 Today we're going to tell you a story about a scandal. It's a time when sports met science, when pop culture met the ideal gas law. Basically the term for this, the title for this video is deflate gate. So let me tell you a little bit about the story. Flashback to January 18th, 2015. The New England Patriots, a team in the National Football League, is playing in their stadium in a playoff game against the Indianapolis Colts. Now during that game, at half time, there are some footballs that are being used in the game, and one of the rules about the game is that each team is allowed to have their own set of footballs that are filled up with air prior to the game, and then they can use them on the field when they're on offense. And at the time, at half time of the game, some of the Indianapolis Colts players made some claims that the football felt a little bit soft, that it was under-inflated. And as a result, there was some measurements taken, and the claim was that the footballs were at least two pounds per square inch under-inflated. The rules say in the NFL that the footballs need to be inflated from, let's see, your NFL rules, state that the football should be inflated from 12.5 to 13.5 PSI as measured with a gauge by officials before the game. Now the Patriots, it is well known that the quarterback of the Patriots at that time, Tom Brady, liked to have the footballs as under-inflated as possible. He liked them a little bit softer when he threw them, and that was the way he liked it. The Indianapolis Colts, they actually chose to have their footballs somewhere in the middle of that range, but the assumption was that Brady would like his as under-inflated as possible, so he would usually have them inflated at 12.5 PSI. The claim, however, was that when the footballs were taken and measured later, that they were way under-inflated, that they were down to around 10.5 PSI. And therefore, there must have been some nefarious form of removing some of the air from the balls after they had been measured. Now this caused a great scandal. As it turns out, the Patriots went on to handily win that playoff game and went on further into the playoffs. But this caused a big scandal and a big problem that the Patriots were caught cheating in this particular case. And then there was lots of questions. This was one of the first times when people in sports radio started talking about things like, well, could this have happened naturally or was this cheating? And then things like the ideal gas law came into play and you actually heard people on sports talk radio talking about something like the ideal gas law. So this lends a particularly good opportunity for us to study an application of the ideal gas law in the context of this deflate-gate scandal. So let's consider for a moment, let's apply the ideal gas law to what we know. First of all, we're going to need the ideal gas law. Well, if you recall the ideal gas law, PV equals NRT. That's the standard memory that the standard formula that many people use. Although in my engineering classes, I teach my students to use a slightly different version. P1V1 over N1T1 equals P2V2 over N2T2. In other words, we say that both of these sections, if you contain everything, they're equal to this constant R. And that mostly what we're interested in comparing two states. In this case, it might be the state of the footballs at one period of time and the state of the footballs at another period of time. Now you will notice that if we actually removed air from the football, that would change N. N is the number of molecules or basically the amount of whatever fluid, in this case air, is present. And so our first assumption here is going to be that N1 is equal to N2. That there wasn't anything removed from the football. If there was, if there definitely was something removed from the football, then yes. The Patriots and Tom Brady or whoever was involved were guilty of something that was cheating based on the rules of the NFL. But we're going to look and see is it possible that there might have been under inflation as a result of other things. So our first assumption is we can cancel out the N1 and the N2, at least for now, working on the assumption that, okay, is it possible this could have occurred without any removing of air from the football? Another assumption that we'll make now, but maybe we won't make a little bit later, is that the volume, the football is a football. We're not expanding or contracting the football significantly assuming it's already full of air. So at least initially we'll make the assumption that the volume of the football before and after stays pretty constant as well. Well that makes our ideal gas law significantly easier. Now we basically have a relationship that says, and we'll do some solutions over here, P1 over T1 equals P2 over T2. Well what are these other things? Temperature and pressure. Is it possible that the pressure change as a result of a temperature change? Well we do know based on this relationship that when something gets colder, when the temperature goes down, the pressure would also go down. And we notice January 18th, actually let me write the date here. January 18th, 2015. That's the date of the game. In Foxboro, Massachusetts. Foxboro, Massachusetts is in New England. In January in New England it tends to be cold. In fact this was actually a relatively warm day in New England. It turns out that our temperature outside, so we're going to need some information. We're going to say our temperature, we have an inside temperature and an outside temperature. Our temperature inside the building, we're going to use a number, a reasonable number of 71 degrees Fahrenheit. That was the temperature inside the building or a reported temperature. And then if we look outside and we go look at some weather reports. Now there were some changes in the weather that we'll talk about a little bit later. But if we just take a number for about the time of day, a reasonable number for the outside temperature was 48 degrees Fahrenheit. Again relatively warm for January in Massachusetts, but it's still significantly colder than it was inside where the balls were originally measured. Now we know for a fact, just looking at this here, if we start the lowest end of our range and then