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I'm studying chemical kinetics in high school. We are studying Integrated Rate Equation of first order reactions - their derivations and graphs. Our teacher showed us a graph of:

Concentration of Reactant [R] at time t vs. Time (t)

In the graph showed to us, The graph line slowly becomes parallel to x-axis and stretches onto infinity which is the Time.

We were explained that first order reactions are never ending because as the concentration of reactant decreases, the Rate of reaction decreases at very large amount and hence the reaction keeps going on and never reachers zero concentration and thus doesn't end.

He gave us the example of radioactive decay, Chernobyl, how the nuclear recations there will never end as the radioactive decay is a first order reaction.

However I am not able to understand the explanation given to us:

first order reactions are never ending because as the concentration of reactant decreases, the Rate of reaction decreases at very large amount and hence the reaction keeps going on and never reachers zero concentration and thus doesn't end.

It would be helpful if someone could help me understand this, maybe some mathematical correlation in the formula or something.

Any help is appreciated.

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    $\begingroup$ Think of the infinite series 1/2+1/4+1/8+1/16.... You will get closer and closer to 1, but only in the limit of an infinite number of steps will you reach it--because each step only goes half as far as the last one. Of course, matter isn't infinitely divisible. If you have a reaction that only proceeds in the forward direction, like radioactive decay, eventually you will reach the point where you have 1 atom and then, when that decays you don't get 1/2 atom, then 1/4 atom , etc. Rather, when that atom decays you are left with 0 radioactive atoms of that isotope—so it does reach 0. $\endgroup$
    – theorist
    Commented Apr 19, 2022 at 6:01
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    $\begingroup$ This is simply an artefact of modeling the reaction with a perfect mathematical expression: exponential decay. In reality, matter is not infinitely divisible, so there will come a point where any first order reaction will run out of molecules. $\endgroup$
    – Hayden S
    Commented Apr 19, 2022 at 6:03
  • $\begingroup$ Ever heard about Achilles and the turtle? $\endgroup$
    – Ivan Neretin
    Commented Apr 19, 2022 at 6:18
  • $\begingroup$ Also related: chemistry.stackexchange.com/questions/57075/… $\endgroup$
    – Ivan Neretin
    Commented Apr 19, 2022 at 6:27

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Note that basics of reaction kinetics were not created with the quantized nature of matter in mind. In this context, reactions of the 1st order do not end for the same reason why the function $\exp{(-x)}$ does not reach zero for any positive real $x$.

Due matter quantization, such reactions would end (unless there are other reactions producing the reactant) when the last molecule or a nucleus reacts or decays. But some time before that, it would not be a reaction of any particular order any more, as there would be strong quantum fluctuations for small counts of particles and the reaction rate would become more and more random.

For many scenarios it may be considered as if matter were not quantized and if the reaction does not end, as the eventual ending does not affect the kinetics at observed scope.

The differential equation of the reaction kinetics of the 1st order is

$$\frac{\mathrm{d}c}{\mathrm{d}t}=-Ac$$

and the function of the respective reaction kinetics is

$$c = c_\mathrm{t0} \exp{(-kt)}=c_\mathrm{t0} \exp{(-\ln{(2)}\frac{t}{T _{1/2}})},$$

where $T_{1/2}$ is the half-time (or half-life) of the reaction or radioactive decay, with the meaning of the time the concentration or amount of radioactive isotope decreases to one half, regardless of the initial value.

BTW, only reactions of the zeroth order ends (in original continuous matter context), as their rate is linear. It applies also on the equilibrium state, where the forward nor backward reaction rates do not cease, they just become equal.

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  • $\begingroup$ You wrote: "... as there would be strong quantum fluctuations for small counts of particles. For most scenarios it may be considered as it does not end." How would quantum fluctuations prevent a sample of radioactive atoms from decaying completely? $\endgroup$
    – theorist
    Commented Apr 19, 2022 at 7:23
  • $\begingroup$ @Theorist I have not said it prevents that. By the quote, I have tried to say that the reaction kinetic progressively stops following the simple exponential function ( which was derived with statistics of large numbers in mind ) and becomes more and more random. I have added some text. $\endgroup$
    – Poutnik
    Commented Apr 19, 2022 at 7:27
  • $\begingroup$ I was referring specifically to your statement that "it does not end." With radioactive decay, it does end -- eventually you reach the point where you have no more radioactive atoms. And likewise, if there is no reverse reaction (or say the products are always swept away) eventually all the reactants will be used up, and that reaction would end as well. I don't see how quantum fluctuations would prevent this. $\endgroup$
    – theorist
    Commented Apr 19, 2022 at 7:32
  • $\begingroup$ No, I have said "For most scenarios it may be considered as it does not end." It implies many macroscopic measurements which end above the detection limit, or below it, implying it still goes on. $\endgroup$
    – Poutnik
    Commented Apr 19, 2022 at 7:33
  • $\begingroup$ But for the scenarios you were describing, it does end. $\endgroup$
    – theorist
    Commented Apr 19, 2022 at 7:34

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