If the diameter of the aluminum atom is 0.21 nm, estimate the number of atoms in a grain with an ASTM grain size of 5. (Calculate volume of each grain from ASTM grain size.
As aluminum is a FCC material, volume of the grain taken by atoms = (volume of grain × APFfcc).
Calculate volume of one atom.
Then calculate the no. of atoms in the grain]
The correct answer and explanation is :
To estimate the number of aluminum atoms in a grain with an ASTM grain size of 5, given that the diameter of an aluminum atom is 0.21 nm, we can follow these steps:
1. Understanding ASTM Grain Size Number:
The ASTM grain size number (G) is related to the number of grains per unit area. The relationship is given by:
[ N = 2^{(G-1)} ]
For G = 5:
[ N = 2^{(5-1)} = 2^4 = 16 ]
This means there are 16 grains per unit area.
2. Volume of Each Grain:
Assuming the grains are roughly equiaxed (having similar dimensions in all directions), the volume of each grain can be estimated. However, without specific information about the material’s thickness or the exact area considered, it’s challenging to provide a precise volume. For the sake of estimation, let’s assume a hypothetical grain volume.
3. Atomic Packing Factor (APF) for FCC Aluminum:
Aluminum crystallizes in a face-centered cubic (FCC) structure, which has an APF of 0.74. This means that 74% of the volume is occupied by atoms, while the remaining 26% is void space.
4. Volume Occupied by Atoms in a Grain:
If we denote the volume of a grain as V_grain, the volume occupied by atoms within that grain is:
[ V_{atoms} = V_{grain} \times APF_{fcc} = V_{grain} \times 0.74 ]
5. Volume of One Aluminum Atom:
The volume of a single aluminum atom can be approximated using the formula for the volume of a sphere:
[ V_{atom} = \frac{4}{3} \times \pi \times r^3 ]
Given the diameter (d) is 0.21 nm, the radius (r) is 0.105 nm:
[ V_{atom} = \frac{4}{3} \times \pi \times (0.105\, \text{nm})^3 \approx 4.88 \times 10^{-3} \, \text{nm}^3 ]
6. Estimating the Number of Atoms in a Grain:
The number of atoms in a grain can be estimated by dividing the volume occupied by atoms in the grain by the volume of a single atom:
[ \text{Number of atoms} = \frac{V_{atoms}}{V_{atom}} = \frac{V_{grain} \times 0.74}{V_{atom}} ]
Without a specific value for V_grain, we cannot calculate an exact number. However, if we assume a hypothetical grain volume, say 1 nm³, the calculation would be:
[ \text{Number of atoms} = \frac{1\, \text{nm}^3 \times 0.74}{4.88 \times 10^{-3}\, \text{nm}^3} \approx 151 \, \text{atoms} ]
Conclusion:
The number of aluminum atoms in a grain with an ASTM grain size of 5 depends on the actual volume of the grain, which is influenced by factors such as the material’s thickness and the area considered. By estimating the grain volume and applying the APF for FCC aluminum, we can approximate the number of atoms per grain. For instance, assuming a grain volume of 1 nm³, there would be approximately 151 atoms per grain. However, without precise measurements of the grain volume, this remains a rough estimate.