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The assertion that the second half of a radioactive substance's mass should need no longer than its half-life to fully expire its radioactivity is clearly not applicable here. As explained, the second half of the radioactive mass will be lost at an ever reducing rate and thus take much longer. In fact, to be exact about it, the remaining mass will take forever to lose full radioactivity as its decay rate slows down to near nothing towards the end.

For a given radioactive substance, \lambda is constant, so its half-life is also constant.

The assertion that the second half of a radioactive substance's mass should need no longer than its half-life to fully expire its radioactivity is clearly not applicable here. As explained, the second half of the radioactive mass will be lost at an ever reducing rate and thus take much longer. For example a half of the remaining half of the mass (= 1/4 of the initial mass) will expire in another half-life. After that, half of the remaining mass (= 1/8 of the initial mass) will decay over the next half-life period and so on. In fact, to be exact about it, the entire remaining mass will take forever to lose full radioactivity as its decay rate slows down to near nothing towards the end.

The kinetics of irreversible processes (chemical reactions, deformation, heat flow, etc) are not always linear, i.e. the second half of the process won't complete as quickly as the first. In general there is no way to predict what the kinetics of a process will be ahead of actually observing the process occur and measuring either the rate of creation of "products" (in this case, radioactive particles or radiation) or else the rate of disappearance of "reactants". So let's look at the observed facts on radioactivity first.

The kinetics of irreversible processes (chemical reactions, deformation, heat flow, etc) are not always linear, i.e. the second half of the process won't complete as quickly as the first. In general there is no way to predict what the kinetics of a process will be ahead of actually observing the process occur and measuring either the rate of creation of "products" (in this case, radioactive particles or radiation) or else the rate of disappearance of "reactants" (here, radioactive nuclei). So let's look at the observed facts on radioactivity first.

Let's look at the observed facts on radioactivity first.

The kinetics of irreversible processes (chemical reactions, deformation, heat flow, etc) are not always linear, i.e. the second half of the process won't complete as quickly as the first. In general there is no way to predict what the kinetics of a process will be ahead of actually observing the process occur and measuring either the rate of creation of "products" (in this case, radioactive particles or radiation) or else the rate of disappearance of "reactants". So let's look at the observed facts on radioactivity first.