An unknown metal is found to have a density of 7.8748 g/cm3 and to crystallize in a body-centered cubic lattice. The edge of the unit cell is 0.28664 nm. Calculate the atomic mass of the metal.

Video transcript

All right. Hi everyone. So this question says that an unknown metal is found to have a density of 7.20 g per centimeter cubed and to crystallize in a body centered cubic lattice, the edge of the unit cell is 0.301 nanometers, calculate the molar mass of the metal. And here we have four different answer choices labeled A through D proposing different values for the molar mass in units of grams per mole. So let's go ahead and get started right. In this case, we're provided the value for the density which gives us a relationship between mass and volume. So before we can go ahead and use the value of the density to get to our mass, let's first go ahead and find the volume of the cubic lattice. In this case, V that's the volume is equal to a cubed or A is the length of each edge. So first, I'm going to plug in the value for a that's 0.301 nanometers. And then I'm going to cube this quantity. Now this is going to give us a value for the volume in nanometers cubed. But the density provided is in units of grams per centimeter. Cute. So here I'm going to use a conversion factor to convert nanometers into centimeters. And then I'm going to cube or apply that quantity that conversion factor to the third power. Now because nanometers are the starting units, nanometers should go in the denominator of my conversion factor to make sure those quantities cancel out or those units. So here one multiplied by 10 to the power of negative seven centimeters corresponds to one nanometer. And so I'm going to take this conversion factor and cube it so that my units match this completely cancels out nanometers. Cute. And so this gives us a volume of 2.727 or rather 2.7271 multiplied by 10 to the power of negative 23 centimeters. Cute. So now we can go ahead and find the mass of the lattice itself. So we can take the volume that's 2.7271 multiplied by 10 to the negative 23rd cubic centimeters and multiply this by the value of our density. So that's 7.20 g per centimeter. Cute. So now after multiplying these two quantities together, the mass is equal to 1.9635 multi by 10 to the power of negative 22 g per unit sub. So lastly, right, to convert the mass into the molar mass, we're going to convert grams per unit cell into grams per mole. So now let's go ahead and find the molar mass. I am going to take my value for the mass thats 1.9635 multiplied by 10, the negative 22nd grams per unit. So and first, I'm going to relate the unit cell to how many atoms are present. Now recall that we were given information about the edge of a body centered cubic lattice. That means that two atoms are present for every edge of the cube itself, one on each corner. This means that one unit cell corresponds to two atoms. And because the unit cell was placed in the numerator of my conversion factor, this cancels out those original units. And so our answer would be reported in grams per atom, which we can then convert into grams per mole using Avogadro's number. Now avocados number states that there are 6.022 multiplied by 10 to the 23rd power atoms or other particles per one mole. So this cancels out my units of atoms. And the answer is going to be reported in grams per mole corresponding to the molar mass. So now after multiplying all of my numerators and dividing by all of my denominators, the molar mass equals 59.1211 g per mole. But if I rounded to three significant figures, my answer becomes 59.1 g per mole and there you have it. So our answer 59.1 g per mole corresponds to option C in the multiple choice. And with that being said, thank you so very much for watching. And I hope you found this helpful.