Problem 143 - A Complete Resource Book in Chemistry for JEE Main

Question

For the following reaction \(\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})\) The initial pressure was \(\mathrm{P}_{0}\) while pressure after time ' \(\mathrm{t}\) ' was \(\mathrm{P}_{t}\). The rate constant \(\mathrm{k}\) will be (a) \(\mathrm{k}=\frac{2.303}{\mathrm{t}} \log _{10} \frac{\mathrm{P}_{0}}{\mathrm{P}_{0}-\mathrm{P}_{1}}\) (b) \(\mathrm{k}=\frac{2.303}{\mathrm{t}} \log _{10} \frac{\mathrm{P}_{0}}{2 \mathrm{P}_{0}-\mathrm{P}_{1}}\) (c) \(\mathrm{k}=\frac{2.303}{\mathrm{t}} \log _{10} \frac{\mathrm{P}_{0}}{\mathrm{P}_{\mathrm{t}}}\) (d) \(\mathrm{k}=\frac{2.303}{\mathrm{t}} \log _{10} \frac{\mathrm{P}_{0}}{\mathrm{P}_{0}-2 \mathrm{P}_{\mathrm{t}}}\)

Step-by-Step Solution

Answer

The rate constant \( k \) is (b) \( \frac{2.303}{t} \log_{10} \frac{\mathrm{P}_0}{2\mathrm{P}_0 - \mathrm{P}_t} \).

Step 1: Understand the Reaction and Initial Information

The given reaction is \( \mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g}) \). Initially, only \( \mathrm{A} \) is present with pressure \( \mathrm{P}_0 \). The total pressure at any time \( t \) is \( \mathrm{P}_t \). Our task is to find the correct expression for the rate constant \( k \).

Step 2: Determine Reaction Order

The reaction \( \mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g}) \) involves decomposition of a single reactant without any stoichiometric coefficients on its decomposition. This suggests a first-order reaction.

Step 3: First-Order Kinetics Formula

For a first-order reaction of the form \( \mathrm{A} \rightarrow \mathrm{Products} \), the rate constant \( k \) can be expressed as \( k = \frac{2.303}{t} \log_{10} \frac{[\mathrm{A}]_0}{[\mathrm{A}]_t} \), where \([\mathrm{A}]_0\) is the initial concentration, and \([\mathrm{A}]_t\) is the concentration at time \( t \).

Step 4: Relate Pressure to Concentration

In gaseous reactions, pressure is directly proportional to concentration. Therefore, \([\mathrm{A}]_0\) becomes \( \mathrm{P}_0 \) and \([\mathrm{A}]_t\) becomes the partial pressure of \( \mathrm{A} \) at time \( t \), which is \( \mathrm{P}_0 - \Delta P \), where \( \Delta P \) is the change in pressure due to decomposition.

Step 5: Calculate Total Pressure Change

Since the reaction produces one molecule of \( \mathrm{B} \) and one molecule of \( \mathrm{C} \) per molecule of \( \mathrm{A} \), the total pressure increase is \( \Delta P = \mathrm{P}_t - \mathrm{P}_0 \). At time \( t \), the change in pressure is from the formed products, hence \( \Delta P = 2x \), where \( x \) is the pressure change from \( \mathrm{A} \) decomposition.

Step 6: Solve for \( k \) Using Pressure Values

For a first-order reaction in terms of pressure, \( [\mathrm{A}]_t = \mathrm{P}_0 - x \). The pressure of the products implies \( x = \mathrm{P}_t - \mathrm{P}_0 \), hence \( [\mathrm{A}]_t = 2\mathrm{P}_0 - \mathrm{P}_t \). Therefore, the correct expression for \( k \) is \( k = \frac{2.303}{t} \log_{10} \frac{\mathrm{P}_0}{2\mathrm{P}_0 - \mathrm{P}_t} \).

Key Concepts

First-order reactions

First-order reactions play a crucial role in chemical kinetics. These reactions are characterized by a rate that is directly proportional to the concentration of a single reactant. It means that as the concentration of the reactant decreases, the reaction rate slows down proportionally. This is a common type of reaction and includes processes such as radioactive decay and certain biochemical reactions.
For first-order reactions, the rate equation can be expressed mathematically as \( r = k[A] \), where \( r \) is the reaction rate, \( k \) is the rate constant, and \([A]\) is the concentration of the reactant. This simple relationship allows us to predict how the reaction progresses over time.
For a first-order reaction involving gaseous reactants, it is also common to use pressure instead of concentration because pressure and concentration are directly relatable in gases. When you follow the progress of a reaction by measuring pressure, applying first-order kinetic equations provides a robust understanding of how far and how fast your chemical processes are happening.

Rate constant calculation

Calculating the rate constant \(k\) in a first-order reaction is essential to understanding how quickly a reaction occurs. The rate constant is a unique value that indicates the speed of a specific reaction. Hence, it can vary with aspects like temperature and the presence of catalysts.
In the context of gaseous reactions, the rate constant can be calculated using pressures instead of concentrations. The general formula for a first-order reaction is

• \(k = \frac{2.303}{t} \log_{10} \frac{[A]_0}{[A]_t}\)

where \([A]_0\) is the initial concentration (or pressure) and \([A]_t\) is the concentration (or pressure) of the reactant at time \(t\).
This transformation into a logarithmic function allows for easy calculation by simplifying the often exponential decay of a reactant's concentration. Being able to calculate \(k\) empowers chemists to compare the rates of different reactions and optimize conditions to achieve desired reaction speeds.

Pressure and concentration relationship

In chemical reactions involving gases, it is vital to understand how pressure relates to concentration. This relationship is grounded in the ideal gas law, which asserts that pressure is proportional to concentration for a given volume and temperature.
Consider a reaction like \( A(g) \rightarrow B(g) + C(g) \). Initially, only \( A \) is present with a pressure \( P_0 \). As the reaction progresses, \( A \) decomposes to form \( B \) and \( C \), increasing the total pressure in the container.
The initial pressure \( P_0 \) represents the starting concentration of \( A \), while the pressure at any time \( P_t \) includes contributions from the products \( B \) and \( C \) as well. This total pressure change is directly related to the extent to which the reaction has progressed. Specifically,

• The change in pressure \( \Delta P \) is due to the formation of products.
• The pressure of \( A \) at time \( t \) reduces by amount \( \Delta P \).

Thus, pressure measurements become a practical way to monitor the progress of a reaction and calculate the rate constant \( k \) in terms of pressure changes.