I have a unit cube question to which the answer is given that I'm trying to figure out. The question is:
If under conditions of high pressure & temperature, the crystalline structure rearranges to form a face-centered cubic unit cell, what is the new density of the substance?
Answer: 11.20 g/cm^3
Obviously this is a multi-part problem; I have solved the other parts and here is the releveant info:
As a body centered cubic unit cell:
side length= 3.14 Angstroms (3.14*10^-8 cm)
density: 10.28 g/cm^3
radius 1.36 Angstroms (1.36*10^-8 cm)
molar mass= 95.94 g/mol
And of course for body centered cells:
4r=3^(1/2)s
AND
2 atoms/1 cell
face centered cubic unit cells have 4 atoms/cell
I tried [mass(element in g/mol)*(# elements/cell)(1/Avogadro's #)]/(s(cm)^3)
in the sense that this would (I thought) give me:
mass of cell(grams)/volume cell (cm^3)
but I get an answer of over 700 so I obviously went wrong somewhere.
mass unit cell = 95.94*4/6.022E23 = ??
volume = a^3
a = 4r/sqrt 2.
I get 11.19
To find the new density of the substance after the crystalline structure rearranges to a face-centered cubic (FCC) unit cell, you can follow these steps:
1. Calculate the volume of the FCC unit cell:
In an FCC unit cell, there are 4 atoms per cell. Each atom is located at the corners of a regular tetrahedron, with one atom at each corner and one in the center of the face. The side length of the unit cell can be found using the relationship you mentioned: 4r = √3s, where r is the radius of the atom and s is the side length of the unit cell. From the given information, r is 1.36 Å (1.36 * 10^-8 cm). Solving the equation for s, we have: s = (4r) / √3 = (4 * 1.36 * 10^-8 cm) / √3 = 3.74 * 10^-8 cm. The volume of the unit cell (V_FCC) can be calculated as the cube of the side length: V_FCC = (3.74 * 10^-8 cm)^3.
2. Calculate the density of the FCC unit cell:
To find the mass of the FCC unit cell, we need to know the number of atoms in one cell and the molar mass of the substance. From the given information, there are 4 atoms per cell in the FCC structure. The molar mass of the substance is 95.94 g/mol. Using the formula: mass = (molar mass) * (# atoms in the cell) / Avogadro's number, we can calculate the mass of the FCC unit cell (m_FCC).
3. Calculate the density of the substance:
The density (ρ) is defined as the mass (m) divided by the volume (V): ρ = m / V.
Now, let's plug the values into the formulas and calculate the new density:
V_FCC = (3.74 * 10^-8 cm)^3
m_FCC = (95.94 g/mol) * 4 atoms / 6.022 × 10^23 atoms/mol
ρ_FCC = m_FCC / V_FCC
Using these calculations, you should be able to find the new density of the substance in grams per cubic centimeter (g/cm^3).