Compute the percent volume change associated with the transformation of iron from a BCC to an FCC phase. Does the volume increase or decrease?
Understand the Problem
The question is asking us to compute the percent volume change associated with the allotropic transformation of iron from the BCC to the FCC phase, taking into account the given changes in atomic radius and the impact on density and volume. We need to determine whether this results in an increase or decrease in volume.
Answer
The percent volume change during the transformation from BCC to FCC is approximately $19.33\%$, indicating an increase in volume.
Answer for screen readers
To find the exact values, you'll need to compute the volumes based on the formulas:
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Volume of BCC: $$ V_{BCC} \approx \frac{4(0.12584 \times 10^{-9})^3}{\sqrt{3}} \approx 1.50 \times 10^{-29} , \text{m}^3 $$
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Volume of FCC: $$ V_{FCC} \approx \frac{4(0.12894 \times 10^{-9})^3}{\sqrt{2}} \approx 1.79 \times 10^{-29} , \text{m}^3 $$
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Percent Volume Change: $$ \text{Percent Change} \approx \frac{1.79 \times 10^{-29} - 1.50 \times 10^{-29}}{1.50 \times 10^{-29}} \times 100 \approx 19.33% $$
Thus, the volume increases by approximately 19.33%.
Steps to Solve
- Calculate the Volume of BCC Unit Cell
The volume of a body-centered cubic (BCC) unit cell is given by the formula:
$$ V_{BCC} = \frac{4R_{BCC}^3}{\sqrt{3}} $$
Substituting the value of $R_{BCC} = 0.12584 , \text{nm}$ (note convert to meters for calculations):
$$ V_{BCC} = \frac{4(0.12584 \times 10^{-9})^3}{\sqrt{3}} $$
- Calculate the Volume of FCC Unit Cell
The volume of a face-centered cubic (FCC) unit cell is given by the formula:
$$ V_{FCC} = \frac{4R_{FCC}^3}{\sqrt{2}} $$
Substituting the value of $R_{FCC} = 0.12894 , \text{nm}$:
$$ V_{FCC} = \frac{4(0.12894 \times 10^{-9})^3}{\sqrt{2}} $$
- Calculate the Percent Volume Change
The percent volume change is calculated using the formula:
$$ \text{Percent Change} = \frac{V_{FCC} - V_{BCC}}{V_{BCC}} \times 100$$
Now, plug in the values of $V_{BCC}$ and $V_{FCC}$ calculated in the previous steps to find the percent change.
- Determine Volume Increase or Decrease
Based on the volume values calculated, if $V_{FCC} > V_{BCC}$, the volume increases, otherwise, it decreases.
To find the exact values, you'll need to compute the volumes based on the formulas:
-
Volume of BCC: $$ V_{BCC} \approx \frac{4(0.12584 \times 10^{-9})^3}{\sqrt{3}} \approx 1.50 \times 10^{-29} , \text{m}^3 $$
-
Volume of FCC: $$ V_{FCC} \approx \frac{4(0.12894 \times 10^{-9})^3}{\sqrt{2}} \approx 1.79 \times 10^{-29} , \text{m}^3 $$
-
Percent Volume Change: $$ \text{Percent Change} \approx \frac{1.79 \times 10^{-29} - 1.50 \times 10^{-29}}{1.50 \times 10^{-29}} \times 100 \approx 19.33% $$
Thus, the volume increases by approximately 19.33%.
More Information
The allotropic transformation of iron is significant in materials science, influencing its mechanical properties. The BCC structure is denser while the FCC is more ductile, crucial for applications in construction and manufacturing.
Tips
- Incorrect unit conversion: Always ensure units are consistent, especially when converting nanometers to meters.
- Misapplying volume formulas: Ensure the correct unit cell volume formula is used for BCC and FCC structures.
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