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Everyone,
My earlier post went through exactly how the gas law calculations are made, in full detail. I have certainly learned that a lot of people tend to freak out just when they see a math equation! This is unfortunate, but it is real. In the interest of being a communicator of science, even to people somewhat hesitant to even THINK about the science itself, I put together this lay person's description of what I did, why, and what it may mean. This might prove more useful in making people more comfortable about the science of deflategate, which is only coming to a clearer and clearer consensus (well, at least unless more sources, unnamed or otherwise, give info to contradict our assumptions). Thanks!
--------------------------
Part 1
We have all heard about “deflategate” by now, the allegation that somehow footballs used by the New England Patriots must have been tampered with, because they were shown to have lost some of their pressure during the first half of the AFC championship game. There have been many varied explanations thrown out there about what it was that caused the pressure drop, though “cheating” has been the most often-heard allegation.
Last weekend I read claims that the loss of pressure in a football by natural causes alone just could not have happened, that it violated the laws of nature, and that the temperature drop, from the time when the footballs were inflated up until halftime, would have had to have been very large, even 100 degrees or more, to see the pressure drop that the referees apparently measured. The first analysis that I read online mentioned that other people had reached this same conclusion. Something about the claims bothered me, though. I grew up in the North. I remember playing basketball outside on cold days, even just slightly cold days. I remembered how the basketball noticeably cooled down and didn’t bounce as well. I am also a scientist, so I thought that I should look into this a bit more, since it just didn’t make sense to me.
Any gas or any mixture of gases, like air, that is kept in a closed container (a balloon, a football, a glass jar, or anything like that) expands when it is heated and contracts when it cools. This was first described over 400 years ago, when early scientists were using glass containers to try to make inexpensive thermometers. Over time scientists found that pressure and temperature move proportionately in the same direction: as temperature goes up, pressure goes up; as temperature goes down, pressure goes down. It became part of what we now call the “ideal gas law”, which is taught in high school and college chemistry and physics classes.
The temperature/pressure connection is something that almost all of you have likely noticed before. Have you bought balloons at the grocery store? They generally are inflated in the store, while you watch, at room temperature. If it happens to be a winter day (and Valentine’s Day is coming up), and if you take the balloons outside to your car, the balloons will shrivel up noticeably! Then when you take them home they will fill out again as they warm up. There are many YouTube videos showing this happening, and here is one:
So the idea that air inside a warm football would lose some pressure as it cools makes sense from our everyday experiences. Why were the calculations that I saw on the internet saying that the temperature had to drop enormously in order to see a significant pressure drop? I decided to look into it. One of the persons making the claim of a necessarily large temperature drop thankfully gave a link to the math that was involved. I reviewed it. Sure enough, they did use the ideal gas law. They did convert the temperatures to the Kelvin scale, which is the temperature scale that is used in the ideal gas law. They used the supposed air pressures that were measured for the footballs in units called psi, for “pounds per square inch”. But then something went badly wrong in their calculations, and it is easy to understand.
A pressure gauge measures something called “gauge pressure” or “relative pressure”. Simply put, this is the pressure inside of the football relative to the outside air. Gauge pressures are what were used in the calculations that I was critiquing. The ideal gas law, however, uses a different pressure measurement for a simple reason: even the air outside of the football is under pressure too, the atmospheric pressure, also called the barometric pressure, which is one of the pieces of information that your weatherman likely tells you every morning. Thus the air that is inside the football is really under a total pressure that is the sum of the gauge pressure and the atmospheric pressure.
Confusing the gauge pressure with the total pressure and using the wrong one in the gas law calculation turns out to be a common mistake. Even the famous astrophysicist Neil deGrasse Tyson made this same error, which he corrected a day later (yesterday)! What happens when you do the calculations the correct way? I will spare you the actual calculations and equations, though they are most definitely not a secret and I have described them in detail elsewhere. Here I instead want to give a layperson’s level description.
The first piece of information that I needed to have was the temperature of the footballs before the game, when they were inflated and inspected. We don’t know that temperature for certain, but we are told that the inflation & inspection was done inside, at room temperature. An educated guess for room temperature is 72 °F. For my calculation I converted this temperature to the Kelvin temperature scale.
