91Ó°ÊÓ

In chemical kinetics, the reaction rate constant is crucial for understanding how quickly a reaction proceeds. For first-order reactions, this constant (\( k \)) describes the proportionate rate at which the concentration of a reactant decreases over time. Imagine the reactant molecules rushing to react with each other; the rate constant helps us mathematically capture the speed of this process. The formula for the reaction rate in such reactions is:

• \( \frac{d[A]}{dt} = -k[A] \)

Here, \([A]\) is the concentration of the reactant, and \(k\) represents the rate constant. A helpful relation for students to remember is that the half-life equation for a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \). This formula tells us how long it takes for half of the reactant to decompose. At a given temperature, smaller values of \(k\) mean the reaction is slower because it takes longer for half of the substance to react. Conversely, a larger \(k\) indicates a faster reaction.

The term half-life refers to the time required for half of a given amount of a substance to decompose or react. In the context of first-order reactions, calculating the half-life helps predict how quickly a chemical species will diminish to half its original amount. This calculation is essential in reactions involving unstable compounds like hypofluorous acid (HOF). To determine the rate constant, we use the relationship:

• \( k = \frac{0.693}{t_{1/2}} \)

In the exercise given, the half-life of HOF is 30 minutes. Plugging this value into the formula, we calculate the rate constant:

• \( k = \frac{0.693}{30} \, \text{min}^{-1} \approx 0.0231 \, \text{min}^{-1} \)

This value allows us to predict how the concentration of HOF decreases over time, informing other calculations, such as the partial pressure after certain time intervals.

Partial pressure is a measure used to describe the pressure exerted by a single type of gas in a mixture of gases. In our exercise, hypofluorous acid (HOF) decomposes, altering its partial pressure in a sealed flask over time. Understanding how partial pressures change is essential when dealing with reactions that generate or consume gases.

The formula for first-order decay, used in calculating partial pressures, is given by:

• \( [A] = [A]_0 e^{-kt} \)

Where \([A]_0\) is the initial concentration and \(t\) is the time elapsed. By knowing the rate constant \(k\), you can calculate how the partial pressure changes. For example, after 30 minutes in the problem, the partial pressure of HOF changes from 100 mm Hg to approximately 50 mm Hg. Similarly, after 45 minutes, it reduces further to nearly 35.4 mm Hg.

Interestingly, even though individual gas pressures change, the total gas pressure in the flask remains 100 mm Hg. This constancy occurs because any decomposed HOF molecules convert into an equivalent number of HF and O2 molecules, maintaining the total number of moles and thus the total pressure.