

Volume of Solvent Plus Solute
Name: David
Status: student
Grade: 912
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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Update: June 2012

