Question

The first order reaction, \(2 \mathrm{~N}_{2} \mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{~N}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})\), has a rate constant equal to \(0.76 \mathrm{~s}^{-1^{-1}}\) at \(1000 \mathrm{~K}\). How long will it take for the concentration of \(\mathrm{N}_{2} \mathrm{O}\) to decrease to \(42 \%\) of its initial concentration? a. \(3.1 \mathrm{~s}\) b. \(0.18 \mathrm{~s}\) c. \(1.1 \mathrm{~s}\) d. \(2.4 \mathrm{~s}\)

Short answer

It will take approximately 1.1 seconds.

Step-by-step solution

Understanding the Reaction Type

First, identify that we are dealing with a first-order reaction. The reaction given is: \[2 \mathrm{~N}_{2} \mathrm{O} \rightarrow 2 \mathrm{~N}_{2} + \mathrm{O}_{2}\] The problem states it is a first-order reaction with respect to \(\mathrm{N}_{2} \mathrm{O}\), which means the rate depends linearly on the concentration of \(\mathrm{N}_{2} \mathrm{O}\).

Identify the First-Order Reaction Formula

For first-order reactions, the time it takes for the concentration to decrease to a certain percentage of its original value can be calculated using the formula: \[\ln \left( \frac{[A]_0}{[A]} \right) = kt\] where \(k\) is the rate constant, \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(t\) is the time.

Substitute Known Values into Equation

You're given that the concentration of \(\mathrm{N}_{2} \mathrm{O}\) decreases to 42% of its initial concentration. Therefore, \([A]_0/[A] = 1/0.42\). The rate constant \(k\) is given as \(0.76 \mathrm{~s}^{-1}\). Substitute these known values into the equation: \[\ln \left( \frac{1}{0.42} \right) = 0.76t\].

Solve for Time \(t\)

First, calculate the natural logarithm: \[\ln \left( \frac{1}{0.42} \right) = \ln(2.38) \approx 0.867\]. Next, solve for \(t\) by dividing both sides by \(0.76\): \[t = \frac{0.867}{0.76}\approx 1.14 \mathrm{~s}\].

Find Closest Value to Calculated Time in Options

The calculated time is approximately \(1.14 \) seconds. Review the provided options to find the closest value, which is option (c) 1.1 seconds.

Key concepts

Rate Constant

The rate constant, often denoted as \( k \), is a fundamental concept in the study of chemical kinetics. It provides the proportionality factor that connects the reaction rate to the concentrations of the reactants in a given reaction. In a first-order reaction, the rate constant has units of inverse time, \( s^{-1} \), indicating how quickly the reaction progresses per unit time.
The rate constant is unique to each reaction and depends largely on factors such as temperature and the presence of a catalyst. A higher rate constant indicates a faster reaction, while a lower one suggests a slower reaction. In our problem, the rate constant is \( 0.76 \; s^{-1} \), providing a quantitative measure of how the concentration of nitrous oxide changes over time at a given temperature of 1000 K.

Natural Logarithm

The natural logarithm, represented as \( \ln \), is a mathematical function often used in reaction kinetics to describe how reactant concentrations change over time. For a first-order reaction, the relationship between the concentration and time is typically expressed through the natural logarithm in the equation: \[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \]This equation shows how the concentration changes exponentially with respect to time, with the natural log creating a linear relationship when plotting \( \ln([A]) \) versus time.
Natural logarithms are particularly useful because they allow complex exponential relationships to be expressed in a form that is more straightforward to manipulate mathematically. In the given problem, we use the natural logarithm to quantify the decrease in concentration of \( \mathrm{N}_2\mathrm{O} \) from its initial value to 42% of the initial concentration, which simplifies the calculation of time elapsed.

Reaction Kinetics

Reaction kinetics is the area of chemistry that deals with the speed, or rate, of chemical reactions and factors that affect those rates. For a first-order reaction, like the one being considered, the rate depends on the concentration of one reactant only.
This dependency is reflected in the formula for first-order reactions, \( \ln ( \frac{[A]_0}{[A]} ) = kt \), providing insight into how reactant concentration changes over time. It helps predict how long a reaction will take to reach a certain point or how the reactants will behave over time, which is critical for applications ranging from industrial synthesis to pharmaceutical formulations.
By examining reaction kinetics, chemists can understand and utilize the mechanisms of reactions. This understanding enables the optimization of conditions to achieve desired reaction rates, which can enhance efficiency and effectiveness in chemical processes.

Concentration Change

Concentration change refers to the variation in the amount of a substance present in a chemical reaction over time. In a first-order reaction, this change is described by an exponential relationship, indicating that the concentration decreases over time according to the first-order kinetics equation.
In the provided example, we calculate how the concentration of \( \mathrm{N}_2\mathrm{O} \) decreases to 42% of its original concentration. This involves using the known rate constant and the properties of logarithms to solve for the time required for this concentration change to occur.
Understanding concentration change is vital because it helps determine the progress and extent of a reaction at any given time. By applying the first-order rate law, we can accurately predict when a specific concentration target will be met, a crucial factor in both academic study and practical applications.