A comparison of density values obtained by X-ray diffraction reveals that the percentages of vacant sites in aluminum (Al) are 0.0700% at 923 K and 0.0090% at 757 K.
Given this data, determine the number of vacant sites per cm3 at 550 K.
Express your answer as vacancies/cm3.
To determine the number of vacant sites per cm3 at 550 K, we can use the relationship between density and the percentage of vacant sites.
Here's how to approach the problem:
1. Convert the given percentages of vacant sites at 923 K and 757 K to decimal form:
- 0.0700% = 0.0700/100 = 0.000700
- 0.0090% = 0.0090/100 = 0.000090
2. Consider the relationship between the percentage of vacant sites and density:
- Density1 / 100 = Vacancies1
- Density2 / 100 = Vacancies2
We want to determine Density3 (density at 550 K). So, the relationship can be expressed as:
- Density3 / 100 = Vacancies3
3. Rearrange the equation to solve for Density3:
- Density3 = (Vacancies3 * 100)
4. Substitute the given values of Vacancies1, Vacancies2, and their respective temperatures to find the relationship between temperature and the percentage of vacant sites:
- (Vacancies2 / Vacancies1) = (T2 / T1)
5. Rearrange the equation to solve for Vacancies3:
- Vacancies3 = (Vacancies1 * (T2 / T1))
6. Substitute the given values of Vacancies1, T1, and T2 to find Vacancies3:
- Vacancies3 = (0.000700 * (757 K / 923 K))
- Vacancies3 ≈ 0.000572
7. Finally, substitute Vacancies3 into the equation found in step 3 to calculate Density3:
- Density3 = (0.000572 * 100)
- Density3 ≈ 0.0572
Therefore, at 550 K, the number of vacant sites per cm3 is approximately 0.0572 vacancies/cm3.