What is achieved when each isotope in a decay chain is decaying at the same rate?

When each isotope in a decay chain is decaying at the same rate, it signifies that a state of secular equilibrium has been reached. In secular equilibrium, the number of parent isotopes decaying per unit time matches the number of daughter isotopes decaying, despite the different half-lives of these isotopes. This means that the activity (or decay rate) of the parent and the daughter isotopes stabilizes over time, leading to a consistent level of radioactivity in the decay chain.

In contrast, decay equilibrium generally refers to situations where the rate of decay reaches a balance, but it does not specify that this balance is maintained indefinitely. Dynamic equilibrium involves systems that can change but still maintain overall balance, which does not directly correlate to radiation decay processes. Thermal equilibrium relates to temperature differences within a physical system and is not relevant in the context of decay chains.

Therefore, secular equilibrium specifically addresses the condition of continuous decay at the same rate for all isotopes in a decay chain, making it the correct term for the situation described.