The density of a sample of metal was measured to be 6.84 g>cm3. An X-ray diffraction experiment measures the edge of a face-centered cubic cell as 350.7 pm. What is the atomic weight, atomic radius, and identity of the metal?

Video transcript

Hey everyone, we're told that a sample of metal was determined to have a density of 8.96 g per cubic centimeters. The metal crystallizes in a face centered cubic cell. And using X ray diffraction, the edge length was measured to be 362 PICO meters determine the atomic weight, atomic radius and identity of the metal. First let's go ahead and determine the radius. So for a face centered cubic cell we know that our edge length, which we symbolize with a is equal to two times the square root of two times our radius. Now, when we plug in these values we get 362 PICO meters which was told to us in our question stem is going to be equal to two times square root of two times our radius, Dividing both sides by two times the square root of two. We end up with a radius of 128 km. Now let's go ahead and calculate for the volume of the cell in cubic centimeters. Now we know that the volume of our unit cell is equal to the cube of our edge link. So first let's go ahead and take Our edge length of 362 PICO meters and let's convert this into centimeters. So per one pick a meter. We know that we have 10 to the negative 12 m. And we also know that per one centimeter we have 10 to the negative two m. So here we're just using our dimensional analysis. So when we cancel out our units And calculate this out, we end up with an edge length of 3.62 times 10 to the negative eight cm. Now, since we want our volume in cubic centimeters, we're going to take 3.62 times 10 to the negative eight centimeters and cube this. This gets us a volume of 4.7438 times 10 to the negative 23rd cubic centimeters. Now, let's go ahead and use this value to determine our atomic weight. Now an important thing to know for a face centered cubic cell Is that we have four atoms per unit cell. So first let's go ahead and determine our atomic weight. So we have 4.7438 times 10 to the negative 23rd cubic centimeters. And this is going to be per unit cell. Now we're going to use our dimensional analysis here Using our identity, we know that we have 8.96 g per one cc. And since we have a face centered cubic cell, we have one unit cell. And this contains for atoms. Using avocados number, we know that we have 6.22 times 10 to the 23rd atoms per mole. So calculating this out and canceling out all of our units, we end up with an atomic weight of 63.99 g per mole, Which estimates to about 64.0 g Permal. Now let's go ahead and refer to our periodic table. And it looks like the identity closest to this atomic weight is going to be our copper. So this is going to be our final answers. Now, I hope this made sense and let us know if you have any questions.