Amount of salt needed to achieve a specific density of soln?

[AqibH]

Everything as follows:
I'm looking to calculate the amount of NaCl (rho=2.19 g/cm^3) needed to add to water (rho=0.999 g/cm^3) to achieve a solution with density=1.071 g/cm^3. Ideally, I want to achieve an Atwood Number of 0.035 and a density of 1.071 g/cm^3 is what I need for the solution. However, the problem is that I'm not sure as to the exact concentration of salt that I need to have in the solution. I measured the density of the solution with 3 g/liter NaCl, 6, 10, 20, 30, 50, 70, 100, 150, and 190 g/liter NaCl. The 100 g/liter NaCl gave me a density of 1.067 which is far off since I need to achieve an Atwood number of 0.035 (extremely small number so even smaller room for error). I'm sick of the trial and error method to achieve a solution density of 1.071 g/cm^3, is there any mathematical way I can go about figuring out the exact amount of salt I need? (theoretically at least, I will still measure the density)

Equations
Atwood number =[ rho(saltwater) - rho(water) ] / [ rho(saltwater) + rho(water) ]
Atwood number is a dimensionless number used in fluid dynamics to relate the densities of two fluids. It can help to predict what instabilities will be present in a flow.

 

[Chestermiller]

The density of brine solutions is a function of salt concentration and temperature. What temperature are you interested in? Have you found any data on the density of salt solution in the literature? This should be handbook info.

Chet

 

[AqibH]

The density of brine solutions is a function of salt concentration and temperature. What temperature are you interested in? Have you found any data on the density of salt solution in the literature? This should be handbook info.

Chet

The experiment I'm conducting is done in a lab which is usually between 20 and 21 deg C. I have found some data in literature, however many texts use only one set of density of just plain water. As in, they say the density of the water they used is 1 g/cm^3. Whereas the water that I have been using the density is consistently measured at 0.998 g/cm^3. This throws off the Atwood Number significantly. I have made a salt water density table based on all the measurements I have already made with the various saltwater solutions. I then interpolated the data to find the concentration as well as fitting the curve after plotting the data points. Neither of these are as accurate as I would like to be.

 

[Chestermiller]

The experiment I'm conducting is done in a lab which is usually between 20 and 21 deg C. I have found some data in literature, however many texts use only one set of density of just plain water. As in, they say the density of the water they used is 1 g/cm^3. Whereas the water that I have been using the density is consistently measured at 0.998 g/cm^3. This throws off the Atwood Number significantly. I have made a salt water density table based on all the measurements I have already made with the various saltwater solutions. I then interpolated the data to find the concentration as well as fitting the curve after plotting the data points. Neither of these are as accurate as I would like to be.

How accurate does it have to be. The literature seems to indicate that a 10% solution should give your desired density at 25 C. Is that not good enough?

Chet

 

[AqibH]

How accurate does it have to be. The literature seems to indicate that a 10% solution should give your desired density at 25 C. Is that not good enough?

Chet

It needs to be fairly accurate. As in my original post, you can see Atwood number is a ratio of the densities of two fluids. I need an Atwood of 0.035. If water is one of my fluids (density of approximately 0.999 g/cm^3) and saltwater solution is my other fluid, I can calculate what the density of the saltwater solution should be (1.071 g/cm^3) to achieve Atwood of 0.035. If the saltwater solution has a density of 1.068 g/cm^3 then I get an Atwood number of 0.033 giving me an error of roughly 6%. Also this Atwood number helps to predict flow instabilities and needs to be precise or the experiment might not be repeatable. The flows are very sensitive I guess.

 

[mfb]

If 100g/l salt increase the density by 0.069 g/cm², you should get an increase of 0.072 g/cm² with approximately 100 g/l * 72/69 = 104.3 g/l. This is not exact as the ratio is not completely linear, but it should give 0.035 with two significant figures. You can get an even better estimate with some curve fitting and interpolation.

 

[AqibH]

If 100g/l salt increase the density by 0.069 g/cm², you should get an increase of 0.072 g/cm² with approximately 100 g/l * 72/69 = 104.3 g/l. This is not exact as the ratio is not completely linear, but it should give 0.035 with two significant figures. You can get an even better estimate with some curve fitting and interpolation.

