Question

For an elementary reaction, \(X(g) \rightarrow Y(g)+Z(g)\) the half life period is \(10 \mathrm{~min}\). In what period of time would the concentration of \(X\) be reduced to \(10 \%\) of original concentration ? (a) \(20 \mathrm{Min}\). (b) \(33 \mathrm{Min}\) (c) \(15 \mathrm{Min}\) (d) \(25 \mathrm{Min}\)

Short answer

Approximately 33 minutes.

Step-by-step solution

Understanding the Concept of Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration. For an elementary reaction, which follows first-order kinetics, the half-life is constant and doesn't depend on the initial concentration.

Calculating the Number of Half-Lives

To find out the time it takes for the concentration of X to reduce to 10% of its original concentration, determine how many half-lives are needed to reach 10% remaining concentration. Since each half-life reduces the concentration by half, we need to solve the equation \(\frac{1}{2^n} = 0.1\), where \(n\) is the number of half-lives.

Solving for the Number of Half-Lives

Using the equation \(\frac{1}{2^n} = 0.1\), take the logarithm on both sides to find \(n\). The equation becomes \(\text{log}_{2}(0.1) = -n\). Calculate the value of \(n\).

Finding the Total Time

After calculating \(n\), multiply it by the period of the half-life to find the total time. Since the half-life is 10 minutes, the total time, \(T\), is given by \(T = 10 \times n\).

Selecting the Correct Answer

Based on the calculated time, select the closest matching answer from the offered options.

Key concepts

First-Order Reaction Kinetics

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.

Elementary Reaction

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.

Concentration Reduction in Reactions

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.