Question

In Lindemann theory of unimolecular reactions, it is shown that the apparent rate constant for such a reaction is \(k_{\text {app }}\) \(=\frac{k_{1} C}{1+\alpha C}\), where \(C\) is the concentration of the reactant, \(k_{1}\) and a are constants. The value of \(C\) for which \(k_{\text {app }}\) has \(90 \%\) of its limiting value at \(C\) tending to infinitely large is \(\left(\alpha=9 \times 10^{5}\right)\) (a) \(10^{-6}\) mole/litre (b) \(10^{-4}\) mole/litre (c) \(10^{-5}\) mole/litre (d) \(5 \times 10^{-5}\) mole/litre

Short answer

\(10^{-5}\) mole/litre

Step-by-step solution

Understanding the Problem

We need to find the concentration of the reactant, C, at which the apparent rate constant, \(k_{\text{app}}\), is 90% of its maximum value. We know that as C tends towards infinity, \(\alpha C\) will be much greater than 1, making \(\frac{1}{1 + \alpha C}\) tend towards 0, and hence \(k_{\text{app}}\) tends to its limiting value of \(k_{1}\). We are given \(\alpha = 9 \times 10^5\).

Expressing the Limiting Value

Write the limiting value of \(k_{\text{app}}\) as C tends to infinity. Since \(1 + \alpha C\) tends towards infinity as well, the fraction \(\frac{1}{1 + \alpha C}\) will tend towards 0, making the limiting value of \(k_{\text{app}}\) equal to \(k_{1}\).

Setting up the Equation for 90% of the Limiting Value

Since we're looking for the concentration C where \(k_{\text{app}}\) is 90% of its limiting value, we write the equation \(\frac{k_{1} C}{1 + \alpha C} = 0.9k_{1}\). Now, we can cancel \(k_{1}\) from both sides as it's a non-zero constant.

Solving for C

After cancelling \(k_{1}\) our equation becomes \(\frac{C}{1 + \alpha C} = 0.9\). Now, multiply both sides by \(1 + \alpha C\) to get \(C = 0.9 + 0.9\alpha C\). Move all terms involving C to one side to get \(C - 0.9\alpha C = 0.9\).

Simplify and Solve for C

Factor out C from the left side of the equation to get \(C(1 - 0.9\alpha) = 0.9\). Inserting the value for \(\alpha\) we have \(C(1 - 0.9 \times 9 \times 10^{5}) = 0.9\). Solving for C gives \(C = \frac{0.9}{1 - 0.9 \times 9 \times 10^{5}}\). Calculating that gives \(C = 10^{-5}\) mole/litre.

Key concepts

Rate Constant in Unimolecular Reactions

In the realm of chemical kinetics, the rate constant is a critical factor which governs the speed at which a chemical reaction proceeds. Specifically, in the context of unimolecular reactions described by the Lindemann theory, the rate constant (\( k_{1} \) in the given equation) is a measure of the probability of the reaction's progress as one molecule undergoes transformation into products without the necessity of colliding with another molecule.

Understanding this concept is pivotal because the rate constant encapsulates the essence of the reaction's intrinsic kinetics under specific conditions, like temperature and pressure. A noteworthy point here is that the rate constant is independent of the reactant concentration, C, which is a separate factor impacting the reaction rate. This independence allows us to make predictions about the behavior of the reaction over a range of concentrations. When we examine the equation \( k_{\text{app}} = \frac{k_{1} C}{1 + \alpha C} \), it becomes evident that the apparent rate constant, \( k_{\int{app}} \), varies with C and approaches the true rate constant \( k_{1} \) as the concentration becomes extremely high.

Understanding Reaction Concentration in Kinetics

Reaction concentration plays a versatile role in impacting the rate of a chemical reaction. In the Lindemann theory, the concentration of the reactant, denoted as C, serves as a variable in determining the apparent rate constant \( k_{\text{app}} \). At lower concentrations, the value of \( k_{\text{app}} \) changes significantly with small variations in C, implying that the reaction is sensitive to concentration changes.

As we raise the concentration, however, we start reaching a point where changes in the apparent rate constant become less pronounced, and it trends towards a limiting value. This behavior is a result of the mechanism postulated by the Lindemann theory, where the initial excitation of a molecule is followed by a step that releases the stored energy, leading to product formation. This theory suggests that at sufficiently high concentrations, the limiting step is not the reaction itself but the dissipation of the energy acquired by the molecule during its excited state.

The Limiting Value in Lindemann's Theory

In the scope of chemical kinetics, the limiting value of the rate constant is essentially the maximum rate at which a unimolecular reaction can occur. According to the equation provided, \( k_{\text{app}} = \frac{k_{1} C}{1 + \alpha C} \), the limiting value is achieved when the concentration, C, tends toward infinity, simplifying this equation to the value of \( k_{1} \), when the denominator becomes overwhelmingly large in comparison to 1.

In our exercise, we were tasked to find the concentration at which \( k_{\text{app}} \) is 90% of this limiting value. The significance of the '90%' condition is that it helps us determine the concentration where the reaction rate has effectively reached near-maximum efficiency, heralding the onset of reaction conditions under which further increases in concentration yield negligible gains in the reaction rate. This particular concentration is critical in practical applications where optimizing reactant concentration is necessary for efficiency and cost-effectiveness.