Sciencemadness Discussion Board - Distilled water density

Distilled water density

sbreheny - 11-2-2014 at 21:30

Hi all,

I keep noticing that when I weigh samples of the distilled water that I use (mostly from Target but sometimes from McMaster-Carr, gives similar results), I get slightly less weight per volume than I would expect. Today I did an experiment where I took three different 25mL volumetric flasks, weighed each one three times empty, then three times when full of water (bottom of the meniscus on the cal line), and averaged the readings for each flask and took the difference. I get the following densities for the water in the three flasks: 0.9943, 0.9935, and 0.9930 g/mL.

These are class A volumetric flasks calibrated for 20 C. The room was 19 C and the water was 19.7 C. My analytical balance shows a maximum possible error of 0.016% on a 150g cal weight (including the tolerance of the weight).

Pure water should be 0.998 g/mL. The difference between this and the result I get, while small, is several times larger than can be accounted for due to my sources of error.

Any idea what is up? Would there be enough impurities in distilled water to cause this much density change?

Thanks!

Sean

chemrox - 11-2-2014 at 21:42

dissolved CO2, N2 & O2

sbreheny - 12-2-2014 at 00:34

Thanks, chemrox. Based on a paper by NIST (Effect of Dissolved Air on the Density and Refractive Index of Water by Harvey, Kaplan, and Burnett), at 20 C this should only make 0.002 g/mL difference, which means an expected density of 0.996 - still 0.002 away which is still more than the sum of the effects of air buoyancy and analytical balance error, although it is getting much closer so maybe I need to be more careful with my error budget.

sbreheny - 12-2-2014 at 00:36

OOPS, no, I just made a big mistake. That paper says that the dissolved gasses should only make 0.002 mg/mL difference - totally in the noise, so no, the dissolved gasses do not explain what I am seeing.

hissingnoise - 12-2-2014 at 02:36

Don't forget the influence of pressure, temp. and R/H?

blogfast25 - 12-2-2014 at 11:48

Seems to me the error is in line with what to expect from using volumetric flasks, as opposed to picnometers. Your average is 0.9936 or about 0.0044 off the listed value of 0.998. Times that by 25 and you get 0.11 g or about 0.1 ml.

I don't think you can expect much better even with Class A volumetric flasks.

sbreheny - 12-2-2014 at 13:24

hissingnoise: I have accounted for temperature in that I used the 20 C value for water density and my volumetric flasks were being used very close to their rated temp (19 C versus 20 C - should make a difference far less than I am seeing). Air buoyancy effect is not nearly large enough to cause this amount of error, either, so I don't see how pressure and RH would matter.

blogfast25: These volumentric flasks are rated +/- 0.03 mL. Also, the error is consistently the same sign across three flasks and the variation from flask to flask is small-ish compared to the overall discrepancy. It is clear that these volumetric flasks were individually calibrated because when you put them next to each other, they are all the same model number but their calibration marks are not at the same height.

sbreheny - 12-2-2014 at 13:28

I ordered a bottle of ultrapure HPLC-quality water and when it arrives I will try this again with that water.

DJF90 - 12-2-2014 at 14:47

Have you considered the calibration of your balance? You say it reads well with a 150g calibration weight but is it a single point calibration or a span calibration? Have you checked what readings it gives with other standardised masses? Particularly in the region of what the volumetric and 25 mL water weighs (50-60 g?). Balances for cGMP use where I work undergo a seven-point calibration to account for such non-linearity.

Ockham's Razor — User Error

bfesser - 12-2-2014 at 15:18

From the descriptions, it sounds to me like your balance isn't properly leveled. Close one eye and look at the bubble from directly above to avoid parallax error when adjusting the leveling feet. The bubble must be dead center under the printed circle, or your weight values will read consistently low. In the diagram below, you want σ = W, θ = 0.

