Formic acid decomposes at 550^∘ C according to the equation...

Formic acid decomposes at $550^{\circ} \mathrm{C}$ according to the equation $\mathrm{CH}_{3} \mathrm{NC}(\mathrm{g}) \rightarrow \mathrm{CH}_{3} \mathrm{CN}(\mathrm{g})$ The reaction follows first-order kinetics. In an experiment, it is determined that $75 \%$ of a sample of $\mathrm{HCO}_{2} \mathrm{H}$ has decomposed in 72 seconds. Determine $t_{1 / 4}$ for this reaction.

Key Concepts

First-Order Kinetics

This concept refers to reactions in which the rate depends linearly on the concentration of a single reactant. The differential rate equation for a first-order reaction shows that the rate of decomposition or conversion is directly proportional to the reactant’s current concentration. This relationship is essential for understanding and predicting how the system evolves over time.

Integrated Rate Law for First-Order Reactions

The integrated rate law for a first-order reaction expresses the concentration of a reactant as an exponential decay function of time. It is typically written as ln([A]?/[A]) = kt, which allows one to relate the time taken for a given fractional change in concentration directly to the rate constant. This equation is a fundamental tool for calculating reaction times such as half-lives or quarter-lives.

Rate Constant Determination

Determining the rate constant, k, is crucial for quantifying the speed of a reaction. In a first-order reaction, once a certain fraction of the reactant has been converted after a specific time, the rate constant can be calculated using the integrated rate law. This constant then facilitates the prediction of time scales for other extent of reactions under similar conditions.

Time Relationships in Reaction Kinetics

Time relationships, like those for half-life (t?/?) and quarter-life (t?/?), describe the durations required to reach specific conversion levels in a reaction. In first-order kinetics, these times are related logarithmically through the rate constant. Understanding these relationships helps in predicting how long it will take for a reaction to proceed to a certain point, which is particularly important in laboratory and industrial processes.

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