Problem 84 - Chemistry The Molecular Science

Question

Many biochemical reactions are catalyzed by acids. A typical mechanism consistent with the experimental results (in which HA is the acid and \(X\) is the reactant) is Step 1: reversible HA(aq) \(\rightleftharpoons \mathrm{H}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) \quad\) fast Step 2: reversible \(\mathrm{X}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{XH}^{+}(\mathrm{aq}) \quad\) fast Step 3: \(\mathrm{XH}^{+}(\mathrm{aq}) \longrightarrow\) products slow (a) Derive the rate law from this mechanism. (b) Determine the order of reaction with respect to HA. (c) Determine how doubling the concentration of HA would affect the rate of the reaction.

Step-by-Step Solution

Answer

(a) Rate law: \( \text{rate} = k[X][HA] \). (b) First order with respect to HA. (c) Doubling [HA] doubles the rate.

Step 1: Understand the Mechanism

The reaction mechanism provided includes three steps: a fast equilibrium dissociation of the acid (HA) into H⁺ and A⁻, followed by a fast equilibrium reaction between X and H⁺ to form XH⁺, and finally a slow conversion of XH⁺ into products. The slow step will determine the rate law.

Step 2: Identify the Rate-Determining Step

The rate of the overall reaction is determined by the slowest step, which is Step 3: \( \mathrm{XH}^{+} \rightarrow \text{products} \). The rate of this step can be expressed as: \( \text{rate} = k_{3}[XH^+] \).

Step 3: Express Intermediates in Terms of Reactants

Since [XH⁺] is involved in the rate-determining step, we express [XH⁺] using the fast pre-equilibria: - From Step 2, fast equilibrium \( \mathrm{X} + \mathrm{H}^+ \rightleftharpoons \mathrm{XH}^+ \) gives \( K_2 = \frac{[XH^+]}{[X][H^+]} \) or \( [XH^+] = K_2[X][H^+] \). - From Step 1, fast equilibrium \( \mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^- \) gives \( K_1 = \frac{[H^+][A^-]}{[HA]} \). Therefore, \( [H^+] = K_1 \frac{[HA]}{[A^-]} \). Substitute into the equation for [XH⁺].

Step 4: Substitute and Derive the Rate Law

Substituting the expression for [H⁺] from Step 1 into the expression for [XH⁺] from Step 3 gives: \([XH^+] = K_2[X]K_1\frac{[HA]}{[A^-]}\). The rate can then be expressed as: \(\text{rate} = k_3K_2K_1\frac{[HA][X]}{[A^-]}\). If assuming [A⁻]\ remains constant, the rate simplifies to \(\text{rate} = k[X][HA]\) because in a fast equilibrium, [A⁻] is usually constant.

Step 5: Determine Reaction Order with Respect to HA

Since the derived rate law \(\text{rate} = k[X][HA]\) shows direct proportionality to [HA], HA is first order.

Step 6: Effect of Doubling [HA]

If the concentration of HA is doubled, the rate of reaction will also double because the rate law is first order with respect to [HA]. This means \([HA]\) is directly proportional to the rate.

Key Concepts

Reaction Rate Laws

In biochemical reaction mechanisms, understanding how reactants turn into products is crucial. Reaction rate laws help describe this process mathematically and show the relationship between the rate of a chemical reaction and the concentration of its reactants. For the mechanism provided, the reaction rate law is determined by the slowest step, known as the rate-determining step.

• In our example, Step 3 is the slowest, and it determines the overall rate law.
• The rate can be expressed as: \( \text{rate} = k_3[XH^+] \).

To use this, we must express the concentration of the intermediate \([XH^+]\) in terms of the concentrations of the reactants. This involves substituting what we know from the fast, reversible reactions in the mechanism. Ultimately, we find a rate law that describes how the concentration of reactants, like HA and X, affect the speed of the reaction. With these relationships understood, we can make predictions about how changes in concentrations will affect the reaction rate.

Acid Catalysis

Acid catalysis occurs when an acid accelerates the rate of a biochemical reaction. In this process, the acid donates a proton (H⁺) to another substance, enhancing the reaction speed without being consumed by the reaction.

• The mechanism involves the acid (HA) dissociating to contribute H⁺.
• This H⁺ then combines with the reactant \(X\), forming \(XH^+\).

The catalyst (in this case, the acid) is crucial because it provides an alternative pathway with a lower activation energy for the reaction. This means that more molecules have enough energy to react when they collide. As a result, reactions that might otherwise happen slowly or not at all occur more quickly, enhancing the efficiency of the biochemical processes.

Reaction Order

Understanding the order of a reaction with respect to each reactant helps in assessing how concentration changes affect the overall reaction rate. For the given mechanism, we determine the reaction order from the derived rate law, which in this case is first order with respect to HA.

• The order of the reaction provides insight into how sensitive the rate is to concentration changes.
• If doubling the concentration of HA doubles the reaction rate, that indicates a first-order relationship.

Interestingly, reaction orders do not always match stoichiometry but instead reflect how the concentration of a substance affects the rate. By understanding reaction orders, we can make informed predictions about the reaction kinetics, enabling better control over chemical processes in experiments and industrial applications.