Steve Turner wrote a comment in "titration calculation" that
additivity of volumes is not a good generalization.

I wanted to know is there any rule of thumb when not to suppose the
volume are additive when mixing two different solutions? This effect
is demonstrated in most cases as an experiments describing mixing of
equal volumes "V" of ethanol and water in 1:1 ratio does not give a
total volume of 2V. However this property of volume is not even
discussed in detail in introductory analytical chemistry courses. Say
if one is determing an analyte in an unknown matrix by single point
caliration using "standard addition method", how one would be sure
that the volumes added were additive or not fro applying dilution
corrections? For dilute aqueous solutions the difference would be
perhaps undetectable. May be NIST would be able to measure it.
While searching about additivity of volume I came across The
Banach-Tarski paradox according to which if a cube is cut into ten
small cubes the total volume of of ten smaller cubes is "not" equal to
the volume of original cube.
Another example is given [Only registered users see links. ] as
"If the solid A has the volume μ(A) and if solid B has the volume
μ(B), then both solids have the volume: μ(A) + μ(B) -
μ(A∩B). For example, the volume of a house with cellar (H)
and that of the Earth (E) is not μ(H) + μ(E), because then
we would count the volume of the cellar twice."
But we are talking about solutions, so where can I find a
not-so-advanced treatment about "additvity of volumes" with respect to
solution chemistry. I searched J.Chem.Ed from 1924 to date but
couldn't find a single title devoted to this subject.
Would this effect be noticed when mixing gases, say if one mixes 1
liter of oxygen with 1 liter of nitrogen would the total volume be 2
liter at STP?

[Only registered users see links. ] (Mohammed Farooq) wrote:

The only generalization of which I am aware is that you never assume
additivity. In analytical chemistry, therefore, one always dilutes to
a final, precise volume. This avoids having to make any assumptions.

For example, in the original post which started this, the OP stated
that 25 mL of vinegar was diluted with 75 mL of water, presumably
resulting in 100 mL of solution which is 25% of the original
concentration. The assumption is that the volumes are additive. But
an analytical chemist would do the dilution differently: place 25 mL
of vinegar, accurately measured, into a volumetric flask and add water
to the mark. This ensures that the final volume is 100 mL and the
solution is 25% of original concentration. No assumptions are needed;
the only limitation is the precision of the vessels and the operator's
technique.

A related situation arises when making reagent solutions. To make 1M
sodium carbonate, for example, one would weigh 105.99 g of Na2CO3 into
a 1L volumetric, add ~800 mL of water, stir until dissolved, and
dilute to the mark. This is NOT the same as adding 1.00L of water to
105.99 g of sodium carbonate!

I would agree with Mohammed Farooq that, for dilute aqueous solutions,
the difference will be vanishingly small. But why not just do it the
right way, since it's no more difficult than the wrong way?

Farooq is also correct that the effect is pronounced and not at all
trivial when mixing liquids which are chemically different (e.g.,
water and ethanol, or organic solvents). This can come into play when
mixing chromatographic solvents, for e.g.