How Close Together Are Gas Molecules? Consider an ideal gas at 27°C and 1.00 atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap?

Video transcript

Welcome back everybody. We are looking at a light bulb that is filled with argon molecules right now. The pressure of the argon molecules in that space is simply just one A. T. M. And the temperature is kelvin and we are tasked with finding what the average spacing is between the two argon molecules. Now before working with this numbers, I just want us to to visualize this real quick. We can assume that these argon molecules are are touching with one another and they are arranged pretty uniformly pretty, you know, structurally right there. They're structured in a sense. So we can really think of, you know these, these are gone molecules as being in these cubes in a cubic structure. And what we are tasked with finding here is we are tasked with finding what the distance is between the center of the molecules. Well, another way to think about this is if we're looking at the distance between these two, that's just the radius of one of these molecules plus the radius of another molecule and each radius is just one half the sides of each of the cubes. So this a down here is just equivalent to the sides of one of our cubes which is what we are trying to find. Now the volume of a cube is simply just the side length cubed, right? And in order to find a we're just going to take the cube root of both sides and we get that uh the cube root of volume is equal to our side length. But what is the volume of our cubes here? Well, in order to find that we are going to use our ideal gas law that says PV is equal to n R T. I'm actually gonna divide both sides by the pressure here and we get that our volume is equal to the number of moles times are ideal gas constant, times our temperature all over pressure. Now that we have a formula that we can work with, let's go ahead and plug in some numbers here, we have that are volume is equal to Well the number of moles, we're just dealing with one moles but we're really going to be looking at the individual molecules here. So we have to convert within one mole there is six point oh times 10 to the 23rd molecules great times our ideal gas constant which is 8.314 joules, per mole, kelvin times our temperature of 293 degrees kelvin, which we work with kelvin. So we don't have to convert their times are pressure of 1 80 M. But once again we're going to need to convert this over to pascal. So for 1 80 M there is 1.13 times 10 to the fifth past. When you plug all of this into your calculator, you get that the volume of one of our cubic representations of our molecules is going to be 3.99 times 10 to the negative 26 m cubed. Great! Now that we have that, let's go ahead and take the cube root of our volume to find our side length. So this is going to be equal to the cube root of 3.99 times 10 to the negative 26 m cubed, giving us a final answer of 3.4 times 10 to the negative ninth meters as the average spacing between two argon molecules corresponding to our answer choice of B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.