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My textbook states that for most reactions, the reaction rate decreases with time. I wonder what reactions will have increasing reaction rates with time. I know that adding a catalyst will increase the reaction rate but the final reaction rate will finally be zero. So, what reactions will their reaction rates increase with time?

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an explosion !! –  Elements in Space Feb 21 '13 at 14:08
    
This question is nonsensical because you misinterpret the "time". Reactions in general slow down because they run out of reactands, which happens "with time". Time per se will nor alter reaction rates. –  Georg Feb 27 '13 at 17:26
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@Georg I think it is fairly common to refer to extent of reaction as `time'. Moreover, it is possible (though rare) to have reactions that become faster when the reactants run out as shown in the answers below. –  Michiel Mar 25 '13 at 16:06
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2 Answers

Reactions that are autocatalytic can have their rate increase as a function of conversion.

One example is the decomposition of cellulose acetate by hydrolysis, also known as vinegar syndrome. The reaction is acid catalyzed, and it produces acid. In theory, the reaction gets faster. In practice, diffusion and limiting reagents many prevent a rate acceleration. Acetylsalicylic acid decomposes similarly. In the following equation, $\ce{R}$, could be the cellulose backbone or the salicylyl group. The acetyl group $\ce{-COCH3}$ is represented by $\ce{Ac}$. Thus acetic acid is $\ce{HOAc}$.

$$\ce{R-OAc + H2O ->[\ce{H+ catalyst}] R-OH + HOAc}$$

Another example is the alpha halogenation of ketones or aldehydes, which is acid catalyzed and acid-producing.

$$\ce{CH3-CO-CH3 + Br2->[\ce{H+ catalyst}] CH3-CO-CH2Br + HBr}$$

A historically interesting example is the allotropic decomposition of tin, or "tin pest", in which tin metal transforms to a brittle form in cold weather. The book Napoleon's Buttons, which is about the influence of chemistry (wittingly or otherwise) on history, opens with the conjecture that tin pest is what really stopped Napoleon's invasion of Russia.

In addition, as this answer reminds us, the rate of the reverse reaction increases to a maximum during the establishment of dynamic equilibrium.

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I will add. Bromation of acetone must be performed with care, allowing first portion to react over long time. Without it, the reaction has very good chances to go out of control. I want also to note, that some reactions produce heat, and heat is 'universal reaction accelerator.' For example, if you put some $FeCl_3$ into cold 30% $H_2O_2$ the reaction will be very slow. But once reaction speed will exceed some limit, it will run out of control, accelerating because of it's own heat. –  permeakra Mar 25 '13 at 15:03
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There are several, but the rate can only increase in time under certain limits, that is, assuming certain approximations. Obviously at "final time" (for instance, when the equilibrium is reached) the rate will not be increasing continuously.

In general, in order to have rates increasing with time, you need to have negative orders of reaction with respect to some reactant, that is, the velocity must depend with the inverse of the population (or less likely, other exponent). This is typical of chain reactions, the most typical case being explosive reactions.

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