Question

For the consecutive unimolecular-type first-order reaction: \(\mathrm{A} \stackrel{k_{1}}{\longrightarrow} \mathrm{R} \stackrel{k_{2}}{\longrightarrow} \mathrm{S}\), the concentration of component ' \(\mathrm{R}\) ', \(C_{\mathrm{R}}\), at any time, ' \(t\) ' is given by \(C_{\mathrm{R}}=C_{\mathrm{AO}} \cdot K_{1}\left[\frac{e^{-k_{1} t}}{\left(k_{2}-k_{1}\right)}+\frac{e^{-k_{2} t}}{\left(k_{1}-k_{2}\right)}\right]\) If \(C_{\mathrm{A}}=C_{\mathrm{AO}}, C_{\mathrm{R}}=C_{\mathrm{s}}=0\) at \(t=0\), the time at which the maximum concentration of 'R' occurs is (a) \(t_{\max }=\frac{k_{2}-k_{1}}{\ln \left(k_{2} / k_{1}\right)}\) (b) \(t_{\max }=\frac{\ln \left(k_{2} / k_{1}\right)}{k_{2}-k_{1}}\) (c) \(t_{\max }=\frac{e^{k_{2} / k_{1}}}{k_{2}-k_{1}}\) (d) \(t_{\max }=\frac{e^{k_{2}-k_{1}}}{k_{2}-k_{1}}\)

Short answer

The maximum concentration of 'R' occurs at the time given by option (a): \( t_{\max} = \frac{\ln\left(k_{2} / k_{1}\right)}{k_{2} - k_{1}} \).

Step-by-step solution

Understand the given function for concentration of R

The concentration of R as a function of time, represented by \( C_{\mathrm{R}} \), is given by the formula \( C_{\mathrm{R}} = C_{\mathrm{AO}} \cdot K_{1}\left[\frac{e^{-k_{1} t}}{\left(k_{2}-k_{1}\right)} + \frac{e^{-k_{2} t}}{\left(k_{1}-k_{2}\right)}\right] \). To find the maximum concentration of R, we need to analyze this function and find the time \( t \) at which the first derivative of \( C_{\mathrm{R}} \) with respect to \( t \) is zero.

Find the first derivative of \( C_{\mathrm{R}} \) with respect to time \( t \)

The first derivative of \( C_{\mathrm{R}} \) with respect to \( t \) is calculated by differentiating the given equation for \( C_{\mathrm{R}} \). The derivative will help us locate the time at which \( C_{\mathrm{R}} \) reaches its maximum value. Use the properties of the exponential function and the chain rule to differentiate each term separately.

Set the first derivative equal to zero to find critical points

To find when the concentration of R is at a maximum, set the first derivative from step 2 equal to zero. This will provide us with an equation that we can solve for \( t \).

Solve for \( t \)

The critical points obtained from setting the first derivative to zero give potential times at which \( C_{\mathrm{R}} \) could be maximized. Solve the equation for \( t \) to find the time \( t_{\max} \) at which the concentration of R is at its maximum.

Check if the found critical point gives a maximum

To confirm that the critical point found indeed corresponds to a maximum, we can use either the second derivative test or simply check that the first derivative changes sign around the critical point. If the first derivative goes from positive to negative, then the critical point is a maximum.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that focuses on the rates of chemical reactions and the factors that affect them. Understanding chemical kinetics is essential for predicting how quickly a chemical reaction will occur, which is crucial in various applications such as the synthesis of new materials, pharmaceuticals, and in environmental modeling.

The rate of a reaction is often expressed as a change in concentration of a reactant or product per unit time. Factors influencing reaction rates include temperature, pressure, concentration, and the presence of catalysts. In our specific case of a unimolecular first-order reaction, the reaction rate is proportional to the concentration of a single reactant oligomer.

Reaction Rate

The reaction rate is a measure of how quickly the concentration of a reactant or product in a chemical reaction changes over time. It is an essential aspect of chemical kinetics because it directly affects how long a reaction takes to reach completion or to reach a state of dynamic equilibrium.

In the context of a unimolecular first-order reaction, the reaction rate is directly proportional to the concentration of the reactant that is transforming into a product. Also, because it is a first-order reaction, its rate law can be expressed as rate = k[Reactant], where 'k' is the rate constant, and '[Reactant]' refers to the concentration of the starting chemical species.

Concentration-Time Relationship

In chemical kinetics, the concentration-time relationship describes how the concentration of reactants and products changes over time during a reaction. This is especially pertinent in a unimolecular first-order reaction where the rate of reaction depends on the concentration of the single reactant.

To express this relationship mathematically, we use rate equations that are derived from the laws of mass action. For example, in the given exercise, the concentration of the intermediate product 'R' is described by a complex expression involving exponential terms. This equation illustrates how 'R's concentration evolves as the reaction progresses and why at a certain point in time, labeled as 't_max', the concentration of 'R' reaches its maximum value due to the interplay between the rates of its production and consumption.