Question

In a first order reaction the concentration of reactant decreases from \(800 \mathrm{~mol} / \mathrm{dm}^{3}\) to \(50 \mathrm{~mol} / \mathrm{dm}^{3}\) in \(2 \times\) \(10^{4} \mathrm{sec}\). The rate constant of reaction in \(\mathrm{sec}^{-1}\) is a. \(2 \times 10^{4}\) b. \(3.45 \times 10^{-5}\) c. \(1.386 \times 10^{-4}\) d. \(2 \times 10^{-4}\)

Short answer

The rate constant is \(1.386 \times 10^{-4} \, \mathrm{sec}^{-1}\), option (c).

Step-by-step solution

Understanding First Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The formula to calculate the rate constant (k) for a first-order reaction is given by the integrated rate law: \[ k = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \]\ where \([A]_0\) is the initial concentration, \([A]\) is the final concentration, and \(t\) is the time taken for the reaction.

Plug Values into the Formula

Given:- Initial concentration, \([A]_0 = 800 \, \mathrm{mol/dm}^3\)- Final concentration, \([A] = 50 \, \mathrm{mol/dm}^3\)- Time, \(t = 2 \times 10^4 \, \mathrm{sec}\)Substitute these values into the rate constant formula:\[ k = \frac{1}{2 \times 10^4} \ln \left( \frac{800}{50} \right) \]

Calculate the Natural Logarithm

Calculate \(\ln\left(\frac{800}{50}\right)\):First, simplify the fraction:\[ \frac{800}{50} = 16 \]Then find the natural logarithm:\[ \ln(16) \approx 2.7726 \]

Calculate the Rate Constant

Now calculate \(k\) using the natural logarithm found in Step 3:\[ k = \frac{1}{2 \times 10^4} \times 2.7726 \]Perform the multiplication:\[ k = \frac{2.7726}{20000} \approx 1.386 \times 10^{-4} \, \mathrm{sec}^{-1} \]

Determine the Correct Option

The calculated rate constant \(1.386 \times 10^{-4} \, \mathrm{sec}^{-1}\) corresponds to option \(\text{c.}\)

Key concepts

Rate Constant Calculation

In the context of a first-order reaction, calculating the rate constant, often denoted as "k," is an essential step. The rate constant describes how fast or slow a reaction occurs.
To find "k," we employ the integrated rate law, which is a mathematical expression derived from the relationship between the concentration of reactants and time.

The formula used is: - \[ k = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \]

• \([A]_0\) is the initial concentration of the reactant.
• \([A]\) is the concentration of the reactant at time \(t\).
• "t" is the elapsed time of the reaction.

To calculate "k," place the known values into this equation. This requires you to calculate the natural logarithm of the ratio \( \left( \frac{[A]_0}{[A]} \right) \) and divide by the total time \(t\). Converting these into numbers, you can easily determine the rate constant for the reaction in question.

This approach allows you to precisely understand the kinetics of reactions and predict how changes in conditions can alter the reaction speed.

Integrated Rate Law

The integrated rate law is a fundamental principle that helps us understand how the concentration of reactants in a chemical reaction changes over time. In first-order reactions, the concentration decreases at a rate proportional to its current value.
This means that as reactants get used up, the reaction rate decreases proportionally. This relationship makes calculation straightforward, especially when using the integrated form of the rate law.

For first-order reactions, this law is expressed as:- \[ ext{-ln}([A]) = kt + ext{ln}([A]_0) \]Rearranged to find the rate constant "k":- \[ k = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \]

• It's crucial to ensure that time \(t\) and concentrations \([A]_0\), \([A]\) are consistent.
• The logarithmic relationship allows for direct calculation of changes over time, making this method efficient for determining reaction speed.

Thus, the integrated rate law offers a valuable tool for analyzing first-order reactions, providing insights into both reaction dynamics and timescales.

Natural Logarithm

The natural logarithm, often represented as \( \ln\), plays a critical role in the mathematics of reaction kinetics, particularly when using the integrated rate law. It helps transform nonlinear relationships into linear ones, simplifying calculations.

In chemistry, the natural logarithm is used to express the ratio of initial to final concentrations logarithmically. This is essential for calculating the rate constant \( k \) in the context of the integrated rate law for first-order reactions.

• The natural logarithm \( \ln \) is based on the constant \( e \), which is approximately equal to 2.718.
• This transformation helps in directly relating concentration changes to time and reaction rate.

By using the natural logarithm, you can easily engage with exponential rate changes in reactions and make reliable predictions about the behavior of the chemical system over time. It's an indispensable tool for effectively analyzing and understanding the kinetics of reactions.