How to calculate rate of carbon dioxide from rate equation? [closed]








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The concentration of carbon dioxide that changes with time:


$$\ce{CO2(g) + H2O(l) <=>[$k\_\mathrm{fwd}$][$k\_\mathrm{rev}$] HCO3-(aq) + H+(aq)}$$


How to calculate the integral for the following rate equation of carbon dioxide production?


$$\frac{\mathrm d[\ce{CO2}]}{\mathrm dt} = k\_\mathrm{rev}[\ce{H+}][\ce{HCO3-}] - k\_\mathrm{fwd}[\ce{CO2}]$$


Water is ignored in the rate equation due to its very large concentration that doesn't change over time.




Suppose we start with $n$ moles of $\ce{CO2}$ for simplicity. Then, at any point during the reaction, we will have $n-V\_\text{gas}[\ce{CO2}]$ moles of each of the products. This means we could write their concentrations as
$$[\ce{H+}]=[\ce{HCO3-}]=\frac{n-V\_\text{gas}[\ce{CO2}]}{V\_\text{liquid}}$$


With all the concentrations expressed in terms of $[\ce{CO2}]$, we can rewrite your rate equation as


$$\frac{\mathrm d[\ce{CO2}]}{\mathrm dt} = k\_\mathrm{rev}\bigg(\frac{n-V\_\text{gas}[\ce{CO2}]}{V\_\text{liquid}}\bigg)^2 - k\_\mathrm{fwd}[\ce{CO2}]=k\_\mathrm{rev}\bigg(V\_\text{ratio}^2[\ce{CO2}]^2-2nV\_\text{ratio}[\ce{CO2}]+A\bigg) - k\_\mathrm{fwd}[\ce{CO2}]$$


Where $A=(\frac{n}{V\_\text{liquid}})^2$. Since this now only depends on a single variable and a few constants (the starting amount of $\ce{CO2}$ and the volume of each phase), it can be [readily integrated](https://www.wolframalpha.com/input?i=f%27%28t%29%3DC*f%28t%29%5E2-B*f%28t%29%2BA) to solve for how $[\ce{CO2}]$ (and consequently $[\ce{HCO3-}]$ and $[\ce{H+}]$) vary over time.


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