[Edited on 12.2.14 by bfesser]

Pyro - 12-2-2014 at 15:40

there is also the fact that he can never know exactly when the flask is filled to the line (assuming he is human)

sbreheny - 12-2-2014 at 23:22

Thanks for all the replies. I do agree that the most likely reason is some kind of user error or problem with the instruments. However, I do not yet see how.

Balance calibration: Unfortunately, the only cal weights I have are 150g, 500mg, and 100mg. The 500mg and 100mg are expensive class 1 weights and the 150g is a cheap class M1, but still supposed to be within 7.5mg. Here is what I get when I weight each one of these weights in sequence several times:

149.9832g, 0.4999g, 0.0996g, 149.9821g, 0.4998g, 0.0999g, 149.9822g, 0.5003g, 0.1000g

Balance levelling: the bubble is within the inner circle, although the place where I have the balance I cannot get my head directly above it because of cabinets above. However, to cause 0.004 scale error would require 5 degrees of tilt which would be clearly noticeable.

Filling the volumetric flasks: I can very easily tell the difference between being above or below the line to within one eyedropper drop with a dropper which is about 1/20 mL per drop, which would be half of the observed error. It would seem strange that I get it consistently wrong in the same direction, but maybe.

phlogiston - 13-2-2014 at 02:45

To conclusively exclude sources of variation as the reason for your result you should repeat your experiment, but this time use the same flask for all measurements and repeat the entire procedure (each and every step of it from begining to end (filing etc)) for each measurement.
Then, calculate the average and standard deviation of your measurements and show that the difference is really statistically significant.
Once you have unequivocally shown that there is a significant difference in density, the discussion will shift to the WHY question.

bfesser - 13-2-2014 at 04:53

Quote: Originally posted by sbreheny

I do agree that the most likely reason is some kind of user error or problem with the instruments.
. . .
Balance levelling: the bubble is within the inner circle, although the place where I have the balance I cannot get my head directly above it because of cabinets above. However, to cause 0.004 scale error would require 5 degrees of tilt which would be clearly noticeable.

Filling the volumetric flasks: I can very easily tell the difference between being above or below the line to within one eyedropper drop with a dropper which is about 1/20 mL per drop, which would be half of the observed error. It would seem strange that I get it consistently wrong in the same direction, but maybe.

I'll assume that you neglected to account for friction in your calculation of 5° tilt. If you can't get your eye directly above the bulb, use a mirror.

Another simple explanation; are you correctly reading the maniscus? Reading from the sides or from a high angle could account for your consistently low values.

blogfast25 - 13-2-2014 at 05:54

Try using a pycnometer:

https://www.google.co.uk/search?q=pycnometer&rlz=1T4DSGP...

MrHomeScientist - 13-2-2014 at 08:16

An interesting distilled water-related situation I came across: I recently judged a science fair for middle and high schoolers, and one of the projects involved measuring condutivity of different drinks to find the relative amount of electrolytes they contained. In the experiment she claimed she got some conductivity from distilled water. If it was pure water, of course, it shouldn't conduct at all.

I haven't done this experiment myself, and it was a middle-schooler's project, so this is all hardly conclusive. Still, another interesting indication that "pure" distilled water from the supermarket might not be quite so pure.

forgottenpassword - 13-2-2014 at 09:02

Quote: Originally posted by sbreheny

These volumentric flasks are rated +/- 0.03 mL

In which case, your density measurement is accurate to one significant figure of precision.

sbreheny - 13-2-2014 at 09:31

Quote: Originally posted by forgottenpassword

Quote: Originally posted by sbreheny

These volumentric flasks are rated +/- 0.03 mL

In which case, your density measurement is accurate to one significant figure of precision.

I'm not sure I understand what you mean. 0.03 is much more than one sig figure out of 25mL, and certainly my balance is accurate to much more than one sig figure out of 25g, so the end result should have more than one sig figure of accuracy.

sbreheny - 13-2-2014 at 09:36

Quote: Originally posted by bfesser

Regarding the meniscus, I am reading it at eye level by kneeling down, leaving the flask on the table, and aligning my eye with the calibration ring so that the entire ring appears as a line and then lining-up the very bottom of the meniscus with the line.

