Question #44192

Mischmetal is a pyrophoric alloy containing $\sim 95\%$ lanthanides (predominantly \ce{Ce} and \ce{La}) and $\sim 5\%$ \ce{Fe}, hardened by interstitial elements like Carbon. The ability to accommodate interstitial atoms depends on the absolute size of the lattice voids, which is affected by the lanthanide contraction despite the packing fraction remaining constant.\n\nAssume both Cerium ($\gamma$-\ce{Ce}) and Lutetium (\ce{Lu}) crystallize in ideal face-centered cubic (fcc) lattices at $298\text{ K}$. The atomic radii are $R_{\ce{Ce}} = 183\text{ pm}$ and $R_{\ce{Lu}} = 173\text{ pm}$. Carbon atoms (radius $r_{\ce{C}} = 77\text{ pm}$) are inserted into the octahedral voids of both lattices.\n\nAssuming hard-sphere behavior where the lattice expands uniformly along the edge to accommodate the carbon atom, calculate the difference in the required unit cell edge length distortion between Lutetium and Cerium, i.e., $(\Delta a_{\ce{Lu}} - \Delta a_{\ce{Ce}})$.\n\n(Take $\sqrt{2} = 1.414$. Express your answer in pm).