Problem 1 - Chemistry in 30 days

Question

The half-life periods of a reaction at initial concentrations \(0.1 \mathrm{~mol} \mathrm{~L}^{-1}\) and \(0.5 \mathrm{~mol} \mathrm{~L}^{-1}\) are \(200 \mathrm{~s}\) and \(40 \mathrm{~s}\), respectively. The order of the reaction is (a) \(\underline{1}\) (b) \(1 / 2\) (c) 2 (d) 0

Step-by-Step Solution

Answer

The reaction is second order, option (c).

Step 1: Understanding the relationship between half-life and concentration

For reactions other than first order, the half-life is dependent on the initial concentration of the reactant. When we see that the half-life changes as concentration changes, it suggests the reaction order is not 1, as for first-order reactions, the half-life is constant and independent of initial concentration.

Step 2: Comparing the half-life changes between concentrations

We are given two different half-life periods for two different concentrations. At concentration of \( 0.1\,\mathrm{mol}\, \mathrm{L}^{-1} \), the half-life is \( 200\, \mathrm{s} \), and at \( 0.5\, \mathrm{mol}\, \mathrm{L}^{-1} \), the half-life is \( 40\, \mathrm{s} \). We notice the concentration increases five-fold while the half-life decreases five-fold.

Step 3: Determining order from relation of half-life and concentration

For a reaction where the half-life \( t_{1/2} \) is inversely proportional to the initial concentration raised to the \((\text{order} - 1)\) power. Mathematically, this can be represented as \( t_{1/2} \propto \frac{1}{[A]_0^{n-1}} \). If the order \( n \) is \( 2 \), then \( t_{1/2} \) is inversely proportional to the initial concentration, which we observe as consistent with our data, hence suggesting the reaction order \( n = 2 \).

Step 4: Conclude the order of the reaction

The half-life decreases proportionally as the concentration increases. Given this relationship, the order of the reaction is determined to be \( 2 \).

Key Concepts

Half-Life

The concept of half-life in chemistry refers to the time it takes for half of the reactant concentration to be consumed in a chemical reaction. It is an important metric as it gives insight into the speed and behavior of a reaction. When dealing with first-order reactions, the half-life is independent of the starting concentration and remains constant. This is why the half-life for any starting concentration of a first-order reaction will remain similar. However, for reactions that are not first-order, such as second-order reactions, the half-life can vary depending on the initial concentration of the reactants.

• Stable half-life: Indicative of a first-order reaction.
• Variable half-life: Suggests a reaction of order other than one.

Understanding this distinction is key in identifying the order of the reaction.

Reaction Kinetics

Reaction kinetics involves studying the speed of chemical reactions and the factors affecting them. Kinetics provides the framework to understand how quickly reactants turn into products. One important aspect of kinetics is the relationship between concentration and time. A chemical reaction’s rate can differ according to its order, and this rate is fundamentally connected to half-lives.
Reaction kinetics helps in predicting how long a reaction will take under specific conditions and how changes in conditions, like concentration, can affect it.

• Speed and mechanism: Analyzes the pathway and speed.
• Order dependence: Highlights whether the rate changes with concentration.

Through reaction kinetics, scientists can derive rate equations that are key to predicting the course of a reaction.

Chemical Concentration

Chemical concentration is the amount of a substance in a given volume. In reaction kinetics, concentration plays a crucial role as it directly influences the rate of reaction. The initial concentration of reactants can determine how quickly a reaction proceeds, and for reactions of order greater than one, it can also dictate how the half-life changes.
As observed in reaction kinetics, when the initial concentration increases, it can affect the half-life for non-first-order reactions, demonstrating a change in rate.

• Higher concentration can speed up reactions.
• Variable concentration impacts half-life in non-first-order reactions.

Concentration is thus not only central in defining reaction speed but also in comprehending the order of reactions.

First-Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. One of the distinct features of such reactions is their constant half-life regardless of the initial concentration, which makes their behavior easier to predict compared to higher-order reactions.
First-order reaction patterns can be seen in processes like radioactive decay and some simple biochemical processes.

• Rate equation: Rate = k[A], where [A] is the concentration.
• Predictable decay: Consistent half-life pattern.

When analyzing if a reaction is first-order, checking the consistency of half-life across varying concentrations provides a clear indication.