density = m / V
density = m / (a^3)
density = 849.48 g / (5.30 Å)^3
Now, convert Å to cm:
1 Å = 1 × 10^(-8) cm
So, the density can be calculated as:
density = 849.48 g / (5.30 × 10^(-8) cm)^3
(c) Now, let's move on to the high-pressure form of RbI, which has a CsCl-type structure.
In the CsCl-type structure, the cations and anions are arranged in a simple cubic lattice. Therefore, the length of the cubic unit cell edge (a) for the high-pressure form of RbI is equal to the sum of the ionic radii of Rb+ and I-.
Using the same ionic radii as before, we have:
a = ionic radius of Rb+ + ionic radius of I-
Plugging in the values, we get:
a = 1.52 Å + 2.20 Å
a ≈ 3.72 Å
Therefore, the length of the cubic unit cell edge for the high-pressure form of RbI is approximately 3.72 Å.
(d) To estimate the density for the high-pressure form of RbI, we can use the same formula as before:
density = mass of unit cell / volume of unit cell
For the CsCl-type structure, the unit cell contains only 1 formula unit of RbI. So, the mass of the unit cell is the sum of the masses of 1 Rb ion and 1 I ion.
Using the molar masses of Rb and I, we can calculate the mass of the unit cell:
m = (molar mass of Rb) + (molar mass of I)
Plugging in the values, we get:
m = 85.47 g/mol + 126.90 g/mol
m ≈ 212.37 g
Now, we can calculate the density:
density = mass of unit cell / volume of unit cell
density = m / V
density = m / (a^3)
density = 212.37 g / (3.72 Å)^3
Again, convert Å to cm:
1 Å = 1 × 10^(-8) cm
So, the density can be calculated as:
density = 212.37 g / (3.72 × 10^(-8) cm)^3
Finally, compare the densities calculated in part (b) and part (d) to determine how they compare.
To answer these questions, we need to consider the ionic radii of Rb+ and I- ions and their crystal structures. Here's how you can approach each part of the question:
(a) To predict the length of the cubic unit cell edge for the NaCl-type structure of RbI, you can use the ionic radii of Rb+ and I- ions. The NaCl-type structure is a face-centered cubic (FCC) lattice, where the cations are located at the corners of the cube and the anions occupy the face centers.
The ionic radius of Rb+ is approximately 1.52 Å, and the ionic radius of I- is approximately 2.20 Å. In the NaCl-type structure, the cation-anion distance is the sum of their ionic radii.
So, the length of the cubic unit cell edge (a) can be calculated using the formula:
a = 2 * (Rb+ radius + I- radius)
Plug in the values:
a = 2 * (1.52 Å + 2.20 Å)
Calculate the value of a to get the length of the cubic unit cell edge.
(b) To estimate the density, we need to know the molar mass of RbI. The molar mass of RbI is the sum of the atomic masses of Rb and I, which can be found from the periodic table.
Once you have the molar mass, you can calculate the number of moles of RbI in the unit cell by dividing the mass of RbI by its molar mass.
Then, you can calculate the volume of the unit cell by using the formula:
V = a^3
Finally, you can find the density by dividing the mass of RbI in the unit cell by its volume.
(c) To predict the length of the cubic unit cell edge for the CsCl-type structure of RbI under high pressure, you can use the ionic radii of Rb+ and I- ions. The CsCl-type structure is a simple cubic (SC) lattice, where the cation and anion alternate at the corners of the cube.
The ionic radius of Rb+ is still approximately 1.52 Å, and the ionic radius of I- may change under high pressure. Find the ionic radius of I- under high pressure.
The length of the cubic unit cell edge (a) for the CsCl-type structure can be calculated using the formula:
a = 2 * (Rb+ radius + I- radius)
Plug in the values:
a = 2 * (1.52 Å + I- radius under high pressure)
Calculate the value of a to get the length of the cubic unit cell edge for the high-pressure form of RbI.
(d) To estimate the density, follow the same procedure as in part (b), using the length of the cubic unit cell edge for the high-pressure form of RbI.
Compare the density calculated in part (b) with the density calculated in part (d) to see how they differ.
Please note that the calculations provided here are based on the given information and assumptions. The actual values may vary. It is always good to cross-check the results and consider any additional factors if needed.