Regarding friction: are you referring to friction caused by the lateral component of gravity placing a sideways force on the bearings of the weigh platform? If so, then yes, I neglected that, but I think that should be a minimal effect because of the way the balance works. I believe that it applies a constant dithering motion (microscopically - you can hear but not see it) so that dynamic friction cancels out and static friction cannot occur because the platform is never truly stationary.

sbreheny - 13-2-2014 at 09:40

Quote: Originally posted by MrHomeScientist

An interesting distilled water-related situation I came across: I recently judged a science fair for middle and high schoolers, and one of the projects involved measuring condutivity of different drinks to find the relative amount of electrolytes they contained. In the experiment she claimed she got some conductivity from distilled water. If it was pure water, of course, it shouldn't conduct at all.

I haven't done this experiment myself, and it was a middle-schooler's project, so this is all hardly conclusive. Still, another interesting indication that "pure" distilled water from the supermarket might not be quite so pure.

Pure water still has some conductivity due to the equilibrium dissociation of water into OH and H3O ions. Also, the dissolved CO2 will contribute to conductivity. Finally, and I think this is the greatest source of error, you cannot use a normal multimeter to measure water resistance because the DC voltage used will begin to hydrolyze the water and cause an artificially-low resistance reading due to the extra ions produced. I think that you can get a fairly good measurement by using an AC resistance meter. The electrode material may be important, too.

sbreheny - 13-2-2014 at 09:42

Quote: Originally posted by blogfast25

Try using a pycnometer:

https://www.google.co.uk/search?q=pycnometer&rlz=1T4DSGP...

Doesn't that require a liquid of known density for comparison?

sbreheny - 13-2-2014 at 09:45

Quote: Originally posted by forgottenpassword

Quote: Originally posted by sbreheny

These volumentric flasks are rated +/- 0.03 mL

In which case, your density measurement is accurate to one significant figure of precision.

Maybe another way to look at this is that 0.03mL is 30mg mass error. Out of 25mL, that is only about 1mg per mL of water. I am seeing an error of about 3.5 to 4mg per mL error.

blogfast25 - 13-2-2014 at 12:46

Quote: Originally posted by sbreheny

Doesn't that require a liquid of known density for comparison?

Yes. It determines relative density of a liquid, usually relative to distilled water at 20 C.

So you weigh the flask empty, then filled with distilled water, determine the weight of the water, divide that number by itself (so you get 1.0000). Then multiply this by the tabled value for water density, and presto... problem solved!

bfesser - 13-2-2014 at 13:16

<strong>sbreheny</strong>, out of curiosity, are you stoppering the volumetric flask? Also, are you handling it with your bare (oily) hands?

sbreheny - 13-2-2014 at 13:25

Quote: Originally posted by bfesser

<strong>sbreheny</strong>, out of curiosity, are you stoppering the volumetric flask? Also, are you handling it with your bare (oily) hands?

Yes, I am stoppering them and no, I was handling the flasks with nitrile gloves. Also, I cleaned the flasks with acetone before using them and made sure they were completely dry before adding water.

blogfast25 - 14-2-2014 at 05:37

Perhaps you should run some other tests on it, like pH and conductivity?

bfesser - 14-2-2014 at 06:04

<strong><a href="viewthread.php?tid=19355">Why does my Distilled water have a pH of 5.6 ???</a></strong>
<strong><a href="viewthread.php?tid=25352">Conductivity cell: DIY ideas needed!</a></strong>

sbreheny - 15-2-2014 at 15:09

I don't have the proper probe to do conductivity tests. I do have a pH meter - I will try but I'm not sure it is accurate if the ionic content is too low (needs a minimum conductivity I think).

I do have some more information, though.

