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Assuming we have a reaction of

$$\ce{CO2 + H2O -> HCO3- + \color{red}{H+}}$$

then is the proton (in red) is considered a product or a catalyst?

I read the Wikipedia article titled Product (chemistry), but the issue is still not clear to me. The article gives the following illustration:

enter image description here

In my original example above, I assume that the proton is considered a product since the hydrogen was originally part of the water molecule.

But in this case (see here, at 6:45) the hydrogen proton of the product of the reaction goes to the hemoglobin and kicks off an oxygen molecule.

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As you've illustrated with your figure from the Wikipedia article, a catalyst appears as the same compound on each side of a chemical equation. The short answer to this part of your question is that since the proton only appears on one side of the equation, it cannot be a catalyst. A hydrogen atom bound to a water molecule is not the same thing as a proton. Rather, it is the result, or product, of a chemical reaction. Also note that the reaction given in your question would be more correctly expressed as:

$$\ce{CO2 + H2O <=> H2CO3 <=> HCO3− + \color{red}{H+}}$$

In the second part of the question, that regarding the video excerpt, the proton from your reaction actually becomes a reactant in a second completely different reaction. Here, the proton reacts with hemoglobin resulting in two new products: the protonated hemoglobin an an oxygen molecule.

I hope that I have addressed the crux of your question. Please don't hesitate to ask for any clarifications in the comments below.

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I guess the correct equation is:

$$\ce{CO2 + H2O <=> H2CO3 <=> H+ + HCO3-}$$

A catalyst is something that enters the chemical equation just to decrease energy needed for the reaction to begin and in this case the $\ce{H+}$ took part in the final product so I don't think it's a catalyst.

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