I've both interpolated and curve fitting and both these methods are not sufficiently accurate. The result you give seems to be a bit low. From interpolation I get 106.4 g/l and from curve fitting i get closer to 106.8 g/l. Either way both these numbers will give me slightly different densities and therefore quite different Atwood numbers which then may not be repeatable. I was hoping for a mathematical model of some kind on which I can rely and then determine error from that. Also, that way the experiment should be repeatable by anyone not just me

Thank you and chet both for your continuing help with my issue

 

[mfb]

Okay, you have a reliable curve fitting estimate. Where is the problem with 106.4 or 106.8 (or 106.6) then? They should give the required number 0.035 within that precision (and you can test it). If you want to keep the experiment repeatable to an even better degree, use the same amount of salt every time.

 

[AqibH]

Okay, you have a reliable curve fitting estimate. Where is the problem with 106.4 or 106.8 (or 106.6) then? They should give the required number 0.035 within that precision (and you can test it). If you want to keep the experiment repeatable to an even better degree, use the same amount of salt every time.

Yea so if you use the Atwood number equation you will see that I need a density of 1.071 g/cm^3 theoretically. Simply put, I want to achieve this theoretical density and I cannot figure out the concentration of salt needed in a saltwater solution. Rather than having an pretty good estimate, I would like to have a model where I can input a desired density and I can get out the required salt concentration. I could probably go ahead and run experiments with an Atwood number of say 0.0334 or similar but then I would not be able to validate my results with previous experiments. The results might even be valid, but with a 6% error, I do not think anyone will agree that it is valid. Also, it is very difficult to be precise with the amount of salt added. Especially when It gets to be a few kilograms of salt.

 

[Borek]

As far as I am aware there is no mathematical model of the type you are looking for. There are quite precise density tables published, and there is a book containing coefficients of highly precise curves fit to these data, but all I have here is this small note taken eons ago (it should help you see what it is all about):

(ρ₁ is water density, octan sodu - sodium acetate, please note the way we write 1 and 7 here are different from the way they are written in US, and I am not necessarily consistent).

My CASC (http://www.chembuddy.com/?left=CASC&right=concentration_and_solution_calculator) does conversions of the density/concentration using a built in (and editable) density tables (for NaCl these are for 20°C) and using linear interpolation between the data points. There is a free trial version, so you can check if it is helpful.

 

[mfb]

Yea so if you use the Atwood number equation you will see that I need a density of 1.071 g/cm^3 theoretically. Simply put, I want to achieve this theoretical density and I cannot figure out the concentration of salt needed in a saltwater solution. Rather than having an pretty good estimate, I would like to have a model where I can input a desired density and I can get out the required salt concentration. I could probably go ahead and run experiments with an Atwood number of say 0.0334 or similar but then I would not be able to validate my results with previous experiments. The results might even be valid, but with a 6% error, I do not think anyone will agree that it is valid. Also, it is very difficult to be precise with the amount of salt added. Especially when It gets to be a few kilograms of salt.

Why 0.0334? If you measure the density to be 1.0710 +- 0.0005, you get something between 0.03502 and 0.03455. Rounded to two significant figures, both are 0.035.
Every model is an approximation, and making a model based on actual measurements close to the target value is not a bad idea.

 

[epenguin]

The experiment I'm conducting is done in a lab which is usually between 20 and 21 deg C. I have found some data in literature, however many texts use only one set of density of just plain water. As in, they say the density of the water they used is 1 g/cm^3. Whereas the water that I have been using the density is consistently measured at 0.998 g/cm^3. This throws off the Atwood Number significantly. I have made a salt water density table based on all the measurements I have already made with the various saltwater solutions. I then interpolated the data to find the concentration as well as fitting the curve after plotting the data points. Neither of these are as accurate as I would like to be.

As you're having to be accurate I think you have to be sure how do you know, I.e. how did you measure this? If you filled a volumetric flask couldn't the flask be the error? Density of water should be more standard than flask volumes. Maybe it is a bit affected by absorbed CO₂ or other gases. Not saying, just asking.

You'd possibly have to go to original literature to know how water is prepared for volume standards.

If it's any help, I think you could accurately measure to change of volume on adding salt to water? Comparing that to what it's sposed to be might tell you something.

 

[mfb]

As you're having to be accurate I think you have to be sure how do you know, I.e. how did you measure this? If you filled a volumetric flask couldn't the flask be the error?

