Palladium crystallizes with a face-centered cubic structure. It has a density of 12.0 g/cm^3, a radius of 138 pm, and a molar mass of 106.42 g/mol.
Use these data to estimate Avogadro's number.
The fcc has 4 atoms/unit cell.
You know r, use that to calculate the edge of the unit cell, a.
4r = a(2)1/2
Then a3 = volume of the unit cell.
mass of unit cell = volume x density.
Then mass of unit cell is
4 atoms x 106.24/N = mass
Calculate N.
I got lost. :( I got:
a=390
...and when I cubed that, I got a number that was too big so I figured that was wrong. What did I do wrong?
To estimate Avogadro's number, we need to calculate the number of atoms per unit cell in a face-centered cubic (FCC) structure and then determine the volume of each atom.
Step 1: Find the volume of the unit cell.
In a face-centered cubic structure, there are 4 atoms per unit cell. The volume of the unit cell can be calculated using the formula:
V_unit_cell = (4 * r)^3
Where r is the radius of the atoms.
V_unit_cell = (4 * 138 pm)^3
V_unit_cell = (552 pm)^3
Step 2: Convert the unit cell volume to cm^3.
Since the density is given in g/cm^3, we need to convert the unit cell volume from pm^3 to cm^3.
1 cm = 10^10 pm
(552 pm)^3 * (1 cm/10^10 pm)^3
(552 pm)^3 * (1 cm/10^10 pm)^3
(552 pm/10^10)^3 cm^3
(552/10^10)^3 cm^3
Step 3: Calculate the number of atoms per unit cell.
Since there are 4 atoms per unit cell in a face-centered cubic structure, we can determine the number of atoms per unit cell using the formula:
Number of atoms per unit cell = Z
Where Z is the number of atoms per unit cell.
Step 4: Calculate the number of moles of atoms in the unit cell.
The molar mass of palladium is given as 106.42 g/mol. To calculate the number of moles of atoms in the unit cell, we need to convert the unit cell volume to liters (L) and then use the density to determine the mass of the unit cell.
Density = Mass / Volume
Mass = Density * Volume_unit_cell
Number of moles = Mass / Molar mass
Step 5: Calculate Avogadro's number.
Avogadro's number (NA) can be calculated using the formula:
Number of moles = NA / Avogadro's number
Rearranging the equation, we can solve for Avogadro's number:
Avogadro's number = NA = Number of moles
Now we can plug in the values and calculate Avogadro's number.
Note: To ensure consistent unit conversions, we'll use picometers (pm) for the radius, grams per cubic centimeter (g/cm^3) for density, grams per mole (g/mol) for molar mass, and centimeters cubed (cm^3) for the unit cell volume.
Let's calculate it step-by-step:
Step 1: Find the volume of the unit cell
V_unit_cell = (4 * 138 pm)^3
V_unit_cell = 1.627088160 x 10^6 pm^3
Step 2: Convert the unit cell volume to cm^3
V_unit_cell = (1.627088160 x 10^6 pm^3) * (1 cm/10^10 pm)^3
V_unit_cell ≈ 1.627088160 x 10^-8 cm^3
Step 3: Calculate the number of atoms per unit cell
Number of atoms per unit cell = Z = 4
Step 4: Calculate the number of moles of atoms in the unit cell
Mass = Density * Volume_unit_cell
Mass = 12.0 g/cm^3 * (1.627088160 x 10^-8 cm^3)
Mass ≈ 1.952905792 x 10^-7 g
Number of moles = Mass / Molar mass
Number of moles ≈ (1.952905792 x 10^-7 g) / 106.42 g/mol
Number of moles ≈ 1.835526118 x 10^-9 mol
Step 5: Calculate Avogadro's number
Avogadro's number ≈ Number of moles = 1.835526118 x 10^-9
Therefore, the estimated value of Avogadro's number using the given data for palladium is approximately 1.835526118 x 10^-9.
To estimate Avogadro's number using the given data, we can follow these steps:
1. Calculate the volume of a palladium atom:
- The radius of a palladium atom is given as 138 pm (picometers).
- Convert the radius to cm by multiplying by 1E-10 (since 1 pm = 1E-10 cm).
- The volume of a sphere is given by the formula: V = (4/3)πr^3, where r is the radius.
- Calculate the volume of a palladium atom.
2. Calculate the volume of the unit cell:
- In a face-centered cubic (FCC) structure, there are four atoms per unit cell.
- Each corner atom contributes 1/8th of its volume to the unit cell, and each face-centered atom contributes its full volume.
- Calculate the contribution of each atom to the unit cell's volume.
- Multiply the volume contribution by the number of atoms per unit cell.
3. Determine the edge length of the unit cell:
- In an FCC structure, each edge of the unit cell is equivalent to four times the radius of an atom.
- Calculate the edge length of the unit cell using the atomic radius and the formula.
4. Calculate the volume of the unit cell:
- The volume of a cube is given by the formula: V = a^3, where a is the edge length of the unit cell.
- Calculate the volume of the unit cell.
5. Calculate the density of palladium:
- The density is given as 12.0 g/cm^3.
- Divide the molar mass of palladium by the volume of the unit cell to obtain the density.
6. Use the density to estimate Avogadro's number:
- The density can be defined as: Density = (mass of palladium atoms in the unit cell) / (volume of the unit cell).
- Rearrange the equation to solve for the mass of palladium atoms in the unit cell.
- Convert the mass of palladium atoms to moles using the molar mass of palladium.
- The number of moles of palladium atoms in the unit cell is equivalent to Avogadro's number since one mole contains Avogadro's number of particles.
By following these steps, you can estimate Avogadro's number using the given data.