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Author: Subject: How accurate are solution density measurements?
fusso
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[*] posted on 19-6-2019 at 16:42
How accurate are solution density measurements?


I used a measuring cylinder to measure my ammonia solution's density. I calculated the density to be 0.912gcm-3. According to https://wissen.science-and-fun.de/chemistry/chemistry/densit... it should be 23.39%. But how accurate is the result? What would be the range of the actual density?

[Edited on 190620 by fusso]




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mayko
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[*] posted on 19-6-2019 at 17:30


Way too little information to even begin: was this a single measurement, or a composite of multiple measurements? What mass(es) and what volume(s) were measured? What size graduated cylinder did you use? (Most have a small inaccuracy near the very bottom, which means that they have a large percent error for relatively small volumes, but a smaller percent error for large volumes.)

The phrase you probably want to look up is "propagation of error". There's also this thread; fitting a regression to multiple (mass,volume) data will give you confidence bands on the density estimated (though incorporating instrumental error will take a little more work).

And/or, independently measure concentration with gravimetry/titration and compare results.




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fusso
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[*] posted on 19-6-2019 at 17:36


Single measurement, using a 25ml cylinder, 19.6g for 21.5ml.



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[*] posted on 19-6-2019 at 17:47


Graduated cylinders are really not very accurate. Density is quite temperature-dependent. Do you know what grade of graduated cylinder you have or is it chinese stuff? Some brands have the tolerance for the markings printed right on them.

A volumetric flask would be much more accurate. The temperature should at least be very close to 20C. A scale that has actually been calibrated, or at least checked against a set of calibration weights matters.
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[*] posted on 19-6-2019 at 19:22


A good set of hydrometers makes this sort of thing much easier.

I suspect even a fairly broad range hydrometer would be a more accurate than using a graduated cylinder and a scale.




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Sulaiman
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[*] posted on 20-6-2019 at 04:13


various notes and opinions:

. the accuracy of measuring cylinders can be Class-A, Class-B or other (usp, unclassified ...)
- you did not specify the accuracy class of your measuring cylinder

. the density of common liquids varies significantly with temperature
- you did not specify the temperature

. thermometers also have accuracy classes
- you did not specify the expected error range for temperature

. weighing scales also have accuracy limits
- you did not specify the error range for weight

You specified the density as 0.912 with no estimate of error
by default the error is +/- one last digit
so you are implying that your measurement is accurate to 0.1%
- I can't read a 25ml measuring cylinder to 0.025ml
- Class-A 25ml measuring cylinders have up to +/- 0.25ml error
- How accurate is your thermometer ?
- how accurate are your scales ?

and if you are going to specify to 0.1% or better,
then your scales need to be calibrated at your location
- due to geographical variations in gravity,
and you need an estimate of bouyancy due to your local air density !.




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[*] posted on 20-6-2019 at 06:02


The density \(\rho\) of a sample is calculated from its mass \(m \pm \delta m\) and volume \(V \pm \delta V\).
$$\rho = \frac{m}{V}.$$
The uncertainty \(\delta \rho\) on the density is calculated using
$$\delta \rho = \rho \Bigg(\Big(\frac{\delta V}{V}\Big)^{2} + \Big(\frac{\delta m}{m}\Big)^{2}\Bigg)^{1/2}.$$
So you need to figure out the uncertainty on the mass and volume. For the cylinder, the uncertainty should be written on it.
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[*] posted on 20-6-2019 at 06:25


You can get specially calibrated cups that fill to an exact volume. The lids have a hole in them so any excess can be squeezed out of the top. You weight the contents on a decent balance (3dp?) that is accurate enough for your purposes and divide by the volume.

I guess it depends how accurate you need your measurement to be.




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[*] posted on 20-6-2019 at 12:08


Sounds like a pycnometer.

They've got ones that include a thermometer in the design too.




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[*] posted on 20-6-2019 at 16:19


Forgot ... If there is enough volume of liquid then I'd use a hydrometer for quick results
with surprisingly good precision (typically much better than 1%) even using cheap hydrometers and thermometers.

