Problem 199 - A Complete Resource Book in Chemistry for JEE Main

Question

Units of rate constant for the first and zero-order reactions in terms of molarity \(\mathrm{M}\), units are respectively? (a) \(\mathrm{s}^{-1}, \mathrm{Ms}^{-1}\) (b) \(\mathrm{s}^{-1}, \mathrm{M}\) (c) \(\mathrm{M} \mathrm{s}^{-1}, \mathrm{~s}^{-1}\) (d) \(\mathrm{M}, \mathrm{s}^{-1}\)

Step-by-Step Solution

Answer

Option (a): \(\text{s}^{-1}\) for first-order and \(\text{Ms}^{-1}\) for zero-order reactions.

Step 1: Understanding Reaction Orders

The rate law for a zero-order reaction is expressed as \( \text{Rate} = k[A]^0 = k \), where \([A]\) is the concentration of the reactant, and \(k\) is the rate constant. The rate law for a first-order reaction is expressed as \( \text{Rate} = k[A]^1 \). The units of the rate constant \(k\) for each order depend on balancing the rate equation to the units of concentration per unit time, which is typically \( \text{M/s} \).

Step 2: Determining Units for Zero-Order Reaction

For a zero-order reaction, the rate is equal to the rate constant \(k\), meaning \( \text{Rate} = k \) with units \(\text{M/s} = \text{units of } k\). Thus, the units of \(k\) are \(\text{M/s}\). This is because the rate must have units of concentration change per unit time.

Step 3: Determining Units for First-Order Reaction

For a first-order reaction, the rate law \( \text{Rate} = k[A]^1 \) implies \(\text{M/s} = k \times \text{M}\). Solving for \(k\), we get \(k = \text{M/s} / \text{M} = \text{s}^{-1}\). Thus, the units of the rate constant \(k\) for a first-order reaction are \(\text{s}^{-1}\).

Step 4: Matching with Given Options

According to the calculations: for a zero-order reaction, we have \(k\) in \(\text{M/s}\), and for a first-order reaction, \(k\) in \(\text{s}^{-1}\). Comparing these with the provided options, option (a): \(\text{s}^{-1}, \text{Ms}^{-1}\) correctly matches as it implies for first-order \(\text{s}^{-1}\) and zero-order \(\text{Ms}^{-1}\) is actually the same as \(\text{M/s}\) once simplification occurs.

Key Concepts

Reaction Order

Understanding the order of a reaction is crucial as it tells us how the concentration of reactants affects the rate of the reaction. Reaction order refers to the power to which the concentration term of a reactant is raised in the rate law equation. This is derived from experimentally determined rate laws and not just from the stoichiometric coefficients of the chemical equation.

The overall reaction order is determined by summing up the powers in the rate law equation. This helps in understanding what happens to the rate as the concentration of reactants changes.

• It helps to predict how changes in concentration affect the reaction rate.
• It provides insight into the reaction mechanism.

For example, a rate law like \( \text{Rate} = k[A]^2[B]^1 \) means the reaction order is 3 (2 + 1). Knowing the order of a reaction helps chemists understand what factors affect the speed of the reaction.

Zero-Order Reaction

In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant. The rate law is expressed as \( \text{Rate} = k[A]^0 = k \). Here, the reactant's concentration doesn't appear in the expression because its power is zero.

The unit for the rate constant in a zero-order reaction is \( \text{M/s} \), indicating that the rate of change in concentration is constant.

• In practical terms, the reaction proceeds at a constant rate till the reactant is depleted.
• It is typically observed in processes like the catalyzed decomposition of a reactant or when the reactant is in large excess.

Zero-order kinetics is less common in chemical reactions but is quite important in biological and industrial processes where conditions control reaction rates more than concentrations do.

First-Order Reaction

First-order reactions depend linearly on just one reactant's concentration. The rate law is expressed as \( \text{Rate} = k[A]^1 \). This indicates that the rate of reaction is directly proportional to the concentration of a single reactant.

• These types of reactions are common in natural processes, like radioactive decay.
• If the concentration of the reactant decreases, the rate of reaction decreases proportionately.

The unit for the rate constant, \( k \), in a first-order reaction is \( \text{s}^{-1} \). This unit shows that the rate constant has the dimension of time inverse, which corresponds to how often the reaction occurs on a per-second basis.

First-order reactions are handy in kinetics studies as they provide a straightforward relationship between concentration and rate, facilitating predictions about the process dynamics over time.