The second piece of information that I needed to have was the temperature of the footballs at halftime, after they were used in the game for 90 minutes or so. We can assume that the footballs reached the temperature of the outside air, to a first approximation. At kickoff the temperature was 51 °F. Granted, it was raining very hard and the rain may have been colder than that, since raindrops form high in the sky and often reach the ground at a colder temperature than that of the air. Thus the rain may have cooled the footballs to a lower temperature than 51 °C. The temperature also likely dropped a little by halftime, by a degree or two. Since it is generally unwise to make any more assumptions than you have to when doing calculations, however, I used the 51 °F value. For the calculations I also converted this temperature to the Kelvin temperature scale.
The third piece of information that I needed was the barometric pressure during the first half in Foxboro, MA. A website called Weather Underground tracks this data. At around kickoff time, the barometric pressure was 29.75 inches. This can be converted to 14.61 psi.
The fourth piece of information that I needed was the pressure of the footballs before the game, when they were inspected. We know that the Patriots wanted their footballs to be inflated to 12.5 psi. Knowing this, we can add this gauge pressure to the barometric pressure to get the total pressure that we need to use in the gas law calculation.
After knowing those four things, the calculation of what the final pressure of the football must be is really pretty simple. Plugging in the actual values, a football inflated to 12.5 psi at 72° F that is then cooled to 51 °F will have a final gauge pressure of 11.34 psi, thus showing a pressure loss of 1.16 psi. This result made some sense to me, harkening back to my days of playing basketball out in the cold. Even a 21 degree drop would make the ball a little less fully-inflated, to my recollection.
However, rumors at the time in the press said that the Patriots footballs had lost 2 psi in pressure. The ideal gas law predicted a 1.16 psi drop. Thus not everything was lining up, for sure. Then, as we have seen often in this story, another report emerged on Monday from yet another “unnamed source”. This particular report claimed that only one of the Patriots footballs lost 2 psi of pressure, namely the football that had been intercepted late in the half. One Patriots football had also been judged to be “passing” at halftime. The other 10 Patriots footballs, the source claimed, dropped much less in pressure. In fact it was proposed that they dropped closer to 1 psi than 2 psi. So we had two outliers, or two odd Patriot footballs, one that was on the high side and one that was on the low side. The rest of them dropped closer to 1 psi in pressure, apparently, and thus the average drop may have been closer to 1 psi, which would agree with my calculations. This report that a pressure drop closer to 1 psi than to 2 psi, on average, may have occurred changed my perspective on my calculations. It seemed that science might be able to explain the drop, completely.
(see part 2)
My earlier post went through exactly how the gas law calculations are made, in full detail. I have certainly learned that a lot of people tend to freak out just when they see a math equation! This is unfortunate, but it is real. In the interest of being a communicator of science, even to people somewhat hesitant to even THINK about the science itself, I put together this lay person's description of what I did, why, and what it may mean. This might prove more useful in making people more comfortable about the science of deflategate, which is only coming to a clearer and clearer consensus (well, at least unless more sources, unnamed or otherwise, give info to contradict our assumptions). Thanks!
--------------------------
Part 1
We have all heard about “deflategate” by now, the allegation that somehow footballs used by the New England Patriots must have been tampered with, because they were shown to have lost some of their pressure during the first half of the AFC championship game. There have been many varied explanations thrown out there about what it was that caused the pressure drop, though “cheating” has been the most often-heard allegation.
Last weekend I read claims that the loss of pressure in a football by natural causes alone just could not have happened, that it violated the laws of nature, and that the temperature drop, from the time when the footballs were inflated up until halftime, would have had to have been very large, even 100 degrees or more, to see the pressure drop that the referees apparently measured. The first analysis that I read online mentioned that other people had reached this same conclusion. Something about the claims bothered me, though. I grew up in the North. I remember playing basketball outside on cold days, even just slightly cold days. I remembered how the basketball noticeably cooled down and didn’t bounce as well. I am also a scientist, so I thought that I should look into this a bit more, since it just didn’t make sense to me.
Any gas or any mixture of gases, like air, that is kept in a closed container (a balloon, a football, a glass jar, or anything like that) expands when it is heated and contracts when it cools. This was first described over 400 years ago, when early scientists were using glass containers to try to make inexpensive thermometers. Over time scientists found that pressure and temperature move proportionately in the same direction: as temperature goes up, pressure goes up; as temperature goes down, pressure goes down. It became part of what we now call the “ideal gas law”, which is taught in high school and college chemistry and physics classes.
