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Name: David
Status: student
Grade: 9-12
Location: AZ
Country: N/A
Date: August 2006

Question:
Is there a formula for determining the volume of sugar in solution (at standard temperature and pressure)? In other words, if I add (e.g.) one gram of sugar to (e.g.) one litre of water, what will the resultant volume be?

I read somewhere that volume of sugar in solution is 0.645 ml/gm, so one gram of sugar would increase the liquid volume by 0.645 ml --- but this does not appear to be correct in empirical testing.

I have not been able to locate the answer to this question in any Internet or Library source. Yet, it seems that there should be a constant or polynomial equation for what can be empirically measured.



Replies:
A tough question. As you found, it is not as easy as using the densities of pure water and pure sugar. Volumes do not add when you mix things together because the intermolecular forces of a mixture are different than the ones in the pure substances.

The way this is handled is by using a property of mixtures called the "partial molar volume."

In a solution of sugar and water, one can show using thermodynamics that the volume is given by the formular

V = n_s *V_{m,s} + n_w *V_{m,w}

where

n_s = # moles of sugar
n_w = # moles of water
V_{m,s} = partial molar volume of sugar in the solution
V_{m,w} = partial molar volume of water in the solution

Both of the partial molar volumes are functions of temperature (and to a lesser extent, pressure) and are strong functions of the concentrations of sugar and water. They are not simple functions, and you cannot get them from looking up the properties of pure water or the properties of pure sugar at the temperature of interest. They are also specific to a particular solvent/solute combination, so you cannot use the partial molar volume function which works for water in a sugar/water solution for any other aqueous solution.

I did a quick search (nothing serious) and was not able to find sources for this particular data freely available on the Internet. I will try searching the primary scientific literature and if I can find it, I will ask NEWTON to post it.

Best,
Dr. Topper


In general, especially if precise/accurate measurements of volume are being made, the volumes of solution formation are NOT additive. A classic example is mixing ethanol and water. This has a negative volume change upon mixing. Put another way volume change is not conserved under the process of solution formation. Of course, mass is conserved. And if the components are not mutually soluble, e.g. sand and water, then the volumes are additive.

There are a couple of reasons why volumes of solution are not additive. One important one is that some chemical reaction occurs. The second is especially important if one component is water. Liquid water has an open structure, that is, the water molecules even in the liquid form molecule sized structures of lower density than the mean. The addition of ionic and/or polar solutes can "collapse" this structure resulting in an increased density. To complicate matters, the solute may also have structure that may be smaller than or greater than the average density of the solid. This will result in an expansion (or contraction) when the solute is dissolved.

There are no good theoretical models, of a general nature, to predict or estimate whether a solvent/solute combination will have a positive or negative "excess" volume of solution.

Vince Calder


David,

I do not believe that there is a generic equation to calculate the resulting volume. It is possible that someone has taken the time to calculate this for particular solutions (ie like your sugar solution). It is very difficult to predict this information because each solid will dissolve and interact with the solvent in its own unique way and are affected by temperature, solubility and concentration. These different interactions will cause differences in density. The density of sugar is much higher in solid form than it is in solution and the concentration of the solution will affect the density as well.

Your second paragraph starts out by saying that the volume is 0.645 mL/g. The units are units of density and not of volume. Volume would be mL only not mL/g. As I stated above, the density of 0.645 mL/g hold true only for a certain concentration of sugar. Technically the density should be written as g/mL, so the real density number should be 1/0.645 = 1.55 g/mL. Since water is 1 g/mL, this sugar solution is probably a saturated sugar solution (i.e. absolutely no more sugar will dissolve in the water because it has reached the saturation point). You can measure the density yourself for specific concentrations (measure out a certain known volume and weigh it) and then use this density to calculate the resulting volume because you can add the mass of the water and the mass of the sugar together (you don't have to reweigh the solution if you know the weight of sugar added and the weight or volume of water due to conservation of mass). You can calculate the mass of water based on the volume or vise versa since you know that the density is 1 g/mL at STP. Temperature does affect the density and charts are out there to list this data. So if you calculate the density for known concentrations, then you can calculate the volume based on the weight, but in essence, you are still measuring the same thing as your original setup. There is no way to predict this without empirical data (i.e. you measure the actual density of the solution).

It might be possible to gather a large amount of empirical data and then fit an equation to that data, but you will not be able to derive the equation without the data.

Matt Voss



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