Problem 31 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A first-order reaction: \(\mathrm{A}(\mathrm{g}) \rightarrow n \mathrm{~B}(\mathrm{~g})\) is started with 'A'. The reaction takes place at constant temperature and pressure. If the initial pressure was \(P_{0}\) and the rate constant of reaction is ' \(K\), then at any time, \(t\), the total pressure of the reaction system will be (a) \(P_{0}\left[n+(1-n) e^{-k t}\right]\) (b) \(P_{0}(1-n) e^{-k t}\) (c) \(P_{0} \cdot n \cdot e^{-k t}\) (d) \(P_{0}\left[n-(1-n) e^{-k t}\right.\)

Step-by-Step Solution

Answer

The total pressure of the reaction system at any time \(t\) will be \(P_{0}[n+(1-n) e^{-k t}]\), which is option (a).

Step 1: Understand the First-Order Reaction

In a first-order reaction of the type \( \mathrm{A}(\mathrm{g}) \rightarrow n \mathrm{B}(\mathrm{g}) \), the rate of the reaction is directly proportional to the concentration of \(A\). Given that the reaction starts with compound \(A\) only, the pressure due to \(A\) at any time \(t\) will decrease exponentially with time as the reaction proceeds.

Step 2: Determine the Change in Pressure

Since the reaction occurs at constant temperature and pressure, the total number of moles of gas at any time \(t\) will correspond to the total pressure. At \(t=0\), the pressure is \(P_0\) due only to \(A\). As \(A\) reacts to form \(B\), for every mole of \(A\) that reacts, \(n\) moles of \(B\) are formed, increasing the total pressure by \(n-1\) times the decrease in pressure of \(A\).

Step 3: Calculate the Total Pressure at Any Time

The total pressure at any time \(t\) would be the initial pressure \(P_0\) plus the net change due to the reaction. Since the pressure due to \(A\) decreases to \((P_0e^{-kt})\), the increase in pressure due to the formation of \(B\) is \(n(P_0 - P_0e^{-kt})\). Thus, the total pressure \(P\) is given by the sum of the pressure of unreacted \(A\) and that due to \(B\), or \(P_0e^{-kt} + n(P_0 - P_0e^{-kt}) = P_0[1 + (n-1)(1 - e^{-kt})].\)

Step 4: Identify the Correct Expression

Upon rearranging the expression for the total pressure \(P\), we get \(P_0[n + (1-n)e^{-kt}]\). This matches with option (a).

Key Concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates at which chemical reactions occur and the factors that affect these rates. The rate of a chemical reaction is a measure of how quickly reactants are converted into products over time.

For instance, when dealing with a reaction such as \(\mathrm{A}(\mathrm{g}) \rightarrow n \mathrm{B}(\mathrm{g})\), kinetics provides insights into the speed of disappearance of \(A\) the reactant, in the gaseous state, and the formation of \(B\), the product. A key aspect of chemical kinetics is the understanding that the reaction rate is often dependent on the concentration of the reactants, which can change over time as reactants are consumed and products are formed.

Moreover, in chemical kinetics, we learn that reactions can be classified by their order. In a first-order reaction, the rate is directly proportional to the concentration of one reactant. This means, if the concentration of the reactant doubles, so does the rate of the reaction. This relationship can be understood and quantified by using mathematical expressions, which help chemists predict the behavior of a chemical reaction under various conditions.

Reaction Rate Constant

The reaction rate constant, often represented by the symbol \(k\), plays a central role in the equations that describe chemical kinetics. It is a quantitative measure of how quickly a reaction occurs under specific conditions, such as constant temperature and pressure.

In the case of a first-order reaction, the rate constant \(k\) defines the exponential decay of the concentration of the reactant over time. The relationship between the rate constant and the concentration of reactants can be described by the equation rate = \(k\)[\(A\)], where \([A]\) is the concentration of reactant \(A\). By integrating this expression over time, it provides a mathematical model that can predict the concentration of the reactant at any given point during the reaction.

An important feature of the rate constant is that it is determined experimentally and is unique to each reaction, dependent on various factors such as temperature and the presence of catalysts. This constant allows chemists and students alike to understand the timing of chemical reactions and is essential in the design of chemical processes where reaction speed is key.

Gas Phase Reactions

Gas phase reactions are chemical reactions that occur between substances in their gaseous state. Unlike reactions in solution, gaseous reactants interact based on partial pressures and volume, following the ideal gas law.

In our example problem, we explore a gas phase reaction where the reactant \(\mathrm{A}\) is transformed into product \(\mathrm{B}\), with \(n\) being the number of moles of \(\mathrm{B}\) produced per mole of \(\mathrm{A}\) consumed. In such reactions, the pressure changes provide valuable information about the progress of the reaction because they are directly related to the number of moles of gases involved.

As the reaction advances, the pressure contributed by reactant \(A\) decreases, which would typically result in a decrease in total pressure. However, if the reaction produces a greater number of moles of products than reactants, as is the case when \(n > 1\), then the increase in the number of gaseous molecules can lead to an increase in total pressure. Gas phase reactions therefore require close monitoring of pressure changes, as they are indicative of the reaction's progression, offering a practical method to apply chemical kinetics to real-world situations.