Sub-module 3A, Page 9
The notion of the time constant and half-life are common in science and engineering. The half-life is the time that it takes for the concentration to decline by half. This time depends on the rate constant, not what the starting concentration is. The rate constant and the half-life are related by the expression:
Half-life = 0.693 / k.
Half life is frequently written t1/2 . [HW problem 6] In our MEK in the lake example, the flow into the lake and the volume of the lake gave you the rate constant. Biological processes, such as the rate at which the bacteria in the lake metabolize MEK, and chemical processes that change MEK into another chemical, might also be functions of the time and some rate constant. This rate constant might be derived analytically or perhaps measured in the laboratory. If the underlying process is determined by always taking a fixed percentage of what is there, you will always get a rate constant and half-life type of equation.
Of course there are many processes that do not take a fixed percentage of what is there. Sometimes systems are saturated and can only transform a fixed amount, other process are proportional to the square or some exponent of what is there. These processes likewise can be modeled mathematically, if one knows the underling process. Often the first order rate constant is used to model situations that are much more complicated - sometimes it works and sometimes it does not. More often we don't know what is going on and might as well use the simplest model. If we don't know what is going on, that's about all we can do. That's enough philosophy for now.
Besides being able to solve some simple first-order decline with time and understand the rate constant and term half-life, we will not spend much time on this. Any process that changes with time is called unsteady state. Such unsteady state is the norm, but before we model unsteady state for complicated systems, we need to know where things are going. For that we need to have a handle on the concept of partitioning.