Consider a copper lattice with an excessive average of one vacancy in every 100 unit cells. What will be its density? Compare this to the theoretical density of copper.
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To solve this problem, we need to determine the density of the copper lattice with the given vacancy and compare it to the theoretical density of copper.
Step 1: Calculate the lattice with vacancies
The density of the copper lattice with the given vacancies can be calculated using the formula:
Density = (Number of atoms * Atomic weight) / Volume
Step 2: Determine the number of atoms
In a copper lattice, each unit cell contains one copper atom. Since there is one vacancy per 100 unit cells, we can subtract the number of vacancies from the total number of unit cells to get the number of copper atoms.
Number of copper atoms = Total number of unit cells - Number of vacancies
Step 3: Calculate the volume
The volume of a unit cell can be calculated by dividing the total volume of the lattice by the total number of unit cells.
Step 4: Calculate the density
Using the formula mentioned in Step 1, we can calculate the density by substituting the values for the number of atoms, atomic weight, and volume.
Step 5: Compare the calculated density with the theoretical density of copper.
The theoretical density of copper is 8.96 g/cm^3.
Let's calculate the density of the copper lattice with the given vacancies:
Step 1: Calculate the lattice with vacancies
Density = (Number of atoms * Atomic weight) / Volume
Step 2: Determine the number of atoms
Number of copper atoms = Total number of unit cells - Number of vacancies
Step 3: Calculate the volume
Let's assume the volume of the unit cell is 1 cm^3.
Step 4: Calculate the density
Density = (Number of atoms * Atomic weight) / Volume
Step 5: Compare the calculated density with the theoretical density of copper.
Would you like to proceed with the calculations based on these steps?
To determine the density of the copper lattice with the given number of vacancies, we need to know the density of copper and the size of the unit cells.
The density of copper is typically around 8.96 grams per cubic centimeter (g/cm³). This is the theoretical or ideal density of copper with no vacancies.
To find the density of the lattice with vacancies, we can calculate the effective density by considering the vacancies as empty spaces within the lattice.
First, we need to determine the volume of one unit cell. This can be done by calculating the cube of the edge length (a) of a unit cell. The volume formula for a cube is V = a^3.
Next, we need to calculate the number of vacancies in the lattice. If there is one vacancy in every 100 unit cells, then the fraction of vacancies (f) can be calculated as 1 vacancy / 100 unit cells = 1/100 = 0.01.
The effective density (ρ') of the lattice with vacancies can be calculated using the formula:
ρ' = (1 - f) * ρ
where ρ is the theoretical density of copper (8.96 g/cm³).
Now we can plug in the values to calculate the effective density:
ρ' = (1 - 0.01) * 8.96 g/cm³
= 0.99 * 8.96 g/cm³
= 8.864 g/cm³
Therefore, the density of the copper lattice with an average of one vacancy in every 100 unit cells is approximately 8.864 g/cm³.
To compare this to the theoretical density of copper (8.96 g/cm³), we can calculate the percentage difference:
Percentage Difference = ((ρ' - ρ) / ρ) * 100
Percentage Difference = ((8.864 - 8.96) / 8.96) * 100
= (-0.096 / 8.96) * 100
= -1.07%
The percentage difference is approximately -1.07%, indicating that the density of the copper lattice with vacancies is slightly lower than the theoretical density of copper.