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In the realm of chemical reactions, understanding the speed or rate at which reactants transform into products is crucial. When we talk about first-order reaction kinetics, it's all about how the rate of reaction depends linearly on the concentration of one reactant. Imagine a simple scenario: If you have a bunch of A molecules turning into B molecules, the speed at which A disappears is directly related to how much A you have to start with.

In mathematical terms, the rate of a first-order reaction is expressed as \( Rate = k[A] \), where \( [A] \) represents the concentration of your reactant A, and \( k \) is the rate constant, unique to each reaction at a given temperature. The magical thing about first-order reactions is their half-life, a time interval in which the concentration of the reactant is halved, remains constant regardless of the starting concentration. This property is tremendously useful because it gives us a reliable measure to predict how long it will take for the reactant to diminish to a certain level, which in the context of our problem, allows us to estimate when only 10% of the original concentration of X would remain.

An elementary reaction is like a one-step dance—all the reactants come together in a single step to form the products without any intermediate phases. It's the simplest type of reaction on a molecular level, and it involves a direct change from reactants to products in a single collision or event. These reactions are important because their rate laws can be directly deduced from the stoichiometry of the reaction equation.

For instance, our reaction \( X(g) \rightarrow Y(g) + Z(g) \) suggests that one molecule of X is converted directly into molecules of Y and Z. The rate law for this first-order elementary reaction indicates that the rate at which X is consumed is directly proportional to its concentration, which we have already seen is key for determining how the reaction proceeds over time. Additionally, for an elementary first-order reaction, the concept of half-life becomes particularly straightforward, as it does not change even as the concentration of X decreases.

Considering a first-order reaction, the term concentration reduction refers to the decrease in the amount of the reactant over time. It's a bit like watching the number of cookies in a jar dwindle as people take them away—one moment, you have a full jar, and the next, only a few cookies left.

In the context of chemical kinetics, to calculate how long it will take for the concentration to drop to a certain percentage—let's say 10% of the original—we use the concept of half-life. With each passing half-life period, the concentration of our reactant halves. So, by calculating the number of half-lives required to reach the desired concentration (10% in our case), we can find out the total time needed for this reduction.

Calculating the Reduction Time

The equation \( (1/2)^n = 0.1 \) helps us determine the number of half-lives \( n \) it takes for the concentration of X to fall to 10%. Once we've figured out \( n \), multiplying by the half-life gives us the total time until only 10% of X remains. In our exercise, this concept helps us to jump from understanding the theoretical framework to solving a practical problem—finding the time required for the reactant to reduce to a specific percentage of its original concentration.