I bought a class 1 50g calibration weight. On my balance it measures as 49.9930g. The ratio of 50g/49.9930g is 1.000140. The ratio I get when I do the same thing with my class M1 150g weight is 1.000112, so it looks like my balance is consistently reading 0.01% low. However, the ratio of the true density of pure water to what I measure is 1.003. The difference from 1 in this case is almost 22 times the balance calibration error.

Second piece of new info: I repeated my water density experiments with a 100mL volumetric flask and with distilled water from two sources: Target and McMaster Carr. I got 0.99468 g/mL for the McMaster Carr water and 0.99502 for the Target water. Compare this to about 0.994 (+/- 0.0005) for the result I got before with the several 25mL flasks.

Now, I also tried measuring the density of acetone and reagent-grade 99% isopropyl alcohol. Here's where it gets interesting. For acetone, I get 0.7879 g/mL when the ideal value is 0.7910. This ratio is 1.00406. For IPA, I get 0.7806 g/mL, and the ideal is 0.7860. The ratio is 1.0069.

So, here are the ideal value/my value ratios for several scenarios:

These are all for fluid and room air between 19 and 21 deg C.
25mL flask #1, distilled water which is some mix of Target and McMaster origin: 1.0037
25 mL flask #2, same source water: 1.0045
25 mL flask #3, same source water: 1.0050
100 mL flask, Target distilled water: 1.0030
100mL flask, McMaster distilled water: 1.0033
25 mL flask, acetone: 1.0041
25 mL flask, isopropyl alcohol: 1.0069

So - it looks like something is causing me to get consistently low density readings, by roughly the same amount regardless of the fluid measured, which suggests that it is something in my technique or equipment and not the fluids themselves. It also does not seem to be the balance. It's hard to understand how it could be the volumetric flasks because it holds true over several of them, even between 25mL and 100mL.

Chemosynthetic - 15-2-2014 at 15:33

I am very impressed with this thread. While I have nothing to add to the latest development, I would like to note as a reminder that commercial distilled water often has plasticizer leaching if stored for long periods of time in the jugs stores sell them in, and these types of substances are not always removable through distillation without breaking an azeotrope. For this reason, DI water is often preferred where I work.

sbreheny - 15-2-2014 at 17:31

Quote: Originally posted by Chemosynthetic

I am very impressed with this thread. While I have nothing to add to the latest development, I would like to note as a reminder that commercial distilled water often has plasticizer leaching if stored for long periods of time in the jugs stores sell them in, and these types of substances are not always removable through distillation without breaking an azeotrope. For this reason, DI water is often preferred where I work.

I am a stickler for accuracy and I often learn a lot by pursuing tiny unexplained error. My day job is as an electrical engineer and the same philosophy often pays there, too. The experts on here are very good, too, at proposing possible sources of error.

I do also have DI water from McMaster Carr which I use for soldering in electronics (corrosion of the soldering iron tip is only really dependent on ions present in the water). I believe that I have roughly measured the density of that before, too, but I can try repeating that.

[Edited on 16-2-2014 by sbreheny]

sbreheny - 15-2-2014 at 17:39

I am trying yet another volumetric measuring device, a buret. I don't have a result to report with that, yet, but I will post it when I do.

I did discover something else interesting, though. To avoid having to use heat to dry my receiving containers (like the volumetric flasks) I have been using acetone to speed drying. I have been waiting until there is no more visible acetone (or any other droplets) and then weighing to get the empty weight of the container. I discovered that if I do this too quickly, the mass of the acetone vapor adds a measurable amount of weight to the empty container reading, which in turn makes the measured mass of the water smaller since the water displaces the acetone vapor. Acetone vapor at room temp is about twice the density of air.

This is not enough to explain the full error but it is something I will keep in mind (by allowing more time for the acetone vapor to diffuse, as confirmed by no longer being able to smell acetone on the container).

