How can I calculate the Density and Volume of a mechanical mixture (atoms)?

[bumblebee77]

Homework Statement

This is not actually homework, but it doesn't seem to fit in any of the other sections. I am old and should be able to figure it out but can't get my head around it for some reason and would really appreciate help!

- I know the density and volume of 55 molecules of water. I know the density and volume of 48 molecules of NaCl.

- Now I want to know the density and volume of a mixture of the water with 9 molecules of NaCl. I want to assume that there's no reaction between the water and NaCl, so the mixture is just a simple combination of water and NaCl.

Relevant Equations

Please see below in my attempt at a solution.

Is it correct that:
density = [ 0.09 * (density of NaCl) ] + [0.91 * (density of water) ]
volume = [ (volume of water) + ( { (volume of NaCl) / 48} * 9) ]

Thank you so much for any advice.

 

[pasmith]

There is a convenient table of density of saline water as function of concentration by mass here.

Tens of molecules is far too small to be able to treat the mixture as a continuum with a defined density or concentration at each point.

 

[bumblebee77]

Thank you, pasmith. I need to check modeling results, so need the data to do that.

 

[Chestermiller]

There is a volume change when salt is dissolved in water. Getting this theoretically is pretty complicated.

 

[bumblebee77]

Thank you, Chester. The thing I want to do is only use the mechanical mixture volume and density and compare those with the solution properties. I have data on the solution properties, but need to make sure I'm calculating the case where there's no dissolution correctly.

 

[Steve4Physics]

Is it correct that:
density = [ 0.09 * (density of NaCl) ] + [0.91 * (density of water) ]

I can't understand how you got your answer. I think the following problem is equivalent to what you are asking - and gives a different answer.

Say you have 55 wooden cubes and 48 steel cubes. The steel cubes are identical to each other but need not be the same size as the wooden ones.

Cubes are always arranged in a line, with faces in contact, to avoid 'packing' considerations, i.e.there are no gaps to increase the volume.

You know the density of the wood ($d_w$) and the total volume ($V_w$) of the wooden 55 cubes.

You know the density of the steel ($d_s$) and the total volume ($V_s$) of the 48 steel cubes.

You make a ‘mix’ consisting of 55 wooden cubes and 9 steel cubes (faces in contact).

What is the density (D) of the ‘mix’?

EDIT: Answer (in spoiler) removed as I suppose you are meant to try yourself and show your working.

 

[bumblebee77]

Thank you very much indeed, Steve. This does sound analogous to my situation.

You're right to be confused about what I said in my problem statement because I messed up (there are 55 moles in 1 L of water, so I should have said that I can relate that to my 55 molecules of water). It's fine though because I understand where you're coming from. You've clarified the problem for me. Thank you!