A metal crystallizes in a face-centered lattice. The radius of the atom is 198pm and the density of the element is 6.57g/cm^3. What is the identify of the metal?
4r = a(2)^1/2
solve for a.
Then volume = a^3
mass = volume x density. Solve for mass.
mass = 4*atomic mass/6.02E23
Solve for atomic mass.
To determine the identity of the metal, we can start by calculating the atomic packing factor (APF), which is the volume occupied by atoms divided by the total volume of the unit cell.
Step 1: Calculate the volume of the atom
Given that the radius of the atom is 198 pm, we can calculate the volume of the atom using the formula for the volume of a sphere:
V = (4/3) * π * r^3
Converting the radius to cm:
198 pm = 198 * 10^-10 cm
Substituting the values into the formula:
V = (4/3) * π * (198 * 10^-10 cm)^3
V ≈ 65.45 * 10^-24 cm^3
Step 2: Calculate the volume of the unit cell
In a face-centered cubic (FCC) lattice, each unit cell contains 4 atoms. Furthermore, the volume of the unit cell can be determined using the formula:
V_unit cell = a^3
where "a" is the edge length of the unit cell.
For an FCC lattice, the atoms touch along the face diagonals of the unit cell. Thus, the edge length (a) is twice the radius (r):
a = 2r = 2 * 198 pm = 396 pm = 396 * 10^-10 cm
V_unit cell = (396 * 10^-10 cm)^3
V_unit cell ≈ 62.35 * 10^-24 cm^3
Step 3: Calculate the atomic packing factor (APF)
The APF is the ratio of the volume occupied by atoms to the total volume of the unit cell:
APF = V occupied by atoms / V unit cell
APF = (V * 4) / V_unit cell
APF ≈ (65.45 * 10^-24 cm^3 * 4) / (62.35 * 10^-24 cm^3)
APF ≈ 4.18
Step 4: Determine the metal's identity based on APF
Based on the calculated APF value, we can refer to a table of common atomic packing factors to determine the metal's identity. For a face-centered lattice, the closest value is approximately 0.74, which corresponds to metals such as copper (Cu) and aluminum (Al). Therefore, based on the given information, the metal could potentially be either copper or aluminum.
It's important to note that the density of the element (6.57 g/cm^3) could be used as additional information to determine the metal's identity. However, based solely on the information provided, copper and aluminum are possible options.
To determine the identity of the metal, we can use the given information about the crystal structure and the radius of the atom.
In a face-centered lattice, each corner atom is shared by eight neighboring unit cells, and each face-centered atom is shared by two neighboring unit cells. This means that in a face-centered lattice, there are 4 atoms per unit cell.
To calculate the volume of the unit cell, we need to determine the length of the edge of the unit cell. Since the lattice is face-centered, the distance between the centers of two corner atoms along a diagonal can be calculated using the Pythagorean theorem.
diagonal^2 = edge^2 + edge^2
diagonal^2 = 2 * edge^2
edge = diagonal / sqrt(2)
The closest distance between two face-centered atoms can be calculated as half the diagonal length of the unit cell:
distance = diagonal / 2 = (edge * sqrt(2)) / 2 = edge / sqrt(2)
Given that the atomic radius is 198 pm, we need to convert it to cm:
radius = 198 pm = 198 * 10^-10 m = 19.8 * 10^-9 m = 19.8 * 10^-8 cm
The density of the metal is given as 6.57 g/cm^3.
To find the identity of the metal, we can use the formula:
density = (atomic mass * number of atoms in a unit cell) / (volume of the unit cell * Avogadro's number)
We know that the volume of a cube is V = edge^3 and the number of atoms in a unit cell is 4.
Let's calculate the volume of the unit cell:
volume = edge^3 = (edge / sqrt(2))^3 = edge^3 / (sqrt(2))^3 = (edge^3) / 2^(3/2)
Now let's plug in the values into the formula to solve for the atomic mass:
6.57 g/cm^3 = (atomic mass * 4) / [(edge^3) / 2^(3/2)] * 6.022 * 10^23
Simplifying the equation, we get:
atomic mass = (6.57 g/cm^3 * [edge^3 * 2^(3/2)]) / (4 * 6.022 * 10^23)
Now substitute the value of edge from the diagonal calculation and solve for the atomic mass. Once you have the atomic mass, you can identify the metal using a periodic table by matching the atomic mass to the known masses of elements.