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In chemistry lab, we did an experiment with water of various temperatures to extrapolate a value for absolute zero based on experimental pressure values for water at different temperatures. Naturally, the data is a little off -- even though absolute zero is $0 K$, my extrapolated value is about $15 K$.

One of the questions in the post-lab asks us to calculate percent error, and I'm a little confused. I thought (and this post confirms) that percent error was calculated by dividing by the expected or "actual" value. In this case, however, the actual value is $0$, and you can't divide by zero.

How do you calculate percent error for absolute zero?

Using Celsius values doesn't really make much sense to me, because the Celsius scale is not absolute; $-273.15$ is essentially an arbitrary number.

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    $\begingroup$ It indeed doesn't seem to make sense, so I don't see much to add - it was a mistake in assignment. $\endgroup$ – Mithoron Jun 25 '17 at 17:24
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    $\begingroup$ @Mith Though I'm not in a position to answer this myself, I don't feel this was a mistake. What the OP did: 1) Conduct an experiment, 2) Expected (theoretical) value is x, 3) Experiment yields a value x', 4) Percent error must be {| x - x' |} 100/ x. Which is fundamentally no different from typical "percent-error" calculations... although the value of x (in Kelvin) is zero, which is tricky :P $\endgroup$ – paracetamol Jun 25 '17 at 18:06
  • $\begingroup$ @paracetamol And there's the rub. Is the percent error "undefined"? $\endgroup$ – Shokhet Jun 25 '17 at 18:34
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    $\begingroup$ What sort of data did you end up with? Perhaps you could take the percent error in your regression. For example if your data was a linear, take the error in slope and intercept of the line. $\endgroup$ – iammax Jun 25 '17 at 20:29
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    $\begingroup$ @iammax is correct perform a linear regression and use the errors produced to obtain your answer. $\endgroup$ – porphyrin Jun 25 '17 at 20:38
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It sounds like this is one of those canned labs that ask you to do standard things, even when they do not apply.

As you noticed, the percent-error formula needs you to use a value in the denominator. If you use 0K in the denominator, you get a divide by zero. If you use your 15K extrapolation, the percent error will always be 100%, no matter what the error actually was!

In such cases, I would look at the percent error with respect to your data. Why? Well, because somebody said you need to calculate a percent error and you might as well use a real datapoint rather than divide by 0 or some other silly result. If you choose your coldest datapoint, you can say something about the percent error of the extrapolation process at the least. Thus, if your coldest temperature recorded was 2C, you could say "My extrapolation suggests that absolute zero is 260C below my coldest recorded value. The actual absolute zero is 275C below my coldest recorded value. This gives me a 5.7% error on my measure of absolute zero relative to my coldest recorded value."

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  • $\begingroup$ But again, the Celsius scale is not absolute. Why is your proposal different than saying "absolute zero should be -273 C, and I got -258 C"? You can't do percent error with values that aren't absolute. (I may simply misunderstand your recommendation.) $\endgroup$ – Shokhet Jul 11 '17 at 0:04
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    $\begingroup$ In this case I'm not comparing absolute temperatures, I'm comparing relative temperatures. I'm saying "My projected absolute zero is X degrees below my coldest point. The known temperature of absolute zero is Y degrees below my coldest point. Percent error is (Y-X)/Y" By anchoring it to the coldest datapoint I have, I can use relative temperatures which give some meaningful percent error. The truth is that it's kinda silly to ask people for percent errors regarding absolute zero, so all you can do is make it as minimally-silly as possible. $\endgroup$ – Cort Ammon Jul 11 '17 at 0:41
  • $\begingroup$ It's kinda like how the absolute Celsius scale is anchored to the freezing point of water. As you pointed out, it's kinda arbitrary. At least by using the coldest datapoint you pick an anchor that has some meaning. This way it at least shows the error that crops up as you extrapolate (since you extrapolate from the last datapoint) $\endgroup$ – Cort Ammon Jul 11 '17 at 0:43
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You should divide by the water's temperature in K. This answer is pretty close to Cort Ammon's (Cort Ammon seems to be dividing by how much you measured your water to be above absolute zero, rather than how much it actually is). My reasoning:

How can you calculate absolute zero to be 15K? 0K is defined as absolute zero. The only way your calculations makes sense is you're trying to calculate how far above absolute zero you are. For instance, suppose your water is at 2°C, and you're trying to use its properties to find AZ. You say that you found that AZ is at 15K. Well, first I'm going to reinterpret that as "I found that AZ is at -258°C", because, again, "I calculated AZ to be at 0K" is nonsensical. Next, I'll interpret "I found that AZ is at -258°C" as "I calculated AZ to be 260 degrees below 2°C". Note that although these last two statements are logically equivalent, the percent error calculations they suggest are quite different. In comments to Cort Ammon, you ask why the temperature of your water isn't as arbitrary as the Celsuis scale. The answer is that this is how far you're trying to extrapolate out. Hopefully, if you had had water at -100°C (okay, that wouldn't be water anymore, but you get the idea), your error would be smaller. It is reasonable to compare how far off your extrapolation was to how far you extrapolated. You were trying to extrapolate to something that was 275 degrees away, and you were 15/275 = 5.45% of the way off.

So in this example of water at 2°C, presumably at some point in your calculations, you found that you were 260 degrees above AZ, and then took the "known" number that your water is at 275K, and took the difference to find AZ = 15K. Or you did some equivalent calculation. You are looking at the final number of 15K and asking what percent off it is, but the more meaningful number is what percent off your intermediate calculation of 260 is.

Asking "How do I calculate percent error for absolute zero?" is a bit like asking "I tried to calculate the position of the center of the earth in a coordinate system in which the center of the earth is the origin. How do I calculate percent error?" If you calculated that the center of the earth is 6,356 km away, rather than 6,371 km, then your percent error is 15 km/6,371 km. Saying "I calculated that the center of the earth is at r = 15 km, but it's actually at r = 0 km. So the percent error is 15/0." is fallacious. Yes, your distance from the center of the earth is in some sense "arbitrary", but it's still the valid denominator, because that's what determining how far out you have to measure.

Any time you have a number that represents how far away you are from a point, whether it's Kelvin measuring distance from absolute zero, or r measuring distance from the center of the earth, measuring percent error with respect to that number will always get you infinity. So instead of measuring distance from that reference point, you should measure distance from the point you started from.

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