In a gas at standard conditions, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about 7 * 10^9 people)?

Video transcript

Welcome back everybody. We are looking at a spear that we are told that contains 200 billion. Alright, let me redo that billion Helium gas molecules were told that the temperature is Kelvin, 300 Calvin. And we are told that the pressure inside the sphere is tend to the 5th Pascal's and we are tasked with finding what is the radius of our sphere. Well, let's see if we can't find some sort of equation to relate what we're given and the radius. Uh to me, the first thing that pops up is in order to know the radius. We are going to have to figure out what the volume is. And in order to do that we're going to use the ideal gas formula except on a per molecule basis. So the formula for that is pressure times volume times the number of molecules times the bolts man constant times the temperature. If I divide both sides by pressure here you can see that these terms cancel out on the left and I get that the volume is equal to N. K. T. Overpay. Let's go ahead and find our volume first and then we can work on finding our radius though our volume is equal to 200 billion molecules or 200 times 10 to the 9th molecules times the bolts man constant which is 1.381 times 10 to the negative 23rd Times are temperature of 300 Kelvin, All divided by our pressure of 10 to the fifth pascal's. And when you plug this into your calculator, we get a volume of 8. times 10 to the negative 15th meters cubed. Wonderful. So we have the volume and I said that we need the volume in order to find the radius. And here's what I mean by that the volume for a sphere is four thirds pi times r cubed. I'm gonna divide both sides by four thirds pi Here. These are going to cancel out and we are going to get that r cubed is equal to let's see here we have 3/ times the volume over. I Now in order to isolate our by itself, I'm going to take the cube root of both sides giving us that our radius is equal to the cube root of 3/4 times volume divided by I great. We have a formula for it and we have all of the variables and numbers that we need. So let's go ahead and find our radius radius is the cube root here of 3/ multiplied by our volume, which we found to be 8. times 10 to the negative 15th meters cubed. All divided by pi. And when you plug this into your calculator we get a radius of 1.25 times 10 to the negative fifth meters. But as you can see we need our answer in micro meters, I'm gonna convert this real quick per one m. There is tend to the six micro meters, which when we multiply across here and cancel out our units, we get that our final radius is 12. micro meters, which corresponds to our answer choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.