Bdtyktl

I was given an interesting dilema today. A co-worker saw me adding a liquid (Diisopropyl ethylamine AKA DIPEA) to a flask filled with another liquid (Tetrahydrofuran AKA THF). I needed to weigh out exactly 5 grams of DIPEA into the THF and so I zero'd the scale with the flask+THF on it, then proceeded to add the DIPEA until the scale said 5.000g. Since masses are additive I assumed this was fine.

My co-worker, however, stopped and told me that although masses of two liquids are additive, the combined weights would not be, and since the scale measures weight as opposed to mass I had apparently just added an incorrect amount of DIPEA. He explained the reasoning to me but I'm a chemist, not a physicist and certainly not skilled in fluid mechanics, so I would like someone to dumb it down for me a bit or tell me if I'm way off.

From what I understand, the scale measures weight which is a function of gravitational force. But gravitational force is a function of buoyant force (its less if the buoyant force is greater since the bouyant force pushed a liquid up). Finally, buoyant force is a function of density. This means that my THF (which had a density of .9 g/ml) had a greater buoyant force than my THF/DIPEA solution (DIPEA density is only .74 g/ml so the solution would be somewhere between .74 and .90). And this means that technically as I'm adding DIPEA, the added mass is not the only thing causing the weight to increase; but rather the decreased buoyant force is also causing that.

And so, when the scale finally read 5.000g, I had possibly only added 4.950 or maybe 4.990 etc (something less than 5.000). Is my reasoning correct? Any help is appreciated.

Of course, by common sense, if you put together two objects with masses $m_1$ and $m_2$, and nothing comes out, then you end up with mass $m_1 + m_2$.

Weights are a little more complicated because of buoyant forces. All objects on Earth continuously experience a buoyant force from the volume of the air they displace. This doesn't matter as long as volume is conserved: if you stack two solid blocks their weights add because the total buoyant force is the same as before. But when you mix two liquids the total buoyant force can change, because the volume of the mixed liquid might not be equal to the sum of the individual volumes.

To estimate this effect, let's say (generously) that mixing two liquids might result in a change of total volume of $10%$. The density of air is about $0.1%$ that of a typical liquid. So the error of this effect will be, at most, around $0.01%$, which is irrelevant. Thus we can conclude that, rather than trying to help, your coworker just wanted to feel smug for a second.

@knzhou supplied a good answer. I’m going to offer a couple of other interpretations.

The first has nothing to do with the fact that you’re mixing liquids—it’s just that there are difficulties is determining mass precisely by measuring weight. As already pointed out, there is the buoyancy of the air—that produces a mass error of about $-0.0013$ g/l at sea level, or about $-0.14$% in the case of a substance with a density of $0.9$ g/l. Then, the gravitational acceleration varies by location by up to half a percent, largest near the poles and smallest near the equator (also smaller at high elevation). So if your scale measures force (uses springs or load cells rather than counterbalancing mass) and is calibrated for use in Paris, you’ll get an additional ~$ -0.1$% error if you weigh your sample in Mexico City. Of course, these errors would be the same if you weighed each liquid separately as if you weigh the mixture.

Second, I don’t know any chemistry, but if your two liquids react and produce gas which escapes from the top of the container, that is obviously lost mass. If this is not the case, then your buddy is pretty much full of crap, because the only other effect I can imagine which would produce a difference based on weight pre- or post-mixing is the one already alluded to:

In oceanography it’s called cabbeling. Two substances mixed together don’t necessarily have exactly the density you’d compute from a weighted average of the densities of each. For liquids I’m familiar with, the discrepancy is far less than $10$%: more like $1$% at most, which, when combined with air buoyancy makes for an error of 0.001%. You might as well worry about the error caused by the tidal force from the moon.