Problem 24 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The mechanism for the formation of a DNA double helix from two strands \(A \text { and } B \text { is as follows. (Section } 9.6)\) (a) Experiments show that the overall reaction is first order with respect to strand A and first order with respect to strand B. Write the equation for the overall reaction. (b) Write the rate equation for the overall reaction. (c) Assuming the processes involved are elementary reactions, derive an expression for the rate constant for the overall reaction in terms of the rate constants for the individual steps.

Step-by-Step Solution

Answer

(a) \( A + B \rightarrow \text{DNA double helix} \) (b) \( \text{Rate} = k [A][B] \) (c) \( k \) is the rate constant for the elementary reaction.

Step 1 - Write the Reaction Equation

Given the mechanism for the formation of a DNA double helix from two strands \( A \) and \( B \), the overall reaction can be represented by the equation: \[ A + B \rightarrow \text{DNA double helix} \] This represents the combination of two strands forming the double helix.

Step 2 - Formulate the Rate Equation

The reaction is first order with respect to strand \( A \) and first order with respect to strand \( B \). Therefore, the rate equation for the overall reaction can be given by: \[ \text{Rate} = k [A][B] \] where \( k \) is the rate constant for the reaction.

Step 3 - Derive the Rate Constant Expression

Assuming the processes involved are elementary reactions, for an elementary reaction \( A + B \rightarrow \text{DNA double helix} \), the rate constant \( k \) for the overall reaction directly represents the probability of \( A \) and \( B \) combining to form the product. Here, \( k \) is the intrinsic rate constant due to the reaction being elementary, requiring no further breakdown into individual steps.

Key Concepts

Rate Equation

In chemistry, the rate equation is a mathematical expression that relates the reaction rate to the concentration of the reactants. For an overall reaction between two strands, specifically strand \( A \) and strand \( B \), forming a DNA double helix, the rate equation is pivotal in predicting the speed of the reaction.
In our exercise, the rate equation is defined by the relationship: \[ \text{Rate} = k [A][B] \]where:

• \( \text{Rate} \) represents the speed of the reaction or how fast the DNA helix is formed.
• \( k \) is the rate constant, a factor that is intrinsic to the reaction and describes its likelihood under specific conditions.
• \([A] \) and \([B] \) are the concentrations of strands \( A \) and \( B \), respectively.

Here, the product of the concentrations reflects the multiplicative nature of the reaction rate concerning each reactant. Changes in either concentration can directly affect the rate, emphasizing the delicate and dynamic nature of chemical reactions.

Order of Reaction

The order of reaction describes how the rate is affected by the concentration of the reactants. In the provided exercise, we are told it is first order with respect to both strands \( A \) and \( B \).
This means:

• A reaction is first order with respect to \( A \) if doubling \([A]\) doubles the rate of reaction. Similarly, it is first order with respect to \( B \) if doubling \([B] \) doubles the rate.
• The overall reaction is second order because the orders of reaction for \( A \) and \( B \) are additive, resulting in a total reaction order of 1 + 1 = 2.

The order of reaction helps make predictions about how the concentration changes will impact the reaction's speed. It also plays a crucial role in determining the mechanism by which reactions occur, aiding in identifying potential steps involved in the transformation from reactants to products.

Elementary Reactions

Elementary reactions are individual steps in a reaction mechanism that occur in a single event at the molecular level. In the context of the formation of a DNA double helix, assuming that the reaction is elementary implies a direct interaction between molecules \( A \) and \( B \).
Characteristics of elementary reactions include:

• Simplicity, often involving a straightforward collision of molecules, atoms, or ions.
• The rate law for an elementary reaction can be derived directly from the stoichiometry, simplifying the understanding and calculation of reaction kinetics.
• The rate constant \( k \) for an elementary reaction reflects the probability and complexity of the molecular interactions needed to form the product.

The presumption that a process is elementary provides a simpler framework to comprehend its kinetics, as the rate constant directly corresponds to observable molecular behavior without needing further mechanistic breakdown.