Question

The rate of the reaction: $$ \mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) $$ shows first-order characteristics-that is, rate \(=\) \(k\left[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\right]\) -even though this is a second-order reaction (first order in \(\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\) and first order in \(\mathrm{H}_{2} \mathrm{O}\) ). Explain.

Short answer

The reaction appears first-order because \(\mathrm{H}_{2} \mathrm{O}\) is in excess, making its concentration effectively constant.

Step-by-step solution

Define the Reaction Order

The reaction between acetic acid ethyl ester (\(\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\)) and water (\(\mathrm{H}_{2} \mathrm{O}\)) is described as a second-order overall reaction. This is because each reactant contributes a first-order dependence to the overall rate, leading to a rate law of the form \(\text{rate} = k[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}][\mathrm{H}_{2} \mathrm{O}]\).

Analyze Reaction Conditions

The first-order reaction rate observed is a simplified condition. It happens when one of the reactants is in excess, in this case, water \(\mathrm{H}_{2} \mathrm{O}\). Under such conditions, the concentration of water remains essentially constant because it is much higher than that of the ester.

Derive the Apparent Rate Law

Because the concentration of water \([\mathrm{H}_{2} \mathrm{O}]\) remains constant, it can be included in the rate constant; thus, the rate law modifies to \(\text{rate} = k'[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}]\), where \(k' = k[\mathrm{H}_{2} \mathrm{O}]\). This results in a first-order reaction with respect to \(\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\).

Conclusion

Therefore, even though the reaction is second-order overall with respect to both reactants, it exhibits first-order kinetics in experimental conditions where \(\mathrm{H}_{2} \mathrm{O}\) is in large excess. The rate seems first-order because the water's contribution is absorbed into a modified rate constant.

Key concepts

First-Order Reactions

In chemical kinetics, a first-order reaction is one where the rate of reaction is directly proportional to the concentration of a single reactant. This means that if you increase the concentration of the reactant, the reaction rate will increase proportionally. The mathematical representation of a first-order reaction is given by the rate law:\[\text{rate} = k[A]\]where \([A]\) is the concentration of the reactant and \(k\) is the rate constant. The unit of the rate constant \(k\) in a first-order reaction is \(\text{s}^{-1}\). The characteristic of first-order reactions is their exponential decay, which can be graphically represented by a straight line when plotting the natural logarithm of the concentration of reactant versus time.

• Proportionality: The rate depends on one reactant.
• Mathematically simple: Uses a straightforward linear equation.
• Exponential decay: Concentration halves in equal time periods.

First-order reactions are quite common and are typically found in various natural and industrial processes, such as radioactive decay and certain biochemical processes.

Second-Order Reactions

Second-order reactions involve the rate being proportional to either the square of the concentration of a single reactant or the product of the concentrations of two different reactants. This variability can make second-order reactions more complex. The general rate law for a second-order reaction that depends on two reactants can be written as:\[\text{rate} = k[A][B]\]Here, the unit for the rate constant \(k\) is often \(\text{M}^{-1}\text{s}^{-1}\), reflecting the dependence on two concentration terms. If a second-order reaction depends on a single reactant, the rate law is:\[\text{rate} = k[A]^2\]The concentration-time relationship for second-order reactions is nonlinear, and plotting \(1/[A]\) versus time yields a straight line.

• Includes interaction of two molecules or looks at second power if single.
• Complex behavior: More factors affecting rate.
• Graphically distinct: Linearity achieved through inverse concentration plot.

Second-order reactions often occur in reactions involving gases or in solution where two substrates must collide.

Rate Law

The Rate Law is a fundamental concept in reaction kinetics, allowing chemists to express the relationship between the reaction rate and the concentrations of reactants. Each reaction has a unique rate law that must be determined experimentally. The general form of a rate law is:\[\text{rate} = k[A]^m[B]^n\]where \(k\) is the rate constant, and \(m\) and \(n\) indicate the order of the reaction with respect to each reactant. These exponents are not simply related to the stoichiometric coefficients of the reaction but are determined based on actual experimental data.

• Unique for each reaction: Empirically determined.
• Includes rate constant \(k\): Essential for calculations.
• Determines kinetic behavior: Essential for understanding how reactions proceed.

Understanding the rate law is crucial for predicting how changes in concentration affect the reaction rate, making it an invaluable tool in industrial, laboratory, and academic settings. By identifying the rate law, chemists can optimize conditions to enhance reaction speed and efficiency.