The temperature/pressure connection is something that almost all of you have likely noticed before. Have you bought balloons at the grocery store? They generally are inflated in the store, while you watch, at room temperature. If it happens to be a winter day (and Valentine’s Day is coming up), and if you take the balloons outside to your car, the balloons will shrivel up noticeably! Then when you take them home they will fill out again as they warm up. There are many YouTube videos showing this happening, and here is one:
So the idea that air inside a warm football would lose some pressure as it cools makes sense from our everyday experiences. Why were the calculations that I saw on the internet saying that the temperature had to drop enormously in order to see a significant pressure drop? I decided to look into it. One of the persons making the claim of a necessarily large temperature drop thankfully gave a link to the math that was involved. I reviewed it. Sure enough, they did use the ideal gas law. They did convert the temperatures to the Kelvin scale, which is the temperature scale that is used in the ideal gas law. They used the supposed air pressures that were measured for the footballs in units called psi, for “pounds per square inch”. But then something went badly wrong in their calculations, and it is easy to understand.
A pressure gauge measures something called “gauge pressure” or “relative pressure”. Simply put, this is the pressure inside of the football relative to the outside air. Gauge pressures are what were used in the calculations that I was critiquing. The ideal gas law, however, uses a different pressure measurement for a simple reason: even the air outside of the football is under pressure too, the atmospheric pressure, also called the barometric pressure, which is one of the pieces of information that your weatherman likely tells you every morning. Thus the air that is inside the football is really under a total pressure that is the sum of the gauge pressure and the atmospheric pressure.
Confusing the gauge pressure with the total pressure and using the wrong one in the gas law calculation turns out to be a common mistake. Even the famous astrophysicist Neil deGrasse Tyson made this same error, which he corrected a day later (yesterday)! What happens when you do the calculations the correct way? I will spare you the actual calculations and equations, though they are most definitely not a secret and I have described them in detail elsewhere. Here I instead want to give a layperson’s level description.
The first piece of information that I needed to have was the temperature of the footballs before the game, when they were inflated and inspected. We don’t know that temperature for certain, but we are told that the inflation & inspection was done inside, at room temperature. An educated guess for room temperature is 72 °F. For my calculation I converted this temperature to the Kelvin temperature scale.
The second piece of information that I needed to have was the temperature of the footballs at halftime, after they were used in the game for 90 minutes or so. We can assume that the footballs reached the temperature of the outside air, to a first approximation. At kickoff the temperature was 51 °F. Granted, it was raining very hard and the rain may have been colder than that, since raindrops form high in the sky and often reach the ground at a colder temperature than that of the air. Thus the rain may have cooled the footballs to a lower temperature than 51 °C. The temperature also likely dropped a little by halftime, by a degree or two. Since it is generally unwise to make any more assumptions than you have to when doing calculations, however, I used the 51 °F value. For the calculations I also converted this temperature to the Kelvin temperature scale.
The third piece of information that I needed was the barometric pressure during the first half in Foxboro, MA. A website called Weather Underground tracks this data. At around kickoff time, the barometric pressure was 29.75 inches. This can be converted to 14.61 psi.
The fourth piece of information that I needed was the pressure of the footballs before the game, when they were inspected. We know that the Patriots wanted their footballs to be inflated to 12.5 psi. Knowing this, we can add this gauge pressure to the barometric pressure to get the total pressure that we need to use in the gas law calculation.
After knowing those four things, the calculation of what the final pressure of the football must be is really pretty simple. Plugging in the actual values, a football inflated to 12.5 psi at 72° F that is then cooled to 51 °F will have a final gauge pressure of 11.34 psi, thus showing a pressure loss of 1.16 psi. This result made some sense to me, harkening back to my days of playing basketball out in the cold. Even a 21 degree drop would make the ball a little less fully-inflated, to my recollection.
However, rumors at the time in the press said that the Patriots footballs had lost 2 psi in pressure. The ideal gas law predicted a 1.16 psi drop. Thus not everything was lining up, for sure. Then, as we have seen often in this story, another report emerged on Monday from yet another “unnamed source”. This particular report claimed that only one of the Patriots footballs lost 2 psi of pressure, namely the football that had been intercepted late in the half. One Patriots football had also been judged to be “passing” at halftime. The other 10 Patriots footballs, the source claimed, dropped much less in pressure. In fact it was proposed that they dropped closer to 1 psi than 2 psi. So we had two outliers, or two odd Patriot footballs, one that was on the high side and one that was on the low side. The rest of them dropped closer to 1 psi in pressure, apparently, and thus the average drop may have been closer to 1 psi, which would agree with my calculations. This report that a pressure drop closer to 1 psi than to 2 psi, on average, may have occurred changed my perspective on my calculations. It seemed that science might be able to explain the drop, completely.
(see part 2)
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