The hydrolysis of sucrose: Sucrose +H2 O ⟶glucose + frutose takes the following time course.

step 1 We are given the hydrolysis reaction of sucrose that creates glucose and fructose. We are also g

The hydrolysis of sucrose: Sucrose +H2 O ⟶glucose + frutose...

The hydrolysis of sucrose: Sucrose $+\mathrm{H}_{2} \mathrm{O} \longrightarrow$ glucose $+$ frutose takes the following time course. $$\begin{array}{cc} \text { Time (min) } & \text { [Sucrose] }(M) \\ \hline 0 & 0.5011 \\ 30 & 0.4511 \\ 60 & 0.4038 \\ 90 & 0.3626 \\ 130 & 0.3148 \\ 180 & 0.2674 \end{array}$$ Determine the first-order rate constant and the half-life of the reaction. Why does this bimolecular reaction follow a first-order rate law? How long will it take to hydrolyze $99 \%$ of the sucrose initially present? How long will it take if the amount of sucrose initially present is twice that given in the table?

The hydrolysis of sucrose: Sucrose $+\mathrm{H}_{2} \mathrm{O} \longrightarrow$ glucose $+$ frutose takes the following time course. $$\begin{array}{cc}
\text { Time (min) } & \text { [Sucrose] }(M) \\
\hline 0 & 0.5011 \\
30 & 0.4511 \\
60 & 0.4038 \\
90 & 0.3626 \\
130 & 0.3148 \\
180 & 0.2674
\end{array}$$
Determine the first-order rate constant and the half-life of the reaction. Why does this bimolecular reaction follow a first-order rate law? How long will it take to hydrolyze $99 \%$ of the sucrose initially present? How long will it take if the amount of sucrose initially present is twice that given in the table?

Key Concepts

First-Order Kinetics

In first-order kinetics, the rate of reaction depends linearly on the concentration of one reactant. This relationship is typically expressed using the integrated rate law, which shows that the natural logarithm of the concentration decreases linearly with time. This simplifies data analysis and allows for straightforward determination of the rate constant from a plot of ln[concentration] versus time.

Pseudo-First Order Reactions

A bimolecular reaction can appear to follow a first-order rate law when one of the reactants is present in large excess. Under these conditions, the concentration of the excess reactant remains nearly constant throughout the reaction, effectively reducing the reaction to a pseudo-first order process with respect to the other reactant. This simplification allows the reaction kinetics to be analyzed using first-order methods.

Integrated Rate Law for First-Order Reactions

The integrated rate law for a first-order reaction is given by ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, k is the rate constant, and t is the time elapsed. This law provides a direct way to determine the rate constant by rearranging the equation and linearizing experimental data.

Half-Life in First-Order Reactions

The half-life of a reaction, defined as the time required for half of the reactant to be consumed, is constant for first-order reactions and is given by t1/2 = ln(2)/k. This constant half-life, independent of the initial concentration, is a key characteristic of first-order kinetics.

Time Calculation for Specific Conversion

Using the integrated first-order rate law, it is possible to calculate the time required to achieve a specific extent of reaction, such as the time needed to hydrolyze 99% of the reactant. By substituting the fraction of reactant remaining into the integrated rate law, one can determine the elapsed time corresponding to that level of conversion.

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