How to rationalize independence of half-life time from the initial concentration for the first order reaction?

Using the rate expression for the first order kinetics and expressing the half-life time, it can be proven the half-life time $t\_\frac 1 2$ of the first order reaction is independent of its initial concentration:


$$t\_\frac 1 2 = \frac{\ln 2}{k}$$


Can we *intuitively* explain that the first order half-life time is independent of its starting concentration and remains constant *without* the aid of the derived expression for the half-life time said above?


For the zeroth order reaction, it can intuitively be explained, but for the first order reaction it seems a little hard.




The basic assumption is that the probability of reaction of a given molecule (or disintegration of a radioactive atom) in a given time interval is independent of past history, and depends only on the time interval. If it is assumed that the time interval we choose is sufficiently small the chance of reaction is *proportional* to this time interval and the exponential decay law can be derived from this. The constant of proportionality is called the rate constant $k$. These assumptions mean that we cannot predict exactly when a given molecule will react but because of the proportionality (the rate constant) we can after *many measurements* know the form of the decay. Of course we assume that all the molecules of a given type are identical and behave in the same way and independently of one another.


We can measure the average time our type of molecule takes to decay by making making many repeated measurements and this average is the reciprocal of the rate constant and is related to the half life as $\ln(2)/k$.


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A particle has some probability to decay (Radioactive decays are ideal illustrative examples.) or react during some time interval. With more particles, the count of particles is directly proportional to particle count that will do so during that interval.


Therefore, the relative portion of particles reacting during that period does not depend on the amount nor concentration (assuming very high particle numbers converging to matter continuity). When the time interval leads to a portion being the half, it is the half life time.


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