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Given the reaction between $\ce{CH_{3}COOH {(l)}}$ and $\ce{NaHCO_3 {(s)}}$, which happens at the same pressure but at different temperatures, is there any way to mathematically compare the rate of reactions?

I thought using the rate equations of order 2, but I use the same concentration of materials. Suppose I could have found the connection between temperature of he material and its concentration, I would be able to find my required rate-indicators.

How to find the connection between temperature and concentration, or how to describe the rate of the reaction more exactly?

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Yes, there is, it has been done by others. Typical acid-base reactions that occur in solution are diffusion controlled reactions. Such reactions do not follow second order kinetics because at virtually every encounter of the acid and base, rapid proton transfer occurs and the reaction is complete. The reaction rate is limited only by how fast the acid and base can diffuse and encounter each other. It is usually easy to identify diffusion controlled reactions because they are extremely dependent upon the rate of stirring or mixing. They are also very sensitive to the solution viscosity. Rate constants for these reactions are typically $>\pu{10^11 sec^-1}$. Here is a link to a full paper that provides considerable experimental detail on one way to measure the rate of diffusion controlled acid-base reactions.

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