we cool the ball down. We're automatically going to get some reduction in pressure and the balls are going to be illegal. That's the case. The question is, are they so illegal as what they were claimed to have been measured down to this minus 2 PSI, so were they at 10.5 PSI or lower? So let's apply the ideal gas law, shall we? Here we go. Let's go ahead and put in the numbers. We know our original, we're going to say 12.5 PSI, that's our original P1. So we'll write 12.5 PSI for our original pressure 1. And then for our starting temperature, we'll use the inside temperature of 71.0 degrees Fahrenheit. We're trying to find out what the second pressure might be. And so we'll put in our outside temperature, 48.0 degrees Fahrenheit. Alright, looks like there's going to be a change there. Multiply both sides by 48 and we can solve for P2. And when we do, we get a number that looks something like this, 8.45 PSI. Well, obviously this could drop a huge amount. As soon as we take that outside, the pressure goes way down. There is no way that the Patriots were cheating and Tom Brady were cheating. Free Tom Brady, let's just forget about this. Obviously can go down like that. However, if you are a student who studied the ideal gas law, you might remember there are some rules about how we apply the ideal gas law. And our very first rule says that the temperatures that you use for the ideal gas law must be absolute. We cannot use Fahrenheit as legitimate temperatures for this particular case. This is an incorrect application of the ideal gas law. Okay, so that's done incorrectly. And at the time, there were some references of people showing that on social media or elsewhere or discussions where that application was debunked. That's not how you apply the ideal gas law. Alright, so let's see if we can fix that, shall we? In order to do this, we need to turn our temperatures into absolute temperatures. In this case, Fahrenheit translates to an absolute scale of something called Rankine. And the conversion is to add 459.67, but because I only have a .0 as my closest degree of precision here for this, I'm going to write 459.7. In both of these cases, we're going to add 459.7 degrees to get a measure in degrees Rankine. So that will give us 530.7 degrees Rankine and 507.7 degrees Rankine. Now you can see the relationship, the ratio between those two is significantly different than the ratio between those two. So let's go back, ignore this and reapply it, try it again. 12.5 PSI over, this time we have 530.7 degrees Rankine is equal to P2 over 507.7 degrees Rankine. Now we put that into our calculator, multiply the top, or multiply both sides of the 507.7, the Rankines cancel out, and we get a value here, P2 equals 11.95, so I guess I'll put that as 12.0 PSI. Oh, the Patriots are dirty cheaters. Right, look, even now that we've done that correctly, there is no way that it dropped two whole PSI just based on that temperature piece. Okay, however, once again, maybe you're paying attention because you're an astute student of the ideal gas law, there is a second rule for the application of the ideal gas law that comes into play here. The second rule says that our pressure must be absolute, in the same way that our temperature must be absolute. And in this particular case, our pressure is measured, we take a football, and we insert a gauge, and that gauge measures the pressure, and that pressure is relative to the atmospheric pressure, to the outside pressure. So when we say that it has a measurement of 12.5 PSI, that means 12.5 PSI more than the atmosphere. So that means we are not using absolute pressures here, we are using what are called gauge pressures. And that's not appropriate in this particular case. So, we've got to do this again. So, and let's see here, we need to make some adjustments. First of all, we need to recognize that my initial pressure started at 12.5 PSI gauge, and then we're going to add to it a number for atmospheric pressure. Well, a standard number to use for atmospheric pressure is 14.7 PSI for the atmosphere. So, if we take those two numbers together, we get a new initial pressure, an absolute initial pressure of 2627.2 PSI absolute. So now I'm going to take a moment to erase a few things, and then we'll come back and solve there. Alright, let's continue. So, we have a new correct, we understand this to be correct pressure, an absolute pressure of 27.2 PSI absolute. So let's go back over here, we'll put in that new 27.2 PSI absolute, and we'll once again put in our absolute temperatures. 530.7 Rankine is equal to P2 over 507.7 degrees Rankine. When we solve that, we get a value for P2 of 26.0 PSI absolute. Notice that's not the number we had originally, that would be significantly more, but now we can go back and we can subtract off the atmospheric value, and when we do so we get a value of 11.3 PSI gauge. Okay, so now that we've seen that number, we've taken into account both the absolute pressure and the absolute temperature. We look here and we can see there is a substantial drop, it's a drop of about 1.2 PSI, but that isn't enough of a drop to take into account the entirety of the claimed under inflation. So it isn't looking particularly good for the Patriots. However, once again there's another however, there are some potential assumptions of things that could come into play here. And there have been a number of analyses in social media and different videos and things along those lines that demonstrate that there are some other possibilities that could lower this gauge pressure. We're going to look at a couple more assumptions that could be made based on information that we have about what happened at the time. There are two things we're going to consider. Thing number one is that the weather changed during the course of the game. During the course of the game it rained for much of the first half, and also when rain comes through, when there's rain coming through, that usually means that there's a difference in the different fronts. The atmospheric pressure will tend to change over the course of the game. So we're going to clear this again, write down some new assumptions, and see if under reasonable assumptions, how low can we go? How low is it possible to go with reasonable assumptions to get an underinflated