Which reaction involves the product or by-product of the reaction initiating further reactions?

The reaction that involves the product or by-product of the reaction initiating further reactions is a chain reaction. In a chain reaction, certain products formed during the initial reaction can cause additional reactions to occur, often at an exponential rate. This is particularly notable in nuclear reactions, where the fission of heavy nuclei releases neutrons that can then trigger the fission of other nuclei, sustaining the process and leading to a rapid release of energy.

This concept is critical in nuclear physics and is also observable in chemical reactions under specific conditions, such as polymerization processes, where the products formed can further react to produce more of the product. The nature of chain reactions is that they can escalate and potentially become uncontrollable, which has implications in both energy production and safety in nuclear reactors.

The other concepts mentioned do not involve this characteristic of initiating further reactions in the manner that chain reactions do. For example, half-life pertains to the time it takes for half of a radioactive substance to decay, while activity refers to the rate at which a sample of radioactive material decays. Nuclear fusion, while involving reactions, does not lead to subsequent reactions based on its products in the same self-sustaining manner as a chain reaction.