You are given a small bar of an unknown metal X. You find the density of the metal to be 10.5g/cm3. An x-ray diffraction experiment measures the edge of the face-centered cubic unit cell as 4.09 A. (1A=10^-10m) Identify X.
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a = edge of unit cell = 4.09 Ao = 4.09 x 10^-8 cm.
volume of unit cell = (4.09 x 10^-8 cm)3.
mass = volume x density = volume from last step x 10.5 g/cc = xx g mass/unit cell.
There are 4 atoms per unit cell in a fcc; therefore, xx g/4 = mass of one atom and that times 6.022 x 10^23 atoms/mol = atomic mass of the element. Check my arithmetic; I obtained about 108.1 or so. Look that up on the periodic table to identify the element X.
To identify the unknown metal (X), we can use the density and the edge length of the face-centered cubic (FCC) unit cell.
Step 1: Calculate the volume of the FCC unit cell.
The volume (V) of the FCC unit cell can be calculated using the formula V = a^3, where a is the edge length.
Given that the edge length (a) is 4.09 A, which is equal to 4.09 * 10^-10 m, we can calculate the volume:
V = (4.09 * 10^-10 m)^3
Step 2: Convert the volume to cm^3.
Since the density is given in g/cm^3, we need to convert the volume to cm^3.
1 m^3 = (100 cm)^3
V = (4.09 * 10^-10 m)^3 * (100 cm / 1 m)^3
Step 3: Calculate the mass of the FCC unit cell.
To calculate the mass (m) of the FCC unit cell, we can use the formula:
density (ρ) = mass (m) / volume (V)
Rearranging the equation, we get:
mass (m) = density (ρ) * volume (V)
Given that the density (ρ) is 10.5 g/cm^3, we can calculate the mass:
mass (m) = 10.5 g/cm^3 * V
Step 4: Identify the metal (X) based on its atomic mass.
To identify the metal (X), we need to find the elemental atomic mass that corresponds to the calculated mass.
Look up the atomic masses of elements and find the one that best matches the calculated mass. The metal with a similar atomic mass is likely to be metal X.
Note: Without additional information, it may be challenging to determine the exact metal (X). However, you can make an educated guess based on the atomic mass.
To identify the unknown metal X, we can use the formula that relates the density of a material to its molar mass and atomic radius. The formula is given as:
Density = (Molar Mass / Volume of Unit Cell) * (Number of Atoms per Unit Cell * Atomic Mass)
First, we need to determine the volume of the face-centered cubic (FCC) unit cell using the given edge length of 4.09 Å. Since the FCC unit cell consists of eight corner atoms and one atom in the center of each face, the total number of atoms per unit cell is 4.
The formula for the volume (V) of the FCC unit cell is:
V = (a^3) * (N / 4)
where "a" is the edge length of the unit cell and "N" is the total number of atoms per unit cell.
Given that the edge of the FCC unit cell is 4.09 Å, we convert it to meters by multiplying it by 10^-10:
a = 4.09 Å = 4.09 * 10^-10 m
Now, we can calculate the volume of the unit cell:
V = (4.09 * 10^-10 m)^3 * (4 / 4) = 6.825e-29 m^3
Next, we rearrange the formula for density to solve for the molar mass (Molar Mass) of the unknown metal X:
Molar Mass = (Density * Volume of Unit Cell) / (Number of Atoms per Unit Cell * Atomic Mass)
Given that the density is 10.5 g/cm^3 and the volume is 6.825e-29 m^3, we convert the density to kg/m^3 by multiplying it by 10^3:
Density = 10.5 g/cm^3 = 10.5 * 10^3 kg/m^3
Plugging in the values, we have:
Molar Mass = (10.5 * 10^3 kg/m^3 * 6.825e-29 m^3) / (4 * Atomic Mass)
Finally, we need to convert the molar mass from kg/mol to g/mol by multiplying it by 10^3:
Molar Mass = (Molar Mass * 10^3 g/mol)
Now, we can determine the molar mass (Molar Mass) of the unknown metal X by solving the equation.
By substituting the value of the edge length, the number of atoms per unit cell, and the atomic mass into the equation, the molar mass of the unknown metal X can be calculated.