Calculate the fraction of empty space in cubic closest packing to...

Question

Calculate the fraction of empty space in cubic closest packing to
five significant figures.

Accepted Answer

Hello. Everyone in this video trying to find the fraction of volume occupied by atoms in a cubic close packed structure. We have to run off to four significant figures in their final answer. The cubic close packed cell or structure is the same as the face centered cubic structure. So here again we're finding the fraction of volume occupied by the atoms in a cubic close packed structure. So we're finding the percent a volume occupied is equal to the volumes from the atoms divided by the volume of the unit cell, Multiplied by 100%. So the solution is that the cubic close packed structure is the same as the face centered cubic structure. So this means that we have so from just this sentence here, what we can get from that is that we have four atoms for every one unit. So Alright so the volume of the atom is equal to the volume of the sphere per adam. So this is equal to Well the volume us here is 4/3 times pi are cute. Again this per adam. Alright so the volume of the atom is then equal to the number of atoms. Her unit cell multiplied by the volume per adam. So now putting in actual numerical values Or just kind of using this formula for our situation. This is four atoms per unit cell. And then we're gonna go ahead and throw in the ocean right over here. So 4/3 times pi r cubed. Sure this little Benito yes pi r cubed developed by items. So the volume of a sphere developed by adam. Alright so then the volume of the atom is equal to 16/3 pi R cubed. For every unit cell. I'm gonna go and scroll down for more space here for the volume of the unit cell. This is equal to the volume of our cube, the volume of the unit. So as you go to the length multiplied by the length multiplied by the length. So that's just the length cubed. So the length in our case is equal to two times to raise the power of one over to multiply by art. So the volume of the unit cell is equal to Two times two raised to power 1/2 times are. And this is all going to be cute. So putting in our simplifying this equation further than its eight multiplied by to raise the power of 3/ multiplied by r cubed. Alright so now for the grand finale here the percent for the volume of the occupied per unit cell. Physical to the volume of our atoms divided by the volume of the unit cell. All multiplied by 100%. That's the equation that we already said earlier in the beginning. So just filling in the actual numerical values here. So we have 16 divided by three times pi R cubed, divided by eight times two. To raise the power of 3/2 multiplied by r cubed. And this all multiplied by 100 simplify this. We have 16/3 multiplied by pi Divide by eight, multiplied by two, raise the power of 3/ Times are cubed divided by R. Times 100%. And now simplifying this further, we get 0.740480 times 100%. We get that the percent of the volume of the occupied her unit Cell is unequal to again run into four different figures. In the answer is 74.05%. And this right here is going to be my final answer for this problem. Thank you so much for watching.

Calculate the fraction of empty space in cubic closest packing to five significant figures.

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