To answer more directly, if I did the measurement I would expect;
+/- 0,25 ml error using a 25ml Class-A measuring cylinder
+/- 0.1 ml reading error (if I'm really careful)
+ about 0.05 ml for each drop stuck to the glass
let's say about 0.32ml error in 21.5ml = 0.15% error, approximately
(Temperature correction is required, but I'd assume that the errors in temperature correction would be less than overall errors)

after calibration I trust my cheap 300g x 0.01g scales to +/- 0.02g
so the weighing error would be about 0.1%

compensate for bouyancy (about 0.1%) with negligible error

So, I'd hope to achieve about 0.2% error in my density measurement.

Unfortunately many liquids change density slowly vs. concentration change,
so a 0.2% density error could lead to a 1% or more error in concentration calculations.


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[*] posted on 20-6-2019 at 18:58


Hm, I made another measurement and get a density of 0.906(forgot the mass&vol). The cylinder is 25±0.5ml, has markings every 0.5ml but unknown class. The cheap scale is 500x0.01g.



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[*] posted on 20-6-2019 at 22:28


all you need is in this book :

Data analysis for chemistry: an introductory guide for students and laboratory scientists

1/ Measure your density with several essay. Let say 5, so N = 5

Evaluate the mean of your density

Check the standard error



Next you can evaluate the confidence limit associate with 95% certitude with student law



where A is a value associated with a normal distribution for 5 essay and 95% confidence in table here : https://en.wikipedia.org/wiki/Student%27s_t-distribution#Tab...
A = 2.015

By this way you don't have to bother with lecture incertitude associated with your instrument (expect systematic error) because all of them are evaluated with statistic

[Edited on 21-6-2019 by brubei]




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[*] posted on 21-6-2019 at 01:36


Two measurements made,
1) Density 0.912, concentration 23.39%
2) Density 0.906, concentration 25.33%
Density change 0.66%, relative concentration change 8.2%
- notice the relative change in concentration is much greater than the change in density

- your two measurements differ by about 0.66%
so even if your measuring cylinder and scales and thermometer and calculations are perfect,
your own skill is limiting the accuracy of your results.

In my opinion, measuring cylinders are good enough for measuring liquid volumes for quick qualitative experiments,
I like to use either hydrometers or a volumetric flask on scales for density measurement,
and a burette for delivery volumes (e.g. deliver a measured volume into a weighing vessel)
- if I get quantitative..

A pycnometer ( I know that at least one member has one) seems an excellent choice for density measurements if you need that level of accuracy,
(a pycnometer looks like a simple diy project)

P.S. as the ammonia concentration changes much more significantly than density,
using the same equipment, a titration would be more accurate.
(IF you have an acid of accurately known concentration to titrate against, which you could make from water and weighed dry powdered acids such as ascorbic, boric, citric, oxalic, sulphamic, sodium bisulphate .....)

[Edited on 21-6-2019 by Sulaiman]




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[*] posted on 21-6-2019 at 02:21


@brubei: Your approach only takes into account the random error right?

As you said you would also need to estimate the systematic error which is due to the calibration of your instruments. Surely in an amateur setting without a calibrated scale, the systematic error is important to consider.
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[*] posted on 21-6-2019 at 03:40


Aside from class of measuring cylinder, cylinders are usually also labeled TC (to contain) and TD (to dispense) so a TD cylinder is going to have more volume than indicated on the graduations. Just do a quick check with a known substance (water) and see where you fall.



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[*] posted on 18-7-2019 at 07:56


Some more info on doing these calcs in Excel:

Tellinghuisen, J. (2018). Least-Squares Analysis of Data with Uncertainty in y and x: Algorithms in Excel and KaleidaGraph. Journal of Chemical Education, 95(6), 970–977. research-article. https://doi.org/10.1021/acs.jchemed.8b00069



Attachment: Least-Squares Analysis of Data with Uncertainty in y and x.pdf (2.7MB)
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[*] posted on 18-7-2019 at 16:09


I have used a 10 ml Class A volumetric flask and a balance reading to 1 mg, with decent (for me, anyway) results.
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