Question

The rate constant of first-order reaction is \(3 \times 10^{-6}\) per second. The initial concentration is \(0.10 \mathrm{M}\). The initial rate is (a) \(3 \times 10^{-7} \mathrm{Ms}^{-1}\) (b) \(3 \times 10^{-4} \mathrm{Ms}^{-1}\) (c) \(3 \times 10^{-5} \mathrm{Ms}^{-1}\) (d) \(3 \times 10^{-6} \mathrm{Ms}^{-1}\)

Short answer

The initial rate is (a) \(3 \times 10^{-7} \mathrm{Ms}^{-1}\).

Step-by-step solution

Understand the Reaction Order and Formula

We are dealing with a first-order reaction, which means the rate depends linearly on the concentration of the reactant. The general rate equation for a first-order reaction is given by: \[ \text{Rate} = k \times [A] \]where \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.

Identify Given Values

From the problem statement, we know:- The rate constant \( k = 3 \times 10^{-6} \mathrm{s}^{-1} \).- The initial concentration \( [A] = 0.10 \mathrm{M} \).

Substitute Values into the Rate Equation

Now, we substitute the known values into the first-order rate equation:\[ \text{Rate} = (3 \times 10^{-6} \mathrm{s}^{-1}) \times (0.10 \mathrm{M}) \]

Perform the Calculation

Multiply the rate constant by the concentration:\[ \text{Rate} = 3 \times 10^{-6} \times 0.10 = 3 \times 10^{-7} \mathrm{M} \mathrm{s}^{-1} \]

Match Your Answer with Given Options

Compare the calculated rate with the options provided:(a) \(3 \times 10^{-7} \mathrm{Ms}^{-1}\) (b) \(3 \times 10^{-4} \mathrm{Ms}^{-1}\) (c) \(3 \times 10^{-5} \mathrm{Ms}^{-1}\) (d) \(3 \times 10^{-6} \mathrm{Ms}^{-1}\)The correct answer is (a) \(3 \times 10^{-7} \mathrm{Ms}^{-1}\).

Key concepts

First-order reaction

In chemical kinetics, a first-order reaction is quite straightforward. It is a type of reaction where the rate depends on the concentration of a single reactant. This means that if you double the concentration of the reactant, the rate of the reaction will also double. First-order reactions are common in many natural processes. For example, radioactive decay and some biological reactions often follow first-order kinetics. To mathematically express the rate of a first-order reaction, we use the equation:

• Rate = k × [A]

where:

• k is the rate constant.
• [A] is the concentration of the reactant.

Understanding the order of a reaction helps predict how a reaction will behave under various conditions, which is crucial for both practical applications and theoretical understandings of chemical kinetics.

Rate constant

The rate constant, often represented as k, is a critical factor in the rate law equation. It provides a direct measure of how fast a reaction takes place. The units of the rate constant differ based on the order of the reaction. For a first-order reaction, the unit is typically per second i.e.

• s^-1

The magnitude of the rate constant can indicate the speed of the reaction. A larger rate constant means the reaction occurs faster, while a smaller rate constant means it is slower. Factors such as temperature can affect the rate constant. Generally, as temperature increases, the rate constant and hence the reaction rate also increase. This highlights the importance of the rate constant in understanding and controlling reaction conditions.

Initial concentration

Initial concentration refers to the concentration of a reactant at the beginning of a reaction. In the context of a first-order reaction, this initial concentration plays a significant role in determining the rate of the reaction, since the rate is directly proportional to the concentration of the reactant. In calculations, the concentration is often expressed in molarity (M), which is moles per liter. Understanding the starting concentration helps predict how much reactant will remain after a certain period and how long it will take for the reaction to reach completion. When working through problems involving reaction rates, knowing the initial concentration allows us to confidently use the rate law formula to find how fast the reaction proceeds right from the start.

Reaction rate calculation

To calculate the reaction rate, especially in a first-order reaction, you should substitute the values you have into the rate formula. Here’s a simple step-by-step guide:

• Start with the equation: Rate = k × [A]
• Identify your values. In this case:
  • k is the rate constant: 3 × 10^-6 s^-1
  • [A] (initial concentration) is 0.10 M
• Substitute these values into the equation and calculate: Rate = (3 × 10^-6 s^-1) × (0.10 M)
• Do the math, and you get: Rate = 3 × 10^-7 M s^-1

This straightforward calculation shows how fast a first-order reaction occurs given the rate constant and initial concentration. The calculated rate tells us the speed at which the reactant is converting into products at the start of the reaction.