# Are all mixtures linear?

I'm writing a computer program to help a factory produce bottles of mixtures. A mixture is usually alcohol, water and some other stuff, like menthol. But could also be water and NaOH and some oils (think schampoo).

My basic understanding is that if one bottle is 500 ml and contains: 50% water 45% alcohol 5% oils I need: 250 ml water 225 ml alcohol 25 ml oils

Then if I want to make 1000 bottles I just multiply this with 1000: 250000 ml water 225000 ml alcohol 25000 ml oils

However this is with the assumption that all mixtures have linear dependencies.

So my questions is:

1. Is all (possible) mixtures linear?
2. Are all reactions linear?
3. Are there a difference between mixtures and solutions here?

I suspect this is a very easy question for someone who know chemistry, but unfortunately my understanding is basic and my search skills too bad to find this out.

• 1. Not generally. A solution that behaves like that is called ideal. 2. Don't know what you mean, what reaction? 3. "Solution" is used to refer to a particular type of mixture, where one component provides a medium in which others (minor components) are dispersed. Official meaning of a solution in chemistry: goldbook.iupac.org/terms/view/S05746 Commented Oct 13, 2020 at 9:34
• It depends on you somehow... As usual I would be in trouble preparing the above 50 45 5 mixture. Usually it should mean 50 45 5 all in grams, but diffused bad practice is always a source of confusion (indeed you were oriented to volumes...). This said, volumes are generally not additive but others have already answered/commented on that. Commented Oct 13, 2020 at 12:24

Mixtures are never linear. Mixing $$50$$ mL water + $$50$$ mL ethanol gives a $$96$$ mL mixture. The masses are additive, but not the volumes. I agree the discrepancy is not huge. If you can tolerate a $$10$$% error, you may consider that the volumes are additive. Anyway take care of the fact that, usually percentage are indicated in mass, not in volume.

• yes but what is 5000 mL water + 5000 mL ethanol? It should be 9600 mL mixture, right? Commented Oct 13, 2020 at 10:09
• This is basically volume contraction in solvent mixing. A similar question has already been asked previously: chemistry.stackexchange.com/questions/23534/… Commented Oct 13, 2020 at 10:33

The nonlinearity of volumetric additions can be addressed theoretically. However, in practice, calculating how much material is needed to fill 1000 500 mL bottles containing 50% water 45% alcohol 5% oils (% by weight), can be addressed for one formulation by making a small batch and determining specific gravity. In practice, making the theoretical amount of product will not give the exact amount needed to fill 1000 bottles. Weighing errors, evaporation, residue on the mixing vessel, spillage, etc., etc., can easily add up to 2-3% - in either direction!

The specific gravity for each formulation must be specified (I did not say "determined"). Some formulations can be estimated to have the same specific gravity as another very similar product. If you are familiar with the products, you will know what can be estimated and what needs to be measured. A difference of 1% in the actual specific gravity is often not enough to worry about.

The things to worry about are:

1. how many bottles do you need to fill? Is it preferable to insure you make enough product for 1000 bottles, or to make sure there is no extra product to store or save (perhaps only enough to fill 973 bottles)?

2. when your computer printout indicates a number of grams of raw material to add, how many digits are needed if you desire 1% accuracy, and how confusing is it to specify 359.739 grams to be added when your balance/scale is only accurate to 1 gram (or less because it is never calibrated) and

3. using manufacturer's typical values for raw material specifications is frequently +/- 1-2%, so factoring them into some computer program may give you a feeling of control, but you really need to measure the specific gravity of the final product. Measuring the specific gravity could be a quality control test.

Computers can tabulate and keep track and calculate formulations for different size batches, but they do not do product formulation very well.