Titanium metal has a density of 4.506 gcm3 and an atomic
radius...

Question

Titanium metal has a density of 4.506 g>cm3 and an atomic 
radius of 144.8 pm. In what cubic unit cell does titanium 
crystallize?

Accepted Answer

Hey everyone, we're told that the density of aluminum is 2.7 g per cubic centimeter. And it has an atomic radius of 143 m, aluminum will crystallize into what type of cubic unit cell first. Let's go ahead and think about the types of cubic unit cell. We have we have our simple cubic which gives us one atom per unit. So next we have our body centered cubic which gives us two atoms per unit cell. Next we have our face center cubic which gives us four atoms per unit cell. So let's go ahead and calculate the volume in cubic centimeters of our unit cell. And we can do so by taking our edge length which was symbolized by A. And we know that this is two times the square root of two times our radius. So plugging in these values we get two times the square root of two times our radius of 143 km. This gets us to an edge length of 404.465 m. Now, since we do on our volume in cubic centimeters, We're going to use our dimensional analysis and we know that one pick a meter contains 10 to the negative 12 m. And we also know that we have 10 to the negative two m per one centimeter. So when we calculate this out and cancel out all of our units, we end up with 4.4465 times 10 to the negative eight centimeters. And since our volume is in cubic centimeters we're going to take that edge length and cubit. So plugging in our values, we got four. So plugging in our values we get 4.4465 times 10 to the negative eight centimeters. And we're going to cube this value. This gets us to a volume of 6. times 10 to the negative 23rd cubic centimeters. Now let's go ahead and determine the number of atoms. And we can do so by using our dimensional analysis. So taking the edge length of 6.6167 times to the negative 23rd cubic centimeters. This is per one unit cell. Again, we're going to use our dimensional analysis here using our density which they provided to us as 2.7 g per one cc. And taking aluminum's molar mass which is 26.982 g per one mole of aluminum. And next using avocados number, we know that we have 6.022 times 10 to the 23rd atoms per one mole. Now, when we calculate this out and cancel out all of our units, We end up with a value of 3. atoms per unit cell which we can round to four atoms per unit cell. And as we stated earlier, a face centered cubic unit cell will have four atoms per unit cell. So our answer here is going to be that aluminum will crystallize in a face centered cubic cell. And we know this because we calculated four atoms per unit cell. Now, I hope this made sense and let us know if you have any questions.

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