A researcher is studying the effect of incremental carbon addition on the density of…
CHEMISTRY · Multi-part · ISC · Chemistry · Some Basic Concepts of Chemistry · Mole concept
Problem
A researcher is studying the effect of incremental carbon addition on the density of body-centered cubic iron. The researcher records the following data at 298 K.
Textual table:
Fraction of octahedral interstitial sites occupied by C | Measured density / g cm−3
0.000 | 7.87
0.025 | 7.95
0.050 | 8.03
0.075 | 8.11
Assume:
• The lattice parameter of Fe remains constant as carbon is added.
• There is one octahedral interstitial site per Fe atom.
• The atomic masses are 55.85 g mol−1 for Fe and 12.01 g mol−1 for C.
Answer the questions that follow.
(a) Derive an algebraic expression for the theoretical density ρ of the Fe–C alloy as a function of x, the fraction of interstitial sites occupied by carbon. Express ρ in terms of x and the molar masses given.
(b) Using your expression from part (a), calculate the theoretical density when x=0.050 and compare it with the corresponding measured value in the table. Report the percent difference between the two values.
(c) The researcher observes that the measured densities are consistently higher than the theoretical values you calculated. Propose one plausible structural explanation for this discrepancy, based on changes to the metallic lattice that could accompany carbon incorporation.
(d) Suppose the researcher wishes to achieve a target density of 8.25 g cm−3. Using your model from part (a), estimate the interstitial occupancy fraction x required, and comment on any practical limitations that might prevent reaching this occupancy in an actual steel sample.