The mineral wustite is a nonstoichiometric iron oxide with the empirical formula FexO, where x is a number slightly less than 1. Wustite can be regarded as an FeO in which some of the Fe sites are vacant. It has a density of 5.75 g>cm3, a cubic unit cell with an edge length of 431 pm, and a facecentered cubic arrangement of oxygen atoms. (c) Each Fe atom in wustite is in either the +2 or the +3 oxidation state. What percent of the Fe atoms are in the +3 oxidation state?

Video transcript

Hello. In this problem we are told a hypothetical cobalt oxide as the formula cobalt subscript X. O. Where X. Is 00.9 to 2.98. Think of this as football to oxide with some cobalt sides vacant. The density of the oxide is 5.12 g per cubic centimeter. The oxide crystallizes in a face centered cubic arrangement for the oxygen atoms. And the unit cell has an edge length of 454 PICO meters cobalt atoms in this oxide or either cobalt two or cobalt three. We wonder what percent of the cobalt atoms in this oxide have an oxidation state of plus three. We will solve this problem in three steps. Step one will use the density of the oxide and the edge length of the cube to find X. Then we will find the average oxidation state from X. And the third step then we will find the percentage of cobalt three plus lines from the average oxidation state. So starting with step one we're told that we have a face centered cubic unit cell and the oxygen atoms occupy the eight corners of this unit cell. Each of these eight corners is shared with eight other unit cells. So we have 1/8 of an atom To have eight corners. Each have 1/8 of an atom. So we have one adam and then the oxygen's also resided the faces of this cube. We have six faces that are each shared with two unit cells. So we have half of an atom within a single unit cell. So we have a total of three atoms. So altogether there are four oxygen atoms and this face centered cubic, raise it now we'll find the volume. Even the edge ling volume then is equal to 450 for PICO meters cubed. We'll convert our volume from PICO meters cubed two cubic centimeters. We have one times 10 to the 12 PICO meters Is equal to one m. We're cute both of those. And our units PICO meters cancel. one m Is equal to 100 cm. Week you both of those and our units of meters cancel. So we find the volume is equal to 9.35, 8 Times 10 to - cubic centimeters. But then take our volume and convert it to mass. Using the density as a conversion factor. So we were told that the density of the oxide is 5. grams per cubic centimeter. So we set up our density so our units of cubic centimeters cancels. And this works out to four times 10 to the -22 g. So that is the mass of our oxide. We will now determine the mass of the oxygen. So we have four oxygen atoms within this, you know, so one more of oxygen, 26.02, 2 times 10-23 oxygen atoms. So we'll cancel our oxygen atoms and one more of oxygen has a mass of 15.9994 g, roughly 16. And our molds of oxygen cancels. So this works out to 1.063 times 10 to the -22 g of oxygen. But then take the mass of our oxide 4.791 times 10 to the minus g and subtract off the mass of the oxygen 1.63 times 10 to minus 22 g. And we get 3.7 - eight times to the -22g. And that's then our massive cobalt. Well then take our massive cobalt 3.7-8 times 10 to the -22g And determine how many cobalt atoms we have. We have one mole of cobalt has a mass of 58.933g of cobalt. To our massive cobalt cancels One mole of cobalt has 6.022 times 10 to the 23rd cobalt atoms. The molds of cobalt cancel. So we have 3.098 cobalt atoms. So within this unit cell We have three That's right. This is at eight. Go back up here. Sorry Mr eight. So this is 3.8098 cobalt atoms. So we take our number of cobalt atoms 3.8098. Bottoms for every for oxygen atoms This works out to 0.95245 which is equal to Rx. Within the problem statement we were told the extra b between 0.92 And 0.98. So this value of X makes sense. And so we can write the formula for this cobalt oxide. It would be cobalt . oh. The second step then we are now going to find the average oxidation states of cobalt From X. So oxygen has an oxidation state of -2. Cobalt will therefore have an ox stage data plus two since they will have to Some 20. And so that means then the average Oxidation states of cobalt is plus two, divided by 0. 5245, Which is equal to 2. 85. So this is our average oxidation state and step three. Then you're now going to find the percent of the um Coppola ions from this average oxidation state. So we have cobalt two plus, we'll say the fraction of cobalt that is cobalt two plus is X one and the fraction that is cobalt three is X two, X one and X two Need some to one. You can find then the average oxidation state if we take the oxidation state particular ion and the fraction um that is that ion we do that for each different ion. And so we found the average oxidation state was 2.09985. The oxidation state of cobalt two plus is plus two Times the proportion that of cobalt that is about to which we call X one plus the onstage state of cobalt three, which is plus three times the fraction of cobalt that is cobalt three, which we call an X two. So here we have one equation and two unknowns. So we are going to substitute X two for one minus X one since X two and X one equal one now have 2.9985 is equal to two X one plus three times one minus X one. This then works out two X one plus three minus three X one, move the three to the other side. We get 2. -3 is equal to - & X one then Works out to 0. X two, which is equal to 1 -11. Then works out to zero 09985 X two again is the proportion that is cobalt with an oxidation state at plus three. We asked 5%. So we multiply our fraction by 100%. So it is 9. cobalt three plus. So this oxide that we have, the percent of football that is cobalt three plus is pretty small, it's 9.99%. But that makes sense when we look at the average oxidation state, Average oxidation state is close to two. So we would expect that the vast majority of cobalt does exist as cobalt two plus, and that would be 90%. This problem, we are asked to determine what percent was cobalt three plus, and that worked out to be 9.99%. Thanks for watching Hope this out.