The rate equation for the hydrolysis of sucrose to fructose...

The rate equation for the hydrolysis of sucrose to fructose and glucose $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})$$ is " $-\Delta[\text { sucrose }] / \Delta t=k\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right] .$ After $2.57 \mathrm{h}$ at $27^{\circ} \mathrm{C}$ the sucrose concentration decreased from $0.0146 \mathrm{M}$ to $0.0132 \mathrm{M} .$ Find the rate constant, $k.$

The rate equation for the hydrolysis of sucrose to fructose and glucose
$$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})$$
is " $-\Delta[\text { sucrose }] / \Delta t=k\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right] .$ After $2.57 \mathrm{h}$ at $27^{\circ} \mathrm{C}$ the sucrose concentration decreased from $0.0146 \mathrm{M}$ to $0.0132 \mathrm{M} .$ Find the rate constant, $k.$

Key Concepts

Integrated Rate Law

The integrated rate law for a first-order reaction provides a relationship between reactant concentration and time, typically expressed as ln([A]?/[A]) = kt. This equation is derived by integrating the differential rate equation and is used to calculate the rate constant or predict concentrations at any given time. It is a key tool in kinetics for analyzing experimental data and understanding the time evolution of a reaction.

First-Order Reaction Kinetics

First-order reaction kinetics describe reactions where the rate is directly proportional to the concentration of a single reactant. This implies that the rate of change in concentration follows an exponential decay. First-order reactions are characterized by a constant half-life that is independent of the initial concentration, making them simpler to analyze mathematically compared to reactions of higher order.

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates at which chemical processes occur and the mechanisms by which they proceed. It involves analyzing how different factors such as concentration, temperature, and catalysts affect the reaction rate, and it provides the framework to understand and model the dynamic behavior of reacting systems over time.

Rate Constant

The rate constant, denoted as k, is a fundamental parameter in rate equations that quantifies the speed of a chemical reaction under given conditions. It links the concentration of reactants to the rate of reaction and is specific to each reaction and its environment. The value of the rate constant is crucial for predicting how fast a reaction approaches completion and for comparing the kinetics of different reactions.

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