Cesium chloride (CsCl) and sodium chloride (NaCl) adopt different three-dimensional…

CHEMISTRY · Multi-part · AP · Chemistry · Compound Structure and Properties · Compound Structure and Properties

Problem

Cesium chloride (CsCl) and sodium chloride (NaCl) adopt different three-dimensional arrangements of ions even though both substances consist of monatomic +1 cations and −1 anions. A student attempts to rationalize this observation by sketching two separate unit cells. In both sketches, the student draws a simple cubic array in which the cations occupy the eight corners and the anions occupy the centers of the cubes. The student uses identical edge lengths for the two unit cells and labels them “CsCl-type (sketch A)” and “NaCl-type (sketch B),” respectively. You are provided with the following ionic radii:

r_{\text{Cs}^+}=1.74\,\text{Å}

and

r_{\text{Cl}^-}=1.81\,\text{Å}

;

r_{\text{Na}^+}=1.02\,\text{Å}

and

r_{\text{Cl}^-}=1.81\,\text{Å}

. (a) Explain why using the same edge length for both sketches is inconsistent with the observed structural difference between CsCl and NaCl. (b) Using the given ionic radii and assuming ions are hard spheres that just touch along the body diagonal in the CsCl-type arrangement, calculate the minimum edge length,

a_{\text{CsCl}}

, required for the CsCl unit cell. (c) Under the same hard-sphere assumption, calculate the minimum edge length,

a_{\text{NaCl}}

, required for the NaCl unit cell, where cations and anions touch along the edge of the face-centered cube. (d) Compare your calculated edge lengths to evaluate whether the student’s decision to use identical cube sizes in sketches A and B is physically reasonable. Provide a brief justification based on your numerical results and Coulombic considerations.

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