Sodium metal (atomic weight 22.99 gmol) adopts a body-centered cubic structure with a density of 0.97 gcm3. (b) If sodium didn't react so vigorously, it could float on water. Use the answer from part (a) to estimate the density of Na if its structure were that of a cubic close-packed metal. Would it still float on water?

Question

Sodium metal (atomic weight 22.99 g>mol) adopts a body-centered cubic structure with a density of 0.97 g>cm3. (b) If sodium didn't react so vigorously, it could float on water. Use the answer from part (a) to estimate the density of Na if its structure were that of a cubic close-packed metal. Would it still float on water?

Accepted Answer

everyone in this video trying to calculate the density of lithium of the structure was a cubic close packed structure and we had to identify it will still float on water. So let's first calculate the volume of our lithium unit cell. So and the problem is is that lithium has a body centered cubic structure. So we'll simplify that into BCC. So that just means that there's two atoms her unit cell. Alright, so let's go ahead and get our dimensional started, dimension analysis started. So we have a two lithium Adams Multiplying by 6. g. And it's going to be divided. Bye. 6.2 times 10 to 23rd lithium atoms which is odd. Guardo's number. They will multiply this So that one ml per every 0. g. And of course we want to convert this into angstrom Cute. So multiply this by one times 10 - 24. And for every one month later. So you can see here in the first part that the lithium atoms will go ahead and cancel over to the right. The middle layers will cancel. And then finally of course then the grams go ahead and cancel out as well. So then putting all these values into my calculator. My numerical value is going to be 43 points 17 units being ancient. Cute. Alright. Now for the volume it's equal to a cute. So a is equal to 3.51 angstrom. What I basically did was just cube root this number here and I got this a value. Not just scrolling down though here. So we have a little bit more space. Alright, so now we'll go ahead and calculate for the radius. All right, so in a BCC the atom touch along the diagonal. So then that would be just the D two Which goes to four times art And by paid a grand theorem D two is equal to Cube root of three times eight. So the radius is equal to Square it up three Times are eight divided by four. Alright, so again, more of my numerical values square root of three A is what we calculate for in green right on top. So we have 3.51 Angstrom divided by four. This would mean that my radius is equal to 1. extra. Now calculating the side of cubic close packed structure so say side of cubic. Alright, so eight is equal to four times art Divided by the Square Root of two. Again, hugging in my numerical values four times the 1.52 which we have solved for an orange right over here Divided by the square root of two. That gives me my a value to be 4.30 and strong. Now finally settling for the new dead city. Do this in a different color. So new density. So we have for lithium Per 4.30 angstrom. Of course this is going to be cute. You'll multiply this so that You can have 6.941 g of lithium. Her 6.022 times 10 to the 23rd lithium atom. And of course that's off guard this number. That's why it looks so familiar. Then we'll go ahead and multiply this by one times 10 to the 20. For cute. For every one millimeter. So, units wise you have that the lithium will go ahead and cancel out not this lithium but the other lithium grounds. Again, we have the lithium items canceling out. Then of course we have the ancient cubed canceling out. And the only units that should be left is grounds of lithium over milliliters. And that's the unit for density. So putting in those values into my calculator. I'll get the numerical value of 0. units being g per no later. So, we have this new density here and this is still less dense than water and because it's less dense than water, what that means is that it will float. Alright, So scrolling back up. We're being asked if or the new to calculate for the new density of this lithium and if it's going to float. So, new density is 0.580 g per milliliters. And yes, it will float. All right. Thank you all so much for watching

Verified Solution

Key Concepts

Density

Crystal Structures

Buoyancy