Chemosynthetic - 15-2-2014 at 18:14

I never understood why some of my former colleagues didn't "get" engineers. I admire the mindset and training. If possible, you may look into running an inert line, such as nitrogen from a small tank such as those sold for paintball, into your flask to speed up evaporation convectively. Reminds me of Raoult's law problems.

unionised - 16-2-2014 at 07:23

I may have missed it, but I didn't see any reference to correcting for the buoyancy of the air while making these measurements of the density of water.

What corrections, if any, were made for that?

The error is about 0.1% if you don't correct for it.

Chemosynthetic - 16-2-2014 at 07:31

Yep. He really did his homework. I doubt latent enthalpy of evaporation is causing the error propagation either, but I just thought to mention it as an additional potential confounding variable. Container shape (Erlenmeyer vs. beaker, per example) will affect this as well.

Quote: Originally posted by sbreheny

Thanks, chemrox. Based on a paper by NIST (Effect of Dissolved Air on the Density and Refractive Index of Water by Harvey, Kaplan, and Burnett), at 20 C this should only make 0.002 g/mL difference, which means an expected density of 0.996 - still 0.002 away which is still more than the sum of the effects of air buoyancy and analytical balance error, although it is getting much closer so maybe I need to be more careful with my error budget.

blogfast25 - 16-2-2014 at 07:36

Quote: Originally posted by unionised

I may have missed it, but I didn't see any reference to correcting for the buoyancy of the air while making these measurements of the density of water.

What corrections, if any, were made for that?

The error is about 0.1% if you don't correct for it.

Never even crossed my mind. And you're right: at about 0.001 g/cm<sup>3</sup> density of air it amounts to an error of about 0.1 %. I wonder if that's part of the reason to do it relatively, with a pycnometer, as both weight measurements would be off by the same amount due to air buoyancy.

[Edited on 16-2-2014 by blogfast25]

unionised - 16-2-2014 at 08:24

If I had a dollar for every time someone slavishly copies down more than 3 significant figures from the balance, but forgets to account for air density, I'd be a rich man.
Why isn't this taught in schools?

And a pyknometer doesn't solve the problem.
There's still an error in the apparent weight.
Imagine that you were trying to measure the density of mercury and rather than an atmosphere of air, you were working under water.
You would start off weighing the vessel "empty" i.e. full of water (in the same way you normally weigh it full of air).
M1
Then you would weigh it full of mercury,
M2
Then you weigh it full of water to get the reference value.
M3

You find the weight of the mercury M2-M1
and divide by the weight of water
M3-M1
But that last term is zero.
You can't divide by it.

blogfast25 - 16-2-2014 at 09:47

@unionised:

So what's the applied correction? Just based on estimated volume (weighing boat + substance) of what you're weighing?

[Edited on 16-2-2014 by blogfast25]

sbreheny - 16-2-2014 at 12:56

I gave up on the buret because the repeatability I get with it is worse than with the volumetric flasks. Next thing I will try is a 250mL volumetric flask since the relative error is less with larger flasks of the same accuracy class. I didn't do this before because it will not fit inside the draft shield on my balance but I just remembered that I can open the top.

About pH - someone asked if I had measured the pH of my distilled water. I just did that. If I do it with just distilled water, I find the pH meter very slow to respond and it is unclear whether the result is accurate. I get about 6.26.

So, I added several different pure salts (one at a time to separate samples of the distilled water). For each one, I added about 1g to 50mL of water. I got the following results (and with all of these the pH meter settled almost immediately to a result):
Pure NaCl 5.45
Pure NaSO4 5.77
Pure CaCl2 5.45
With the NaCl I tried adding about 20mg, then 100mg, then 1g and it made very little (0.15) difference in the pH so I think that the salt is not greatly affecting the actual pH of the water.
I believe that the salts help the pH meter by increasing the conductivity of the water so that you don't get a localized potential difference between the probe and the rest of the water due to things like static charge.

