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Ch.12 - Solids and Modern Materials
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All textbooksBrown 14th EditionCh.12 - Solids and Modern MaterialsProblem 116c

Chapter 12, Problem 116c

Sodium oxide (Na2O) adopts a cubic structure with Na atoms represented by green spheres and O atoms by red spheres.

(c) The unit cell edge length is 5.550 Å. Determine the density of Na2O.

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Hey everyone, we're given the representation for the cubic structure of potassium oxide, calculate its density. If it's scheduling this 643 PICO meters first, let's go ahead and count how many oxygen atoms we have in order to determine how many potassium oxide units we have. So looking at our cubic structure, We can see that we have eight corners and that will be and eight. So we have a corners and we have 1/ adam per one corner. And now we're going to go ahead and add the six faces. So we have six faces and we're going to multiply that by one half adam per one face. Now when we calculate this out and cancel out our units, we end up with a total of four oxygen atoms. So since we have four oxygen atoms, this must mean that we have eight potassium atoms, which also means that we have or potassium oxide atoms. Since our formula is two potassium ions plus one oxygen ion gets us to potassium oxide. And using that reaction, we're able to get these values. Now let's go ahead and calculate our density. Now we've learned that our density equals mass divided by volume. So first let's go ahead and determine our volume. Now we were told that we had 643 km. Now we want to convert this into volume. Using our dimensional analysis, we know that one PICO meter contains 10 to the negative cm and since volume is in cubic centimeters we're going to have to cube each of these. Now, when we calculate this out, we end up with a value of 2.6585 times 10 to the negative 22 cubic centimeters. And this will be our volume. Now to calculate our mass, we can go ahead and start off by taking the molar mass of potassium oxide, which is 94.2 g of potassium oxide per one mole of potassium oxide. Now converting our mole In two units, we know that one mole of potassium oxide contains 6.02, 2 times 10 to the 23rd units. And as we calculated, we know, we have four potassium oxide units in our cubic structure. So when we calculate this out and cancel out our units, we end up with a mass of 6.25, 7 times 10 to the -22 g. Now, to calculate our density, we're going to take our mass of 6.257 times 10 to the negative 22 g and divide that by our volume of 2.6585 times 10 to the negative 22 cubic centimeters. And this will get us to a density of 2.35 g over cubic centimeters, which is going to be our final answer. Now, I hope that made sense. And let us know if you have any questions
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