This is designed as an in-class activity that steps students through the calculation of the amount of unoccupied (or empty) space and packing fraction in cubic packing structures. Questions imbedded in the exercise are designed to get students to “own” the concept of the empty space more effectively and then use that concept to understand how some ionic solid structures can be visualized as placing smaller ions in the holes available, for example, NaCl can be thought of as Na+ in a face-centered lattice structure with Cl- in every octahedral hole.
A student should be able to calculate volume of a cubic unit cell given the length of a side.
A student should be able to calculate the radius of an atom given the cubic crystal structure and length of a side in a unit cell.
A student should be able to calculate hard-sphere volume of an atom knowing the length of a side of a cubic unit cell.
A student should be able to calculate the unoccupied space within a cubic unit cell.
A student should gain an understanding about the relationship between unit cells, the occupied and unoccupied volume of the cells and a use for the unoccupied volume.
Students need a calculator.
Not required but if had some space filling models of primitive, face-centered and body-centered cubes along with cubic and hexagonal closest packed structures they would be valuable to have available. A model of sodium chloride would also be useful.
If you have software like CrystalMaker available with files for simple, face-centered and body centered cubes along with sodium chloride available for viewing and rotating that may helpful as well.
The calculation for simple cube is mapped out in a step-by-step fashion because of the breadth of skills my students bring to the 100 level course where I plan to use this activity. Subsequent packing structures have hints provided as useful intermediate steps but do not provide an explicit scaffold in attempt to make students more independent learners.
To use in class instructors will want to decide how to edit the handout attachment to provide additional space to complete the work. I teach using the DyKnowTM software on tablet based PC’s so I would cut and paste each step (separated by a blank line in the MSWord document) onto a separate slide in DyKnowTM.
The sections in the handout digging deeper, side distractions, and challenge questions could be completely omitted at the instructor’s discretion or assigned as homework or even extra credit.
One way to build on this exercise for upper level students would be to have students work through the geometry that justifies why the hard sphere radius for a tetrahedral hole can only be 0.225r of the atoms creating the tetrahedral hole, and 0.414r for the octahedral hole.