football that's underinflated without removing any mass or any of the air from the ball? So now I've rewritten our ideal gas law, the engineering form of our ideal gas law, over here, and we're going to start again looking at a couple of assumptions. Our first assumption once again is going to be that N1 equals N2, that there was nothing taken out of the ball. We're going to see what the possibility has when there isn't anything removed. How low can we go? There's a second assumption here that if you take a football and you wet it, you make it very wet, like as a result of the rain, if you're playing the pouring rain for a while, that football is going to soak in some of the water, and when it does so it's going to loosen the leather that the football is made out of a little bit. And if you look at some sources you can see that a football, when it gets wet, can expand. It can change its ability to hold so it naturally expands. So we're going to make an assumption, the numbers that I've seen, is the most you can sort of get a reasonable amount of expansion for a football, is that the football expands by about 3%. So we are going to make an assumption that because we do know that the football's got wet, that they expanded to this maximum limit of 3%. If you don't like that number, if you think it's smaller, you could do a similar calculation with a smaller value. So we'll go ahead and make this assumption. We're going to say that volume 2 is equal to 1.03 times volume 1. 1.03 times volume 1. In other words, we're adding 3% to volume 1, 1.03, there's the multiplier that we put in that place. So when we do that, we can replace this volume 2 with a number of 1.03 volume 1, and then we can cancel out the volume 1. So that means we now have a multiplier of 1.03 on this right-hand side that represents the expansion in volume. That's the first assumption, or that's the second assumption we're going to make after the assumption of equal masses. Another assumption that we can make here that could adjust our gauge pressure, you'll notice these are gauge pressures, is that there is a change in the atmospheric pressure between our two situations. Now, if you look for the changes that are possible in the Foxboro, Massachusetts region, you can see that there is a possible range in pressures from, there's, I think, a maximum, the minimum and maximum run from about 14 psi to somewhere around 15.2 or 15.3 psi. Now, we're not going to assume that those extremes are met on the same day. It seems unlikely that we're going to meet those extremes. However, we're going to say that there's a reasonable range around that 14.7 of plus or minus 3 psi. So we're going to make the assumption that our pressures, that there are two pressures involved here, okay, and that our pressures are going to, our pressure number one will start at, I'm sorry, our atmospheric pressure. Atmospheric pressure number one is going to start at 14.4 psi. And atmospheric pressure number two will go up to 15.0 psi, that there will be a change in the pressure over the course of the time from when the balls are measured to when the balls are measured the second time. And because there was a change in the weather, there was rain and there, so there was generally a change in the weather. Now, I'm not sure exactly how that change in the weather progressed. We're looking for possible scenarios. And this type of scenario, if we increase the atmospheric pressure, we're going to end up with a potential decrease in the gauge pressure. Because if we stay the same inside the football, we increase what's outside, the difference between those things will change. So, that's what we're going to do in this particular case. We're going to go ahead and make some assumptions here. So, to do that, we're going to say, all right, what is our pressure one? Well, we'll say our pressure one was that 12.5, that low pressure started psi gauge. And then we are going to add the atmospheric pressure at the time, which we'll assume was low, maybe lower than the standard average of 14.7. And when we do that, we'll add those together, we'll get 26.9 psi, and there's our absolute, we'll put A for absolute there. Okay, now we have the information we need. We'll get back to this 15.0 in a minute. Now we have the information we need to go back and put this into my ideal gas law. We'll do what we did before. We'll use those same temperatures that we did before with our absolute temperatures. And let's go ahead and put that in here. So, we'll start with 26.9 psi absolute. We'll divide that by our 530.7 degrees ranking. We'll make that equal to our 1.03 times the pressure two that we're looking to solve for, divided by my 507.7 degrees ranking. So, there's our new balance taking into account our new assumptions. When we do that, we get a pressure two value of 25.0 psi absolute. Well, so there is a change. It's a pretty substantial change. It looks like a change of about 1.9 psi. That's partially due to this 1.03 factor and due to the change in the 1.03 factor and due to the fact that maybe we have a little bit lower or a different psi there. But now we're going to go ahead and from that 25, we're going to say, okay, now when we measure the next time, when we did, we had a higher atmospheric pressure. So, we'll subtract off that atmospheric pressure. And when we do, we get a value of 10.0 psi gauge. Notice you don't have to make the assumption that that bumped up. You could say, well, let's leave it at the 14.4, 25.0 psi absolute minus 14.4 or something a little lower, 14.5, 14.6, maybe didn't rise a whole 0.6 psi. But if we do take that 14.4, you still get a value, 10.6 psi gauge. Now those numbers are much closer or do seem to be in that range of that 2 psi underinflated. Is it possible that the measurements were made, 12.5 psi, 10.5 psi, without taking any of the gas out of the ball? Now I'll leave that to you. As a scientist, you'll have to ask yourself, do I really believe the assumptions that are made here or what are reasonable assumptions or can I get more proof of these various numbers that were chosen here? As a fan, you might have a particular bias about whether or not you think the New England Patriots were dirty cheaters or whether or not they were unfairly victimized. I'll leave that decision to you.