Anonymous
Anonymous asked in Science & MathematicsChemistry · 2 months ago

# How do I find the radius of this?

Chromium forms a body centered cubic crystal. If the length of an edge is 2.884 angstroms, and the density is 7.20 g/cm^3 , what is the radius of a chromium atom in angstroms?

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• Dr W
Lv 7
2 months ago

I cover all the variations of these crystal cell problems in my answer here

spend some time and sort through it.

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this problem.  see the image I've attached

in image #1.. note the edges of the cell start in the CENTER of each corner atom

in image #2.. note how the corner atom exists in 8 adjacent cells

in image #3.. note how 1/8th of each corner atom is in any given cell and how the central atom is entirely in the cell

in image #4..

.. (1)  note that for simple cubic cells, the atoms touch on an edge

.. .. ... so that edge length = 2*r.

.. (2) For face centered cubic cells, the atoms touch on the diagonal

.. .. .. of the face.  so that face diagonal length = 1r + 2r + 1r = 4r

.. .. . .then using p-theorem

.. .. .. ... . (edge length)² + (edge length)² = (face diagonal length)²

.. .. .. ... . 2*(edge length)² = (face diagonal length)²

.. .. .. ... . √ 2*(edge length) = (face diagonal length)

.. .. .. ... .√ 2*(edge length) = (4r).

.. .. .. .. ... edge length = 4r/√ 2

.. (3) for body centered cubic cells, the atoms touch on the diagonal

.. .. . .of the CELL.  which has length = r+2r+r = 4r

.. .. ...with legs of 1 edge and 1 face diagonal so that

.. . . ...... . (edge length)² + (face diagonal length)² = (cell diagonal length)²

... .. .. ... . 3*(edge length)² = (cell diagonal length)²

.. .. .. ... . √3*(edge length) = (cell diagonal length)

.. .. .. ... .√ 3*(edge length) = (4r).

.. .. .. .. ... edge length = 4r/√ 3

see the pics

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all that behind us, you don't need density to finish this

.. r = √ 3 / 4 * (edge length) = √ 3 / 4 * (2.884Å ) = 1.249 Å