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I learnt that the chemical symbol for baking soda and baking powder is NaHCO3. However, when I do a simple experiment which is put vinegar into both substances. I can get different observations.

When I put vinegar into baking soda, I get a fast and fuzzy reaction. But when I put vinegar into baking powder, instead of a fast and fuzzy reaction, I get a slow and bubbly reaction.

So, my question is what properties do each of the both substances have to make different reactions with vinegar although their chemical symbols are the same?

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As LDC3 says, Baking soda is simply sodium bicarbonate, $\ce{NaHCO3}$. When one adds an acid, the following reaction occurs: $\ce{HCO3- + HA -> H2CO3 + A-}$ and the carbonic acid releases carbon dioxide: $\ce{H2CO3 -> CO2 + H2O}$ (these are both equilibrium reactions, but as you can observe when you add vinegar, a lot of gas is produced and the gas is generally lost, pushing the equilibrium towards making carbon dioxide).

Baking powder is a mixture of sodium bicarbonate and one or more weak acids. When dry, no reactions occur, but when wet, the components dissolve and can react. The observed differences when mixing vinegar with baking powder probably result from the fact that there is less sodium bicarbonate in a given volume of baking powder than baking soda, and baking powder often contains starch, which can influence the properties of the added liquid, giving the bubbles a different appearance.

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Baking soda and baking powder are not the same.
Baking soda is (almost 100%) sodium bicarbonate ($\ce {NaHCO3}$).
Baking powder is a mixture of sodium bicarbonate, and some acids.

From Wikipedia: Typical formulations (by weight) call for 30% sodium bicarbonate, 5-12% monocalcium phosphate, and 21-26% sodium aluminium sulfate.

When you mix baking powder with water (in a dough), it will cause bubbles to form. Nothing will happen with baking soda.

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