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) \(6 \times 10^{-8} \mathrm{Ms}^{-1}\)

Short answer

The initial rate does not directly match any given option; there may be an error in options or units.

Step-by-step solution

Understand the Formula for Rate

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

Identify Given Values

We know the rate constant \( k = 6 \times 10^{-3} \, \text{s}^{-1} \) and the initial concentration \( [A] = 0.10 \, \text{M} \).

Substitute Values into Formula

Substitute the given values into the rate equation: \( \text{Rate} = (6 \times 10^{-3} \, \text{s}^{-1}) (0.10 \, \text{M}) \).

Calculate the Initial Rate

Calculate the result of the multiplication: \( \text{Rate} = 6 \times 10^{-4} \, \text{M} \cdot \text{s}^{-1} \).

Match with Given Options

The calculated initial rate \( 6 \times 10^{-4} \, \text{M} \cdot \text{s}^{-1} \) should be compared to the given options, but note that it does not match any directly. Check for possible errors in matching units or options.

Verify Correct Option

Re-examine the options carefully and ensure that there are no typographical errors. In case you find no errors, inform that none match exactly.

Key concepts

Rate Constant

The rate constant, often denoted as \(k\), is a fundamental aspect of chemical kinetics that describes how fast a reaction progresses. For a first-order reaction, this is defined as the rate at which the concentration of a reactant decreases over time.
The units for a rate constant in a first-order reaction are typically inverse seconds \(\text{s}^{-1}\). This reflects that the half-life of the reaction, or the time it takes for the reactant concentration to decrease by half, does not depend on the initial concentration.
The rate constant is an intrinsic property of the reaction and is influenced by factors such as temperature and the presence of a catalyst.
To calculate the initial rate of a simple first-order reaction, the rate constant is multiplied by the initial concentration of the reactant.

Initial Concentration

Initial concentration, signified as \([A]\), is the amount of reactant present at the very beginning of a reaction. For our example, it was given as \(0.10 \, \text{M}\).
This initial concentration will greatly affect how quickly the reaction appears to progress, especially in reactions that are not zero-order.
It is crucial in calculating the initial rate because it determines the starting condition from which the reaction starts to deplete the reactants.
The formula for the rate of a first-order reaction, \(\text{Rate} = k[A]\), highlights this by multiplying the rate constant \(k\) by the initial concentration \([A]\). This direct proportional relationship means that higher initial concentrations lead to a larger initial rate, assuming the rate constant remains unchanged.

Initial Rate of Reaction

The initial rate of reaction refers to the speed at which reactants are converted to products as the reaction begins. For a first-order reaction, this is calculated with the formula \(\text{Rate} = k[A]\), integrating both the rate constant and the initial concentration.
Our calculation showed an initial rate of \(6 \times 10^{-4} \, \text{M} \cdot \text{s}^{-1}\).
This initial rate gives a snapshot of the reaction speed and is useful in experimental comparisons. However, it is important to verify the calculated rate against given answer choices accurately.
Small discrepancies can occur due to unit conversions or transcription errors in multiple-choice options, so careful examination is essential when reconciling calculated results against listed options.