Problem 29 - Chemistry in 30 days

Question

Consider the reaction, \(2 A+B \rightarrow\) Products. When concentration of \(B\) alone was doubled, ?he half life did not change. When the concentration of \(A\) alone was doubled, the rate increased by two times. The unit of rate constant for this reaction is (a) \(\mathrm{s}^{-1}\) (b) \(\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}\) (c) unitless (d) \(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The unit of the rate constant is \( \text{s}^{-1} \).

Step 1: Understanding Reaction Order

First, note that when the concentration of B is doubled and the half-life does not change, B is a zero-order reactant. For reactants of zero order, changes in concentration do not affect the rate of reaction.

Step 2: Analyzing Effect of Concentration of A

It is given that when the concentration of A is doubled, the rate doubles. This indicates that the reaction is first order with respect to A, since the rate is directly proportional to the concentration.

Step 3: Writing the Rate Law

Knowing that B is zero-order and A is first-order, the rate law can be expressed as: \[ \text{Rate} = k[A]^1[B]^0 = k[A] \] This shows that the overall order is 1 (from A).

Step 4: Determining Unit of Rate Constant

The unit of the rate constant (k) for a first-order reaction depends only on the reactant order. For a first-order reaction, the unit of the rate constant is \[ \text{s}^{-1}. \]

Step 5: Selecting the Correct Answer

From the provided options, the correct unit corresponding to a first-order reaction is (a) \( \text{s}^{-1} \).

Key Concepts

Rate Law

A rate law is a mathematical equation that describes the relationship between the concentration of reactants and the rate of a chemical reaction. It is written in the form:

• Rate = k[A]^m[B]^n[C]^p...

Here, "Rate" is the speed of the reaction, "k" represents the rate constant, and the exponents "m," "n," and "p" signify the orders of reaction with respect to each reactant. These orders are usually determined experimentally.
The overall reaction order is the sum of these exponents. This tells us how the reaction rate depends on the concentrations of all reactants combined.
The rate law helps chemists understand how changing conditions affect the speed of a reaction, which is useful for controlling and optimizing chemical processes.

First-Order Reaction

A first-order reaction is a type of reaction where the rate of reaction is directly proportional to the concentration of one reactant. In a first-order reaction, when you double the concentration of the reactant, the rate of the reaction will also double.
For example, if you have a reaction Rate = k[A], the rate is dependent on the concentration of A. Doubling the [A] will double the rate.

• This behavior is characteristic of first-order kinetics.
• The half-life of such reactions remains constant and does not depend on the initial concentration.

First-order reactions are common, especially in processes like radioactive decay and some enzymatic reactions. Understanding them is critical as it helps predict how long a reaction will take under various conditions.

Zero-Order Reaction

In contrast, a zero-order reaction is one where the rate of reaction is independent of the concentration of the reactant. This means that no matter how much you increase or decrease the concentration of the reactant, the rate will not change.

• The rate law can be expressed as Rate = k[B]^0 = k, indicating that the reaction rate is constant.
• Zero-order reactions are less common and usually occur under specific conditions like on surfaces in catalyzed reactions.

For such reactions, the concentration of the reactant decreases linearly with time. This is important in reactions where the amount of reactant is so large or its interaction with the catalyst defines the rate.

Rate Constant

The rate constant, represented by "k," is a crucial element of the rate law. It is a proportionality constant that relates the reaction rate to the concentrations of reactants. Each reaction has a specific rate constant, which can be influenced by factors such as temperature and presence of a catalyst.
For a first-order reaction, the unit of the rate constant is \[\text{s}^{-1}\]whereas for zero-order reactions, the unit could differ. The unit of "k" provides insight into the reaction order:

• First-order: \( \text{s}^{-1} \)
• Zero-order: \( \text{mol} \cdot \text{L}^{-1} \cdot \text{s}^{-1} \)

Understanding the rate constant allows chemists to compare speeds of reactions under different conditions and makes it easier to predict reaction behavior in various industrial and laboratory settings.