Iron (Fe) undergoes an allotropic transformation at 912°C: upon heating from a BCC to an FCC phase. Accompanying this transformation is a change in the atomic radius from R_BCC = 0... Iron (Fe) undergoes an allotropic transformation at 912°C: upon heating from a BCC to an FCC phase. Accompanying this transformation is a change in the atomic radius from R_BCC = 0.12584 nm to R_FCC = 0.12894 nm. Compute the percent volume change associated with this reaction. Does the volume increase or decrease?
Understand the Problem
The question is asking to compute the percent volume change associated with the allotropic transformation of iron from BCC to FCC, given changes in atomic radius and density. The goal is to determine whether the volume increases or decreases during this transformation.
Answer
The percent volume change is approximately $3.91\%$, indicating that the volume increases during the transformation from BCC to FCC.
Answer for screen readers
The percent volume change is approximately $3.91%$, indicating that the volume increases during the transformation from BCC to FCC.
Steps to Solve
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Calculate the volume of BCC iron The volume $V$ of a single atom can be approximated using the formula for the volume of a sphere: $$ V_{BCC} = \frac{4}{3} \pi R_{BCC}^3 $$ Substituting for $R_{BCC} = 0.12584 \text{ nm}$: $$ V_{BCC} = \frac{4}{3} \pi (0.12584)^3 \text{ nm}^3 $$
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Calculate the volume of FCC iron Using the same formula for the volume of a sphere for FCC: $$ V_{FCC} = \frac{4}{3} \pi R_{FCC}^3 $$ Substituting for $R_{FCC} = 0.12894 \text{ nm}$: $$ V_{FCC} = \frac{4}{3} \pi (0.12894)^3 \text{ nm}^3 $$
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Determine the percent volume change The percent volume change can be calculated using the formula: $$ \text{Percent Volume Change} = \left( \frac{V_{FCC} - V_{BCC}}{V_{BCC}} \right) \times 100% $$
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Perform calculations Compute the volumes using a calculator:
- For $V_{BCC}$: $$ V_{BCC} \approx \frac{4}{3} \pi (0.12584)^3 \approx 0.00823 \text{ nm}^3 $$
- For $V_{FCC}$: $$ V_{FCC} \approx \frac{4}{3} \pi (0.12894)^3 \approx 0.00855 \text{ nm}^3 $$
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Calculate the percent volume change Substitute the computed values into the percent change formula: $$ \text{Percent Volume Change} = \left( \frac{0.00855 - 0.00823}{0.00823} \right) \times 100% $$
The percent volume change is approximately $3.91%$, indicating that the volume increases during the transformation from BCC to FCC.
More Information
The transformation from BCC to FCC in iron is significant not only in terms of atomic structure but also contributes to changes in physical properties such as density and thermal expansion. The slight increase in volume during this process shows how structural transitions can influence material characteristics.
Tips
- Miscalculating the volume of spheres by using incorrect formulas. Ensure to use $V = \frac{4}{3} \pi R^3$ correctly.
- Not converting units appropriately, which can lead to erroneous results. Always check that all measurements are in consistent units.
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