Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant of a first-order reaction is \(6 \times 10^{-3}\) \(\mathrm{s}^{-1} .\) If the initial concentration is \(0.10 \mathrm{M}\), the initial rate of reaction is (a) \(6 \times 10^{-3} \mathrm{Ms}^{-1}\) (b) \(6 \times 10^{-1} \mathrm{Ms}^{-1}\) (c) \(6 \times 10^{-6} \mathrm{Ms}^{-1}\) (d) \(6 \times 10^{-8} \mathrm{Ms}^{-1}\)

Step-by-Step Solution

Answer

None of the given options match the calculated initial rate of \(6 \times 10^{-4} \text{Ms}^{-1}\).

Step 1: Understand the Reaction Order Formula

For a first-order reaction, the rate of reaction can be calculated using the formula: \( \text{Rate} = k[A] \) where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.

Step 2: Substitute Known Values

We have the rate constant \( k = 6 \times 10^{-3} \ \text{s}^{-1} \) and the initial concentration \( [A] = 0.10 \ \text{M} \). Substitute these values into the formula: \[ \text{Rate} = (6 \times 10^{-3} \ \text{s}^{-1})(0.10 \ \text{M}) \]

Step 3: Calculate the Initial Rate

Perform the multiplication to find the initial rate: \[ \text{Rate} = 6 \times 10^{-4} \ \text{M s}^{-1} \]. This calculation stems from multiplying \( 6 \times 10^{-3} \) by \( 0.10 \) which results in \( 6 \times 10^{-4} \).

Step 4: Determine the Correct Option

The calculated rate \( 6 \times 10^{-4} \ \text{M s}^{-1} \) is not directly listed in the options. Thus, none of the provided options \( (a), (b), (c), \text{or} (d) \) match the calculated initial rate.

Key Concepts

Rate Constant

In chemical kinetics, the rate constant is an essential parameter that helps us understand how quickly a reaction proceeds. For a first-order reaction, the rate constant is represented by the symbol \( k \). It is crucial to note that the units of \( k \) for a first-order reaction are \( ext{s}^{-1} \), which means it is a measure of the rate per second. A larger \( k \) value indicates a faster reaction. Conversely, a smaller \( k \) value implies a slower reaction.

The calculation of reaction rates relies heavily on the rate constant. For first-order reactions, where the reaction rate depends linearly on the concentration of one reactant, knowing \( k \) helps predict how quickly the concentration of reactants will decrease over time. For instance, a rate constant of \( 6 \times 10^{-3} \ \text{s}^{-1} \) suggests that with the right concentrations, the reaction changes notably every second.

Understanding the nature and implications of the rate constant allows chemists to manipulate reaction conditions to optimize processes, either speeding them up or slowing them down based on practical requirements.

Initial Concentration

The initial concentration of a reactant is another significant factor in determining the reaction rate. It is denoted by \([A]\) in the rate formula \( \text{Rate} = k[A]\). For a first-order reaction, the process depends directly on the initial concentration of the reactant.

More specifically, a higher initial concentration usually leads to a higher initial reaction rate. This is because there are more molecules available to react per unit of time, increasing the frequency of effective collisions. This concept is clearly illustrated in the given problem, where the initial concentration is \(0.10 \ \text{M}\). The calculation of the initial reaction rate involves multiplying this concentration with the rate constant, showing its direct impact on how fast the reaction starts.

In practical terms, understanding how initial concentration affects reaction rate is vital in both laboratory and industrial settings. By adjusting the concentration of reactants, chemists can control the speed and extent of chemical reactions to achieve desired outcomes swiftly and efficiently.

Reaction Rate Formula

The reaction rate formula for a first-order reaction is a simple but powerful tool in chemistry. It is given by \( \text{Rate} = k[A]\), where \( k \) is the rate constant, and \([A]\) is the concentration of the reactant. This formula makes it straightforward to calculate the rate at which a first-order reaction proceeds.

To apply the formula, simply insert the known values: the rate constant \( k \) and the initial concentration of the reactant \([A]\). In the specific exercise discussed, substituting the rate constant \( 6 \times 10^{-3} \ \text{s}^{-1} \) and initial concentration \( 0.10 \ \text{M} \) into the formula yields: \\[ \text{Rate} = (6 \times 10^{-3} \ \text{s}^{-1})(0.10 \ \text{M}) = 6 \times 10^{-4} \ \text{M s}^{-1}\] Understanding this formula helps in predicting the behavior of chemical reactions, especially in forecasting how quickly products form or reactants get consumed.

In summary, the reaction rate formula is an essential step in chemical kinetics, providing insights into the dynamics of reactions and aiding in the design of experiments and industrial processes where reaction speed is a critical factor.