Modern vacuum pumps make it easy to attain pressures of the order...

Question

Modern vacuum pumps make it easy to attain pressures of the order of 10^-13 atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

Accepted Answer

Welcome back everybody. We are observing a malfunction in an insulation system and we are told that before this malfunction, the pressure inside of the insulation system is tend to the negative eight A. T. M. And we are told the pressure after is 10 to the negative fourth A. T. M. Now volume and temperature those are just going to stay constant. But we are tasked with finding how many molecules are in the system before compared to the amount of molecules after. And the way that we're gonna compare them is by multiplying the number of molecules after by some constant C. So we really need to find that constant. Now we're dealing with pressure, volume temperature molecules. First thing that pops out to me that we need to use is our idea gas law, ideal gas law states that PV equals N. R. T. But I'm gonna make a substitution here. We know that the number of moles is just the number of molecules divided by avocados number. So I'm gonna go ahead and plug this in for N. And we get that pressure times volume is equal to number of molecules times the ideal gas constant times temperature all over of a God rose number Now a couple more things here since we are looking at the pressure before and the molecules before compared to after. I'm really going to be seeking that ratio. So I kind of want to isolate P over N. On one side of the equation. So here's how we're gonna do that. I'm gonna divide both sides by end. So these terms are going to cancel out right here. And I'm gonna divide both sides by our volume. Right? And so what this is going to yield for us is that the ratio between pressure and the number of molecules in the system, either before or after is going to be equal to the ideal gas constant time temperature. All divided by the number of times volume. But if you remember volume and temperature remain constant before and after the malfunction, this entire term right here is just going to work out to be constant. And we can use this in our favor to set up two equations for both Before and after. We're going to have that our pressure before divided by our molecules before is equal to some constant. And then on the other side we're going to have that our pressure after divided by our molecules after is equal to that same constant. Now I'm going to get like terms on either side here. So we are going to have that our molecules before divided by our molecules after is equal to our pressure before divided by our pressure after all equal to our constant C. That we are looking for. Let's go ahead and fill in some numbers here so that we can find C. Well, the only numbers that were given are these pressures right here. So we'll set that up that way. We have that our pressure before over our pressure after is equal to our constancy. So this is equal to 10 to the negative 80 M. Divided by 10 to the negative fourth A. Tm. These unit will cancel out leaving us. That R. C. Is tend to the negative fourth. So when we plug C back into this spot right here we get that. The comparison between the before and after molecules is as follows the molecules before is tend to the negative fourth time's the molecules after which corresponds to our answer choice of E. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

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