That is another reason to use the own measurements, as a wrong volume of the flask won't influence the result much then. Relative error on volume relates to relative error on Atwood number, so 0.1% uncertainty for the volume is negligible - if you use the flask just for one measurement the relative error goes to the density measurement, and gets massively scaled up by the subtraction of two similar numbers.

 

[Borek]

use the own measurements, as a wrong volume of the flask won't influence the result

One of the first things taught in analytical chemistry when I was a student was how to calibrate volumetric glass (using water and balance). Then you are limited by the balance accuracy, not the volumetric glass accuracy.

Actually we were interested not in the exact values, but in the "commensurability" (that is, how many times the single volume pipette fits the single volume volumetric flask). But in principle it is all the same thing.

 

[AqibH]

Why 0.0334? If you measure the density to be 1.0710 +- 0.0005, you get something between 0.03502 and 0.03455. Rounded to two significant figures, both are 0.035.
Every model is an approximation, and making a model based on actual measurements close to the target value is not a bad idea.

Could you explain where the +-.0005 is coming from? From interpolation I can find an approximate concentration, but then I have no idea how to get the density of the solution without actually measuring it

 

[Borek]

why I do not use the data from linear interpolation or curve fitting. This is because then I am about .03-.08 off.

Doesn't it depend on how close are the points you use for extrapolation? Typical density tables have data points 1% or 2% (concentration) apart, with the density change (for NaCl solution) around 0.015 g/mL, and the plot is quite linear.

 

[AqibH]

Doesn't it depend on how close are the points you use for extrapolation? Typical density tables have data points 1% or 2% (concentration) apart, with the density change (for NaCl solution) around 0.015 g/mL, and the plot is quite linear.

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Yea you are right but the problem is that the density of their water might be slightly different from my water. And the density of their salt might be different from my salt. I honestly though that there was some mathematical way of solving this problem but I guess I'll have to make my table with more data as you have shown with my salt and solution. Thanks a lot!

 

[Borek]

Yea you are right but the problem is that the density of their water might be slightly different from my water. And the density of their salt might be different from my salt.

Density of water and salt (as long as we are talking about reasonably pure samples) doesn't depend on the source of substance. Also density of the solution - as long as we are talking about solution prepared from pure substances and of a well known concentration - doesn't depend on "my", "your" or "theirs".

 

[mfb]

Could you explain where the +-.0005 is coming from? From interpolation I can find an approximate concentration, but then I have no idea how to get the density of the solution without actually measuring it

Well, you didn't give any uncertainties so far, so I was guessing. If you can measure 1.071 with a significant last digit, then the uncertainty shouldn't be much larger than 0.0005.

The density of water depends slightly on the isotopic composition which depends on its origin, but I think this effect is negligible here.

 

[AqibH]

Density of water and salt (as long as we are talking about reasonably pure samples) doesn't depend on the source of substance. Also density of the solution - as long as we are talking about solution prepared from pure substances and of a well known concentration - doesn't depend on "my", "your" or "theirs".

I agree, but how can one be sure that they are using pure substances? In theory, there is no 'mine' or 'theirs', but in this case, there definitely is.

 

[Borek]

how can one be sure that they are using pure substances?

Are we talking about kindergarten, or a serious research? In the latter case, all reagents should be listed, together with at least basic information about purity, so as to make the results reproducible. In the former case, you are wasting time.

 

[AqibH]

Are we talking about kindergarten, or a serious research?

Yea kindergarten for sure. I was trying to say that you can't really know that a substance is pure. All these materials will go through a beaker and then a bucket and then a larger bucket and then a long pipe and then some tubing. The materials will no longer be 'pure'. You can't just use what you find online. You have to measure out everything yourself. Some kind of staff you are

 

[Borek]

I was trying to say that you can't really know that a substance is pure. All these materials will go through a beaker and then a bucket and then a larger bucket and then a long pipe and then some tubing. The materials will no longer be 'pure'.

If you can't guarantee you know what is the composition of your solution, your experiment is badly designed. If whoever did experiments earlier couldn't guarantee they knew what was the solution composition, their experiment is unreliable and shouldn't be used for comparison. That's the way it works, otherwise it is not even stamp collecting.

Published density tables were typically prepared with substances of known purity and in a way that guarantees reproducibility of the results. They are in use for 200 years or even longer, and they were tested tested on many occasions, so they are quite reliable.