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Argon crystallizes in the face-centered cubic arrangement at $40 \mathrm{K}$. Given that the atomic radius of argon is $191 \mathrm{pm},$ calculate the density of solid argon.

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$1.69 \mathrm{g} / \mathrm{cm}^{3}$

Chemistry 102

Chapter 11

Intermolecular Forces and Liquids and Solids

Liquids

Solids

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04:08

In physics, a solid is a state of matter characterized by rigidity and resistance to changes of shape or volume. Solid objects have a definite volume, they resist forces (such as pressure, tension and shear) in all directions, and they have a shape that does not change smoothly with time. The branch of physics that studies solids is called solid-state physics. The physical properties of solids are highly related to their chemical composition and structure. For example, the melting point of ice is significantly lowered if its crystal structure is disrupted.

03:07

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, a liquid is one of the four fundamental states of matter (the others being solid, gas and plasma). A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena.

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Argon crystallizes in the …

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Argon crystallizes in a fa…

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2 Argon crystallizes in a …

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Argon has a cubic closest …

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What is the atomic radius …

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Argon crystallizes at $-23…

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The density of argon (face…

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The radius of an argon ato…

let's start this question by drawing out what an FCC cell looks like. So for an FCC from one face, it's going to look like this where we have one cell in the centre, one Adam in the center of the edge and one in each corner. Now, this is not to scale by any means, but it gives us the information we need to know, which is that these Adams lineup right on this axis right here. And if we counted up, there is exactly one, 23 four radius is because remember that if we split this Adam in half, it has two radius is one on each half. Does this high ponte news which is gonna give us the edge of the cell? This is important because we need to know the volume of ourselves. So the volume of ourselves is going to be equal to the length of our edge. Cute. But we need to know this edges in terms of our atomic radius to be able to do anything with it. So let's go ahead and figure that out. So from this high pot news here, we can say that we have four r squared is going to be equal to one edge here squared, plus one edge here squared, which is just two edges squared. So we can therefore say that one edge equals four are over square root of two. And if we clear the fraction that's equal to root two times to our okay. So now we know that the volume of ourselves is going to be equal to eight square root. Eight are cute if we go ahead and do a substitution of this and Cuba tal to Cube is eight, and then we have in our cubed leftover it's the volume of ourselves is a Route eight and let's go ahead and run this math using are given radius, which is 191 p commuters. So visa equals eight Route eight, 191 Pekka meters cute, and that is equal to 1.5766 times 10 to the eight Pekka meters cubed per cell because this is a single cell that we are considering. Now let's move along and consider the density that we're working with in terms of atoms. So to figure this out, our density atomic, not cellular, just the atomic density is gonna be equal to, well, the molar mass, which is 39 point 948 Rams terminal times, one mole, huh? Avocados number, which is 6.2 to times 10 to the 23 atoms. And that gets us a very large number for a density, which is six point 634 times 10 to the negative, 23 grams her Adam. Now let's figure out our cellular density or de so which is going to be the grams we have per self. So that's just gonna be our atomic density times number of atoms we have Purcell, which we know for a fact in an FCC cell is going to be four because we have one on this face, one on the opposite face, one on this face, one on this face, one on the top face and one of the top based, and each of those are all 1/2 So that gets us ah total of six times 1/2. Plus, we have eight corner pieces, eight corners on our cube, and each corner has 1/8 of a cell, which equals four atoms total in an FCC cell. So we know that we have four atoms, Purcell in that multiplication. And if we multiply 6.634 times 10 to the negative 23 we get an astonishingly large number, which should come as no surprise, which is 2.65 times 10 to the negative 22 grams. Uh, so okay, now, lastly, we can figure out our density not in terms of cell, but in terms of, ah meters, because we do know the dimensions of the cell, as we had previously calculated. So let's go ahead and take the density of ourselves and divide it by the volume of ourselves. So we're gonna have 2.65 times 10 to the negative 22 grams per cell. And we will divide that by 1.577 times 10 to the aid PICO meters Cube. Purcell. But we need to get this into centimeters, not Pekka meters, because that's a much more convenient and common way of reporting density. So if we look up a conversion from Pekka Meters Cube, two Centimeters Cube, we see that it's going to be times 10 to the negative. 30 centimetres cubed Purcell. Okay, go ahead and run that calculation and we get a final number of 1.6 nine grams per centimeter cubed.

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