It's not a good idea to assume the "soda" that people are using in these videos is BAKING soda (sodium bicarbonate). It's not.
Yes, I have publicly said in the thread provided by Mobjack Bay and elsewhere that baking soda (sodium bicarbonate, NaHCO3) can theoretically be used to make soap. That is a correct statement.
I didn't go into any detail n that other thread. Although it can be done, saponification with baking soda also requires HIGH temperatures. Baking soda will never make soap at the usual temperatures that people use for hot or cold process soap making. You have to heat the mixture to a much higher temperature range.
The saponification reaction of baking soda and fat also evolves carbon dioxide gas, so the mixture foams a lot during saponification. There are fire extinguishers that have baking soda in them that are used to fight grease fires in kitchens. The baking soda reacts with the burning grease to form a foamy soap. The CO2 foam helps extinguish the fire by reducing the oxygen going to the fire and also by physically smothering the flames.
That's great for a fire extinguisher, but for everyday soap making, baking soda is not a practical alkali to use.
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There's at least one video on Youtube in a language other than English that supposedly uses (baking) soda to make soap, at least according to the original Englsh translation of the script. A later, more accurate translation of the script shows it's actually sodium carbonate that's being used, not baking soda, sodium bicarbonate. (edit: This video may be the one in Mobjack's link)
While bakers might call baking soda just "soda", that's not what "soda" means in the chemistry manufacturing world. "Soda" as the term is used in chemical manufacturing is sodium carbonate (washing soda, soda ash, NaCO3).
Soda = sodium carbonate, Na2CO3. Caustic soda = sodium hydroxide, NaOH. Potash = potassium carbonate, K2CO3. Caustic potash = potassium hydroxide, KOH.
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Sodium carbonate can be used directly to make soap, but only if the ingredients are simmered and stirred for hours. But it can be done.
A more efficient way to use sodium carbonate for soap making is to first convert it into sodium hydroxide, and then use the resulting NaOH solution for soap making as we normally would use it. Slaked lime (calcium hydroxide) is used to convert sodium carbonate to sodium hydroxide.
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Just because a weak alkali like baking soda happens to make your skin feel slick isn't proof that saponification is happening.