The radius of a xenon atom is 1.3 * 10 - 8 cm. A 100-mL flask is filled with Xe at a pressure of 1.0 atm and a temperature of 273 K. Calculate the fraction of the volume that is occupied by Xe atoms. (Hint: The atoms are spheres.)

Video transcript

Hello everyone in this video we'll be working with the ideal gas law equation. So PV goes N. R. T. So first I want to go ahead and write out what information we are given. So by reading problem I can see that we have a millimeter cylinder. It's filled with a ton of krypton gas At a pressure of 2.6 a. t. m. A temperature of 2 78 calvins As well as our radius of 2.02 Times 10 to the -8 centimeters. So that's all the information that we're given. And let's see. So we said we have the ideal gas law creation being PV equals an R. T. So let's see what pieces of the puzzle we have given. So we have a volume we have a pressure and we have a temperature. So we have our P. V. And T. R. Is always going to be something that we know. So that our is going to be our guest law constant being 0.08206 units of 80 M. Times later. All over Kelvin's times multi. Alright so you can see that we're obviously going to need to solve our end value. So what we're trying to find is our moles which is our end value. Alright so whenever we're working with the pivot her tears since we have this gas law constant, whatever information that we're going to be plugging it into our equation, we want the units to match the Guest law units as well. The guests are constant unit as well. I apologize. Alright. So because we're solving for N. I'm going to go ahead and divide each side bar R. T. So we can get the end by itself because that is what we're solving for. So we get the new equation of N equals PV all over our T. All right. So let's first check to see if our units will match here. If we need to do any conversions. So we have a volume. We have milliliters but we want leaders. So we need to do some conversions. A. T. M. With hmm That's correct Calvin with kelvin. That is great. All right. So just the volume. So we have our 250 millions. We want to convert that into a leader. So let's see we have 1000 millimeter for every leader and then you can see that the male leaders will cancel. Giving us a unit of leaders in the answer. And that's exactly what we need. 0.250 leaders. All right. Now plugging it into our equation, We start off with our pressure being at 2.6 A. T. M. Multiplied with our volume that would just sell for zero 250 leaders all over. Then we have our guest lock constant 0.08206. He does have a tm times leader all over kelvin times more. And then we have our temperature Being at Calvin. So We see that units of 80 M will cancel. Leaders will cancel and Kelvin's will cancel. So our unit our our answer the malls is going to be 0.0 49 moles of krypton. All right. So because we're given that the krypton atom has a radius of this, we have only the moles. We want to find out how many atoms are in this moles. So how can you do that? Is by using avocados numbers? Let me go ahead and scroll down slightly. So 0. 49 moles. Using our gorgeous number. We have one small of krypton And then we have 6.022 times 10 to the 23rd items. And that's a gross number. You see that the most will cancel out nicely giving us the final answer of having the atoms as our units. Which is what we want. So putting that into my calculator, I get 1.7 krypton Adams. Alright, so before we're able to see the fraction of the volume that the krypton atoms is occupying. We need to first solve for the volume of what this is occupying. So we have the radius we have this. What we can do is first we know the volume of sphere. Right? Let's put volume of sphere. That's going to be 4/3 pi times pi times the radius cubed. Let's go ahead and insert our radios giver information. Four thirds times pi the radius is 2.02 times 10 to the negative eight centimeter. That is being cute. Bring that into my calculator. I'll go ahead and get a value a three point four five nine Time is 10 to the negative 23rd centim cubed. All right. So now since we have to find the volume and we have our unit here being centimeter cubed, I want to go ahead and actually change that into milliliters. So we're starting off. Let me scroll down. Okay, so we just solved and we have the Volume of the sphere being 3.459 Times 10 to the -3 centimeter cubed. Want to convert that to milliliters? That's one easy conversion. It's one centimeter cubed for everyone milliliters. We see they're centimeter cubes will cancel. Leaving us with just the unit of milliliters, which is what we want. So that's just 3.4 59 times 10 to negative three milliliters. See so I realized that there's a slight mistake here, volume of the sphere that's actually going to be -23 cm cubed. So Again here we have 23 And here 23. All right now continuing on to get the ratio that we want. We're going to be taking the krypton atoms here Which is 1. krypton Adams. Then we can sort of use a conversion factor to convert that into the male leaders. And why we want to use milliliters. It's because you see here that we have male leaders. So you find how much this occupies with this. Then you can get how much that this occupies as a fraction. So going back down we can use this as a conversion factor. So on the bottom we have one adam And on top we'll have 3.4 59. It was 10 to negative 23rd milliliters, putting that into my calculator. We see that our atoms will cancel. I'll get 0. milliliters. Alright. Like I said, since we have this now we can go ahead and divide With 2 50 to see how much of the krypton atoms is occupying that container. So we divide This by 250 ml then going into my calculator. Okay. The final value of 2.37 times 10 to the -3. And that is going to be my final answer for this problem.