Problem 71 - General Chemistry: Principles and Modern Applications

Question

One proposed mechanism for the formation of a double helix in DNA is given by $$\left(S_{1}+S_{2}\right)=\left(S_{1}: S_{2}\right)^{*} \quad \text { (fast) }$$ $$\left(S_{1}: S_{2}\right)^{*} \longrightarrow S_{1}: S_{2} \quad \text { (slow) }$$ where \(S_{1}\) and \(S_{2}\) represent strand 1 and \(2,\) and \(\left(S_{1}: S_{2}\right)^{*}\) represents an unstable helix. Write the rate of reaction expression for the formation of the double helix.

Step-by-Step Solution

Answer

The rate of reaction expression for the formation of the DNA double helix is Rate = k[S1][S2].

Step 1: Identify the Rate Determining Step

The rate determining step is the slowest step in a reaction mechanism, which here is given as the second step: \( (S_{1}: S_{2})^{*} \longrightarrow S_{1}: S_{2} \). The rate of reaction would therefore be initially defined as, Rate = k[(S1: S2)*] where 'k' is the rate constant.

Step 2: Recognize and Substitute the Intermediate

Note that \((S_{1}: S_{2})^{*}\) is an intermediate for it appears in the mechanism but not in the overall reaction. By using the fast step, \( (S_{1}+S_{2})=\left(S_{1}: S_{2}\right)^{*} \), we can express \((S_{1}: S_{2})^{*}\) in terms of S1 and S2 since intermediates must not appear in the rate law. Thus, replace \((S_{1}: S_{2})^{*}\) with [S1][S2], assuming the fast step is at equilibrium.

Step 3: Write Rate Law

Substitute the intermediate into our rate law expression to obtain the final rate law: Rate = k[S1][S2]. This rate law indicates that the rate of helix formation is proportional to the concentration of strand S1 and strand S2.

Key Concepts

Rate Determining Step

Understanding the 'rate determining step' is crucial when studying chemical kinetics. It's akin to the slowest runner in a relay race — no matter how fast the other runners are, the overall speed of the team is limited by the slowest member. Similarly, in chemistry, the rate of the entire reaction is controlled by the slowest step in the reaction mechanism, often referred to as the rate determining step.

In the context of the given exercise, the DNA helix formation involves a fast step followed by a slower one. The slow step, the transformation of the unstable helix \( (S_{1}: S_{2})^{*} \) into a stable double helix \( S_{1}: S_{2} \) is the rate determining step. Therefore, when writing the rate expression for the reaction, we focus on the concentrations of the reactants participating in this step. It is important to realize that a reaction can comprise multiple steps, but the overall reaction rate is dictated by this single bottleneck — the rate determining step.

Reaction Intermediate

Another interesting aspect of reaction mechanisms is the 'reaction intermediate'. In the storyline of a reaction, intermediates are like characters that play a role in the plot but don't appear in the beginning or the end of the story. They are produced and then consumed during the course of the reaction sequence.

In the given DNA helix formation scenario, the unstable helix \( (S_{1}: S_{2})^{*} \) acts as a reaction intermediate. It is formed in the first, fast step and then used up in the second, slower step. When it comes to writing the rate expression, intermediates should not be included directly. Instead, we express them in terms of the reactants that form them. Here, the assumption that the fast step is at equilibrium allows us to substitute the intermediate \( (S_{1}: S_{2})^{*} \) with the concentrations of \( S_{1} \) and \( S_{2} \) — the reactants that combine to form it. This ensures our rate law includes only species that are present at the beginning of the reaction.

Rate Constant

Delving into the reaction kinetics, the 'rate constant' emerges as a pivotal figure. It is the factor that quantifies the speed of a reaction at a given temperature and is denoted by the symbol 'k'. Think of it as the chef's secret ingredient that influences the pace at which a recipe is prepared. In our exercise, the rate constant 'k' tells us how rapidly the DNA double helix is formed once the reactants \( S_{1} \) and \( S_{2} \) are combined.

The value of the rate constant depends on various factors, such as temperature and the presence of a catalyst, but it is independent of the concentration of reactants. When we write out the rate law, \( Rate = k[S_{1}][S_{2}] \), 'k' multiplies the concentration terms, signifying its direct impact on the rate. It's a unique value specific to each reaction under particular conditions, characterizing the intrinsic reactivity of the process.