So, I think the pH is very slightly acidic which is what we would expect from dissolved CO2. No additional discovery there.

sbreheny - 16-2-2014 at 13:47

OK, I think I may have the answer. I just re-did the test twice with a 25mL volumetric flask but this time, after rinsing the flask with acetone and before I weighed it to get the tare weight, I gave it a much longer time to dry out and I made sure that there was no smell of acetone remaining. This gave me 0.9976 g/mL the first time and 0.9969 g/mL the second time. If I then apply the air buoyancy correction of 0.0012 g/mL, I get 0.9988 g/mL the first time and 0.9981 g/mL the second time. The water temp was about 17C where the ideal density would be 0.9988. These are well within the margin of error for my volumetric flask (+/- 0.03 mL is 0.0012 g/mL for a 25mL flask).

So, it looks like it was due to the flask not being quite truly empty (air filled) when weighed before filling with water, plus the air buoyancy difference. Together, the air buoyancy and acetone vapor could make as much as 0.0024 g/mL difference. Several of my readings were in the 0.993 range, but most were 0.995. The outliers like 0.993 must have actually had a thin layer of acetone on the glass which was in equilibrium with the acetone vapor and did not evaporate. Since the volumetric flask has such a narrow neck, it takes a long time for the acetone vapor to diffuse out. Inverting the flask helps because of the greater density of the vapor.

feacetech - 16-2-2014 at 14:34

Why not use a hydrometer

[Edited on 16-2-2014 by feacetech]

sbreheny - 16-2-2014 at 16:10

Quote: Originally posted by feacetech

Why not use a hydrometer

[Edited on 16-2-2014 by feacetech]

First, because I don't have one. Second because once I tried this measurement and obtained results which were well beyond the limits of error which I should have been able to attain, I wanted to keep at it until I found out what was wrong (was it my equipment? my technique?" Looks like it was my technique but not in any of the expected ways, so I learned something!

Chemosynthetic - 16-2-2014 at 21:38

Excellent troubleshooting and results!

jock88 - 17-2-2014 at 06:07

Look up specifications for water for injections. This is crazy pure stuff.

blogfast25 - 18-2-2014 at 11:39

Quote: Originally posted by sbreheny

Since the volumetric flask has such a narrow neck, it takes a long time for the acetone vapor to diffuse out. Inverting the flask helps because of the greater density of the vapor.

In pro labs compressed air is often pumped through the glass work, with a rubber/glass tube.

Zyklon-A - 18-2-2014 at 12:15

Quote: Originally posted by MrHomeScientist

I know this was posted a while ago (and is slightly off topic).
Pure water does conduct electricity actually, because of the self ionization of water.
However, I doubt that a student had the equipment necessary to measure conductivity that precisely. So it almost certainly had to do with impurity's in the water.

[Edited on 18-2-2014 by Zyklonb]

AJKOER - 20-2-2014 at 05:41

Quote: Originally posted by sbreheny

.....
Today I did an experiment where I took three different 25mL volumetric flasks, weighed each one three times empty, then three times when full of water (bottom of the meniscus on the cal line), and averaged the readings for each flask and took the difference. I get the following densities for the water in the three flasks: 0.9943, 0.9935, and 0.9930 g/mL."

A few comments from the point of view of statistical analysis. In statistical sampling theory for ratio estimators (the density is a ratio estimate), the common measure employed is the ratio of the means of two variate and not the average of individual ratios as the sampling variance of the latter is much greater. It is, however, a bias estimate which can be adjusted. For an in depth discussion of various estimators and their respective standard error see "Advances in Sampling Theory-Ratio Method of Estimation" by Hulya Cingi, Cem Kadilar at http://books.google.com/books?id=ORy83SaeWqgC&printsec=f... .

Now, in the case of your experiments, I will assume that the sample ratio (r), calculated as the sum (or average) of the respective water weights divided by the respective volume measures, would be .9936. This statistic is, however, bias and needs to be corrected (see good discussion at Wikipedia at on Ratio Estimator at http://en.wikipedia.org/wiki/Ratio_estimator ).

