the metal gold, with an atomic radius of 144.2pm crystallizes in a face-centered cubic lattice. what is the density of gold?
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volume=1.442x10^-8cm
4atoms per cell so,
4x (1/6.022x10^23) x 196.97g = 1.31x10^-21 g of gold
cell volume=(1.442x10^-8cm)^3= 2.99x10^-24
density=(1.31x10^-21)/(3.0x10^-24)=438g/cm^3
144.2 pm = 1.442E-8 cm and that's the RADIUS.
volume of sphere is (4/3)*pi*r^3.
See if that gets you off on the right foot?
To determine the density of gold, we need to know its atomic mass and the structure of its lattice.
The atomic mass of gold (Au) is approximately 197 g/mol. It is given that gold crystallizes in a face-centered cubic (FCC) lattice.
In an FCC lattice, each corner of the cube is occupied by an atom, and an additional atom lies at the center of each face of the cube.
The number of atoms per unit cell of an FCC lattice is 4. This can be calculated using the formula:
Number of atoms per unit cell = 1/8 * 8 corner atoms + 1/2 * 6 face atoms
= 1 + 3
= 4
Now, we can calculate the volume of the unit cell:
The face diagonal of an FCC unit cell can be found using the formula:
Face diagonal length = 4 * (Side length)
In an FCC lattice, the face diagonal length is equal to four times the atomic radius (4r).
Here, the atomic radius of gold is given as 144.2 pm, so the face diagonal length will be 4 * 144.2 pm.
Now, we need to convert the atomic radius from picometers (pm) to centimeters (cm). There are 1e10 pm in one cm, so 144.2 pm is equal to 144.2 * (1e-10) cm.
Substituting these values, the face diagonal length (d) becomes:
d = 4 * (144.2 * 1e-10) cm.
The face diagonal length (d) is related to the side length (a) of the unit cell by the formula:
d = √2 * a
Solving for the side length (a), we get:
a = d / √2
Substituting the value of d, we get:
a = (4 * 144.2 * 1e-10) cm / √2
The volume of the unit cell (V) is given by:
V = a^3
Now, we can substitute the value of 'a' in the equation above and calculate the volume of the unit cell (V).
Once we know the volume of the unit cell and the number of atoms per unit cell, we can calculate the density (ρ) using the formula:
ρ = (mass of unit cell) / V
The mass of the unit cell can be calculated by taking the atomic mass (m) and dividing it by Avogadro's number (6.02e23) since we have one atom per unit cell.
Once we have the density, we can express it in g/cm^3.