Which runs out first: baking soda or vinegar?

In one line: with Nico's recipe the vinegar runs out first. A cup of vinegar only has enough acid for about 1 to 1¼ Tbsp of baking soda, so more than half of the 3 Tbsp should be left unused, and only more vinegar would make more gas.

They pair up one to one

Vinegar is mostly water with a little acetic acid in it. Baking soda is sodium bicarbonate. Each acid molecule hands one hydrogen ion to one bicarbonate; that makes carbonic acid, which falls apart into water and one molecule of carbon dioxide gas. The other product, sodium acetate, stays dissolved in the liquid:

NaHCO₃ + CH₃COOH → CH₃COONa + H₂O + CO₂

One of each, every time. When one of them is used up the reaction stops, however much of the other is left. The bubbling tapers off over the next minute or so: the last bit of fizz is dissolved gas coming out, and a stir helps the last of the acid find the powder. So the question is just: which one has fewer molecules?

Counting them

• A level tablespoon of baking soda weighs about 14 g (USDA: 4.6 g per teaspoon, so 13.8 g). Bakers' charts go up to 18 g (King Arthur), depending on how you fill the spoon. Chemists count molecules in "moles"; one mole of baking soda weighs 84 g, so 1 Tbsp is about 0.16 mol (USDA figure).
• A cup of 5% white vinegar (237 mL) holds about 12 g of acetic acid ("5% acidity" means about 5 g per 100 mL). One mole of acetic acid weighs 60 g, so a cup has about 0.20 mol.
• So 1 cup of vinegar is used up by about 1 to 1¼ Tbsp of baking soda, depending on the spoon (0.197 mol × 84 g per mole ≈ 16.5 g).
• Nico's recipe has 3 Tbsp (41 g, 0.49 mol) with 1 cup. The vinegar runs out first, and about 25 g, a bit under 2 Tbsp, of baking soda should be left, unused (more with heavier spoonfuls). Using all 3 Tbsp would take about 2½ cups of vinegar.

The gas

At room temperature (20 °C) and sea level, a mole of any gas takes about 24 litres. Nico's recipe makes about 0.20 mol of CO₂, so about 4.7 litres: about 2.4 big 2-litre soda bottles, or 11 of his 14.3 oz glasses. A little of it (about 5%) stays dissolved in the liquid for a while, like an opened soda that's still fizzy. Up high the same gas takes more room: about 20% more in Denver.

Where does it go? Some of it gets trapped in the soapy foam (below). The rest is invisible and a bit heavier than air (about 1.5 times as dense), so it slowly spills over the rim as it's made and spreads out low before it mixes into the room.

More baking soda can make it fizz a bit faster, but once the vinegar's acid is used up, it can't make more gas. Only more vinegar does that.

Will it come over the top?

Foam is gas wrapped in thin walls of liquid. The foam's liquid comes out of the glass itself, so the top of the foam sits where the liquid and powder plus the gas trapped in the foam would reach. Whatever doesn't fit in the glass spills into the tray:

foam that spills = gas trapped in foam − empty room above the liquid and powder

The dish soap doesn't trap all the gas. The rest is never caught: it escapes at the top while it's being made (that's the invisible gas spilling over the rim, drawn thicker when more is escaping), and a little stays dissolved in the liquid (about 5% with Nico's recipe; more when there's extra vinegar). Foam that does form lasts: Nico saw it survive a long time, so in the sim it stays.

How much gas the soap traps isn't something you can look up (I found no measurement for this kitchen reaction), so the sim is tuned to a real run: Nico's fullest run, which left foam 1 to 2 cm deep in his tray (about 20 × 40 cm). Taking 1.5 cm, about 1.2 litres of foam spilled. Together with the foam that filled the glass, that foam holds about 1.37 litres of gas: about 29% of the 4.7 litres made. (With 1 cm it would be about 20%, with 2 cm about 37%.) That was the most foam he got, so the sim shows about the most you'd expect; a typical run spills less.

The foam is wet: it carries liquid up and out with it. Nico said: "especially if the reaction is balanced, it looks like just a bunch of foam." The sim reads that as all the liquid ending up in the foam, so it is calibrated so the foam carries all the liquid when the vinegar runs out, matching what Nico saw. (His recipe isn't balanced: it has more baking soda than 1 cup of vinegar can use. But the vinegar runs out, and each cup of vinegar makes the same gas either way.) So with his recipe all 237 mL of vinegar ends up in the foam (in the glass and in the tray), which makes the foam about 15% liquid (237 mL of liquid in about 1.6 litres of foam). People who study foam call it "dry" under about 5% liquid and "wet" over about 15%, so this foam is at the wet end. It's one family's observation, not a measurement, and the sim uses the same share of liquid for every mix. It also assumes each bit of vinegar reacts with the baking soda before the foam carries it up and out (if it didn't, the acid carried away would stop fizzing and less gas would be made).

That has a neat result. Whenever the vinegar runs out first, each cup of vinegar makes the same amount of gas, so the foam takes all the liquid: the glass ends up foam to the rim, with the leftover baking soda at the bottom and no liquid left except a little dampening the powder. When both run out together it's nearly the same: a few spoonfuls of liquid can stay. With extra vinegar there's less gas for all that liquid, so some stays at the bottom: with 1 Tbsp of baking soda and 1½ cups of vinegar, about 160 mL.

Small mixes may not make enough foam to fill the empty part of the glass: with ¼ Tbsp of baking soda and ¼ cup of vinegar the foam gets about 79% of the way up, so nothing spills. That depends on how foamy your soap is: at the 2 cm end of Nico's range the same small mix would get about 102% of the way, and just spill. The dish soap makes no gas; it holds some of it in bubbles, so without it far less would spill.

The end picture is the moment the fizzing stops. Left alone, liquid then drains down out of the foam and collects at the bottom again.

The leftover baking soda

This is what the chemistry predicts; nobody has checked Nico's glass yet. The drawing shows it as powder at the bottom, under the foam. Once the foam drains back and you stir, much of it can dissolve: the liquid from Nico's recipe can take up roughly 16 g of the about 25 g (plain water holds about 96 g per litre; the dissolved sodium acetate lowers that). So you may see less powder than the drawing. Unstirred, most of it stays as damp powder for a while.

To check: wait for the fizz to stop, stir, then pour in a splash of fresh vinegar. If it fizzes hard again, there was baking soda left. (Adding more baking soda isn't a fair test: the used liquid still holds dissolved gas, and fresh powder makes it bubble a little anyway.)

The liquid left when the vinegar runs out isn't vinegar any more: its acid is used up. It's water with sodium acetate dissolved in it (salty water, a little on the basic side). The drawing turns it greyer to show that, in the glass and in the foam.

Assumptions: level US spoons and cups, 5% white vinegar, 20 °C, sea level. Heaped spoons just leave more baking soda over.