If mean of the weights and volumes employed are both greater than 10 (that is, the volume of water measured each time is over 10 cc), I would recommend using the bias correction formula specified in Wikipedia (using a somewhat larger sample size) where the density estimate should have an error in the order of at most 1/n cubed.

Any error observed in excess of what is expected per sampling theory should then be attributed to experimental design.

[Edited on 20-2-2014 by AJKOER]

blogfast25 - 20-2-2014 at 05:56

Quote: Originally posted by AJKOER

(the density is a ratio estimate)

Nope. He is measuring absolute density, not relative density. 'g/cm<sup>3</sup>' isn't a ratio, it's a quotient. Ratios are dimensionless, see for instance 'dimensional analysis'.

[Edited on 20-2-2014 by blogfast25]

AJKOER - 20-2-2014 at 08:06

I an assuming that the water's absolute density is being ascertain by a comparison of mass to volume, with both of the latter subject to possible error sources.

For example, even if a vessel's volume is precisely calibrated, is it properly filled per the human eye? Using multiple vessels, are all properly calibrated or is there an equipment error distribution.

Related arguments for weight assessment using one or more scales.

For the degree of precision desired, I am not sure if one can ignore the manner in which the data is processed (like using an inefficient simple average of individual density observations).

macckone - 20-2-2014 at 08:57

This is a most enlightening thread.
The measurement error being due to acetone remaining
in the flask is logical in hindsight but having worked in
a lab where analytical procedures were followed, I can
see where the insufficient drying and diffusion time
could lead to errors as well as the air displacement.
Now living at high altitude it makes it even more complicated
to account for air displacement.

unionised - 20-2-2014 at 12:42

Quote: Originally posted by blogfast25

@unionised:

So what's the applied correction? Just based on estimated volume (weighing boat + substance) of what you're weighing?

[Edited on 16-2-2014 by blogfast25]

Google is your friend
http://metrology.burtini.ca/grav_air.html

unionised - 20-2-2014 at 12:45

Quote: Originally posted by sbreheny

About pH - someone asked if I had measured the pH of my distilled water. I just did that. If I do it with just distilled water, I find the pH meter very slow to respond and it is unclear whether the result is accurate. I get about 6.26.

.

It is difficult to measure the pH of unbuffered solutions (and it's impossible with an ordinary pH meter)

sbreheny - 21-2-2014 at 00:22

Hi everyone,

I received my ultra-pure $25 per liter (!) water and repeated the experiment a few times. I put together a spreadsheet which automatically applies several corrections and tells you the error from the ideal value. I have attached the spreadsheet (LibreOffice format) and here is an image of the sheet:
<a href="http://s13.postimg.org/bwp92sl6v/table.png" target="_blank"><img src="http://s13.postimg.org/bwp92sl6v/table.png" width="800" /></a>

The spreadsheet doesn't handle significant figures very well (or at least I don't know an easy way to get it to do so), so some of the numbers given are ridiculously long.

I have 5 instances of the experiment recorded, three with high purity water and two with Target distilled water. There is no noticeable difference with my equipment between the high purity and simple distilled (as expected). All 5 readings, now that I am much more careful about drying out the flask and I add in the air buoyancy correction, are within the tolerances of my flasks.

Temperature corrected volume is the volume of the flask assuming that it is at the same temperature as the water, using a standard correction formula. Balance correction factor is the ratio of 50 grams to the reading of my balance when I weigh a class 1 50 gram weight. Balance discrepancy and volume discrepancy (at the far right) are values of balance error and flask volume error, respectively, which would explain the entire observed error if that error were solely due to error from that instrument.

Thanks again for your help and for listening!

Sean

Attachment: water density.ods (15kB)
This file has been downloaded 411 times

[<a href="u2u.php?action=send&username=bfesser">bfesser</a>: reduced image width, linked to full-res.]

