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I don't agree at all with this question, and I think my teacher does not understand his topic. I hope you can prove me wrong.

The question was this one:

Potassium crystallises in a Body Centered Cubic way. It's density is 0.853 g/cm3, its molar mass is 39.9 g/mol.

Calculate the number of atoms per cell and its atomic packing factor.

My argument is this:

This question makes no sense because BCC means it has 2 atoms per cell and has a 0.68 Atomic packing factor. You might say: That's the theoretical model, what we want you to calculate is the actual thing.

And I would say: Well you surely can't do it with the regular formula because that formula depends on a perfectly organised crystal and even makes reference to the relation 4r=(3)^.5*a which depends on a regular lattice.

When I do the calculations the way you want it I get 1.929 atoms per cell, which is absurd. Does this mean some cells have more atoms than others? Well if that's implied then theres no way to talk about a unit cell and make reference to the 4r=(3)^.5*a relationship I mentioned. Another interpretation of this cypher (1.929) is that maybe atoms are of varying mass, and the mathematics is telling you atoms are incomplete. But I think that too is absurd.

How does one interpret these things? Or is it in fact nonsense (they question they gave me) ?

Thanks a lot.

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  • $\begingroup$ Hi, I'll assume you did all the numbers correctly. In which case perhaps nature is non-ideal. Maybe there are some defects or impurities in the lattice. $\endgroup$ – George Herold Oct 2 '14 at 0:48
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Yes, that certainly seems like a rather odd question. Given that the crystal is BCC, that information alone means that there are 2 atoms per cell, and the atomic packing factor is $\pi\sqrt{3}/8\approx 0.680$. The density and molar mass are irrelevant additional pieces of information.

A sensible thing to ask for given those three pieces of information (BCC structure, density and molar mass) would be the implied atomic radius.

You don't say how the 1.929 atoms/cell was calculated, but if it was calculated from an atomic radius (which wasn't specified as a part of the problem), it wouldn't make sense to calculate things in that direction, because the 2 atoms per cell implied by the BCC crystal structure is a more precise piece of available information than what a table gives for potassium's atomic radius. Crystals do have crystallographic defects, and a metal like potassium is going to actually be polycrystalline instead of being one solid crystal, but I'm presuming you aren't being expected to make calculations based on those considerations, and even if you were, the calculations presumably would be expressed in a different way than as a non-integer number of atoms per cell.

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  • $\begingroup$ I used the equation ρ = ZM / a³N (where Z=#atoms, M=Molar mass, a=length of a side of the unit cell, N=avogadros number) and the relationship I spoke of in the larger part of my text. Thanks. $\endgroup$ – DLV Oct 2 '14 at 4:10

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