in general a reaction is essentially over after 10 half-lives. prove that this generalization is reasonable
To prove that the generalization "a reaction is essentially over after 10 half-lives" is reasonable, we need to understand the concept of half-life and examine its implications.
The half-life of a reaction is defined as the time it takes for half of the reactant to undergo a chemical change or decay. In other words, after one half-life, only half of the original amount of reactant remains.
Now, let's consider what happens after each half-life:
1. After the first half-life, half of the reactant has transformed, and the remaining half remains unchanged.
2. After the second half-life, half of the remaining reactant from the first half-life has transformed, leaving only one-fourth (or 50% * 50% = 25%) of the original amount.
3. After the third half-life, half of the remaining reactant from the second half-life has transformed, leaving only one-eighth (or 50% * 50% * 50% = 12.5%) of the original amount.
4. This process continues, and after the tenth half-life, only a small fraction of the original reactant (approximately 0.1% or 1/1024) remains.
From this analysis, we observe that with each successive half-life, the remaining amount of reactant decreases exponentially. Therefore, after ten half-lives, the amount of reactant is reduced to less than 0.1% of the original quantity, which is generally considered negligible.
Based on this reasoning, it is reasonable to generalize that a reaction is essentially over after 10 half-lives. However, it is important to note that this generalization may not hold true for all reactions, particularly those with very long half-lives or complex reaction kinetics. Therefore, it is always advisable to consider the specific characteristics of a reaction when applying this generalization.