[Edited on 21.2.14 by bfesser]

bfesser - 21-2-2014 at 05:47

Is this more like what you were going for?

<table><tr><td> Attachment: water_density_2.pdf (25kB)
This file has been downloaded 542 times</td><td> Attachment: water_density_2.ods (15kB)
This file has been downloaded 409 times</td></tr></table>
By the way, simply changing the extension from .jpg to .png doesn't make a PNG—they're totally different compression schemes.

sbreheny - 21-2-2014 at 07:42

Quote: Originally posted by bfesser

Is this more like what you were going for?

<table><tr><td> </td><td> </td></tr></table>
By the way, simply changing the extension from .jpg to .png doesn't make a PNG—they're totally different compression schemes.

Hi bfesser,

Thanks for your suggestion. No, what you did is not exactly what I am looking for. For one thing, the percent error is off by a factor of 100 (100x too high). Also, some of the significant digits are actually hidden now. I know that I could manually adjust the format of each cell to reflect the number of sig figures but what I meant is that I am not aware of a way to make a typical spreadsheet automatically show the correct number of sig figures of a computed value given the number of sig figures in the input values.

JPG vs PNG - yes, I know what you mean. The original image is indeed a PNG file. When I uploaded it to that image hosting site for some reason it converted it to a JPEG format but left the extension as PNG. Sorry for the confusion.

Sean

AJKOER - 25-2-2014 at 10:40

Quote: Originally posted by AJKOER

Based on my prior statement (above) and the fact that your sample size is 5, my suggested density sampling estimate would still have a statistical error in the order of (1/5)^3 or .008, which is very large.

As your current perceived error is less than this, I continue to recommend an increased sample size if you intend to present the results of experiments formally. For example, with 10 samples, the error with respect to purely statistical sampling error (employing the estimator and bias correction factor I suggested) would be in the order of (1/10)^3 or .001.

If, with an increased sample size and your indicated error is too large for sampling error alone, there could be potentially a systematic experimental measuring error (to be determined) accounting for the difference. Else, your done, your estimate is statistically within tolerance levels.
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The simple average of the observed ratio, the so called index method, is generally bias and subject to large variability. Source, please see "Ratios: A short guide to confidence limits and proper use", by V.H. Franz, October, 2007, available at http://arxiv.org/pdf/0710.2024.pdf). To quote from page 12:

"Also, if the mean ratio r-bar is used as a point estimate for rho it shows systematic biases and can be much more variable than the ratio of the means"

The author also notes on pages 23 and 24, to quote:

"The index method is used very often (almost all of the example studies in the supplementary material provided with this article used this method). We can justify the method in the context of a linear model if the denominator [the volume in our case] is bounded away from zero and if the data have a specifc heteroscedastic structure, such that the numerator [measure of mass] has larger variability at larger values of the denominator [or volume, in our case] .....
Because the method is used so often and because it seems unlikely that the data in all these cases show the specific heteroscedastic structure."

Now, in the current context, I am not sure if this is indeed completely correct. I expect the error in measuring the volume to be independent of the magnitude of the volume. The error in weighting may,or may not, be proportional to the weight (and its volume as required by the modeling assumption). Repeated weighting of the same set of water volumes varying in size, with different scales, may give insight as to if and how the error variance in determining mass is related to volume.
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If you continue to believe in small sample based quotients as a reliable indicator of a population mean, try a spreadsheet simulation. The model would be a regression through the origin where x is a selected from a distribution of possible masses, and y is constructed from the product of the x with the known population density plus, say, a Normal distribution error term with specified mean of zero and standard deviation based on your data. Tabulate the generated densities (y/x ratios) and store to compute the mean bias, associated standard error, the median absolute deviation,...over your simulation run (several thousand).

If you need to do some research on constructing such a simulation, the topic is Monte Carlo simulation techniques.

[Edited on 26-2-2014 by AJKOER]