Problem 88 - Chemistry The Central Science In Si Units

Question

The degradation of \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) (an \(\mathrm{HFC}\) ) by OH radicals in the troposphere is first order in each reactant and has a rate constant of \(k=2.1 \times 10^{8} \mathrm{M}^{-1} \mathrm{~s}^{-1}\) at \(10^{\circ} \mathrm{C}\). If the tropospheric concentrations of \(\mathrm{OH}\) and \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) are \(1.0 \times 10^{12}\) and \(7.5 \times 10^{14}\) molecules \(/ \mathrm{m}^{3}\), respectively, what is the rate of reaction at this temperature in \(M / \mathrm{s} ?\)

Step-by-Step Solution

Answer

The rate of reaction is approximately \(4.35 \times 10^{-13} \text{ M/s}\).

Step 1: Identify Required Rate Law

The reaction rate for a reaction that is first order in each reactant is given by the rate law: \[ r = k [A][B] \] where *k* is the rate constant, \([A]\) is the concentration of \(\text{OH}\), and \([B]\) is the concentration of \(\mathrm{CF}_{3}\mathrm{CH}_{2}\mathrm{~F}\).

Step 2: Convert Units of Concentrations

First, convert the concentration of molecules per cubic meter to molarity (M). The conversion is necessary because the rate constant \(k\) is in \(\mathrm{M}^{-1}\mathrm{s}^{-1}\).Using Avogadro's number, \(6.022 \times 10^{23}\):\[ [OH] = \frac{1.0 \times 10^{12}\, \text{molecules/m}^3}{6.022 \times 10^{23}\, \text{molecules/mol}} = 1.66 \times 10^{-12}\, \text{M} \]\[ [\mathrm{CF}_{3}\mathrm{CH}_{2}\mathrm{~F}] = \frac{7.5 \times 10^{14}\, \text{molecules/m}^3}{6.022 \times 10^{23}\, \text{molecules/mol}} = 1.25 \times 10^{-9}\, \text{M} \]

Step 3: Substitute Values into the Rate Law

Substitute the values into the rate law:\[ r = (2.1 \times 10^{8} \, \mathrm{M}^{-1} \mathrm{s}^{-1})(1.66 \times 10^{-12} \, \mathrm{M})(1.25 \times 10^{-9} \, \mathrm{M}) \]

Step 4: Calculate the Rate of Reaction

Perform the calculation:\[ r = 2.1 \times 10^{8} \times 1.66 \times 10^{-12} \times 1.25 \times 10^{-9} \]\[ r \approx 4.35 \times 10^{-13} \, \mathrm{M/s} \]

Step 5: Interpret the Result

The rate of the reaction, when both reactants are at the given concentrations, is approximately \(4.35 \times 10^{-13} \, \text{M/s}\). This indicates the speed at which the concentration of reactants decreases and products form at this specific temperature and conditions in the troposphere.

Key Concepts

rate law

Understanding how fast a chemical reaction occurs can be crucial for many scientific applications. In the provided problem, the degradation of \( \mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F} \) involves a reaction between two reactants in the atmosphere. The rate law for such a process is an equation that expresses the reaction rate in terms of the concentration of reactants and a rate constant. This specific reaction is first order concerning each reactant, formulated by the equation: \[ r = k [A][B] \]Here, \(k\) is the rate constant, while \([A]\) and \([B]\) represent the concentration of the reactants \(\text{OH}\) and \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\), respectively.

• The rate law helps to predict how changes in concentrations affect the reaction rate.
• A first order reaction means if you double the concentration of either reactant, the rate of reaction doubles as well.

This simple but powerful equation is fundamental in understanding reaction kinetics and informs how conditions in the troposphere influence molecular degradation processes.

first order reactions

First order reactions are a special category in reaction kinetics where the rate of reaction is directly proportional to the concentration of one reactant. In our problem, both reactants are first order. Let's break down what this means:

• For a single reactant, the rate of reaction depends linearly on its concentration.
• For a two-reactant reaction like ours, each reactant has a separate first order dependency, leading to a combined effect on the rate.

These reactions feature a constant rate constant \(k\), indicating that the proportionality to concentration stays consistent across the reaction.In simple terms, the rate of degradation increases proportionally to the increase in either the \(\text{OH}\) or \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) concentrations and is understood through the way they combine linearly as expressed in the rate law. Understanding the order of reactions helps scientists in experimental planning and provides insights into mechanism predictions.

concentration conversion

Before applying the rate law, one step in solving the exercise is converting the given reaction concentrations into a usable form, which is molarity (M). This is crucial because the rate constant \(k\) is expressed in units of \(\text{M}^{-1}\text{s}^{-1}\). The original concentrations are given in molecules per cubic meter.To convert this, we use Avogadro's number, \(6.022 \times 10^{23}\), to translate molecule counts into a molar concentration:

• For \(\text{OH}\), we calculated \([OH] = 1.66 \times 10^{-12} \text{ M}\).
• For \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\), it is \([\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}] = 1.25 \times 10^{-9} \text{ M}\).

These conversions are essential for correctly applying the rate law to find the reaction rate. Always remember that accurate conversion of units is foundational in scientific calculations to ensure the consistency and validity of results.

molecular degradation

Molecular degradation in the atmosphere involves the breakdown of complex molecules into smaller fragments, often by interactions with radical species like \(\text{OH}\). These reactions are significant in atmospheric chemistry as they determine the lifespan and effects of chemical species, influencing environmental processes such as air quality and climate change.In our scenario, the reactants \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) and \(\text{OH}\) interact to degrade the compound effectively:

• The measured rate of reaction indicates how quickly this degradation occurs.
• This knowledge helps in assessing the atmospheric impact of halogenated compounds, commonly used in refrigerants and aerosol propellants.

By understanding the process of molecular degradation through reaction kinetics, we can better predict the environmental implications of different chemical species released into the atmosphere and devise strategies to mitigate harmful effects.