The half-life for the first-order decomposition of A is \(355 \mathrm{~s}\). How much time must elapse for the concentration of A to decrease to (a) \(\frac{1}{8}[\mathrm{~A}]_{0} ;\) (b) one-fourth of its initial concentration; (c) \(15 \%\) of its initial concentration; (d) one-ninth of its initial concentration?

Imagine you have a magic cube that disappears piece by piece over time. If someone tells you that its half-life is one hour, it means that in one hour, exactly half of your cube will have vanished. In the realm of chemical reactions, half-life is the period it takes for the concentration of a substance involved in a first-order reaction to be reduced by half its initial amount.

Understanding half-life is crucial because it is a constant value for a given reaction at a constant temperature and does not depend on the initial concentration of the reactant. Therefore, half-life provides a handy reference for predicting how long it will take for a substance to reach a certain level of decay, regardless of its starting concentration. This is especially important when dealing with pharmaceuticals, environmental pollutants, or any situation where the decay rate of a substance impacts health and safety.

Let's think of a first-order reaction as a group of marathon runners who all run at different speeds. In this case, the speed of our slowest runner dictates that of the whole group. In chemistry, a first-order reaction is one where the rate of the reaction is directly proportional to just one reactant's concentration.

Like our slowest runner, it's the concentration of this single reactant that sets the pace for the entire reaction. This proportionality to one reactant means that if you were to double the concentration of this reactant, the reaction rate would also double. First-order reactions are common in nature and are often found in processes like radioactive decay and simple enzyme reactions.

If you have a container of 100 ping pong balls and every minute you take away half of them, how many would you have after one minute? Fifty, right? Now what if you kept going, taking half of whatever's left each minute? Well, this is similar to how the concentration of a reactant decreases over time in many chemical reactions.

In a first-order reaction, the concentration of the reactant falls exponentially, which means that it drops quickly at first, then more and more slowly over time. The reaction 'forgets' its past, in a way, as the rate of decrease is always relative to the current concentration - the fewer reactant molecules there are, the slower they are used up.

Mathematics often seems like a secret language, but it helps us unlock the mysteries of the universe. Logarithms are one such magical key. In chemical kinetics, they allow us to relate the concentrations of reactants in a first-order reaction to time in a clear and concise way.

Through logarithms, we can create simple linear relationships from complex exponential decay patterns, enabling us to calculate how concentrations will change over time without having to tally each reactant molecule. It's like having a map that tells you exactly where you'll be and when, on a long winding road trip where distances don't always match the effort it takes to travel them.