Question

The degradation of \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) (an \(\mathrm{HFC}\) ) by \(\mathrm{OH}\) radicals in the troposphere is first order in each reactant and has a rate constant of \(k=1.6 \times 10^{8} \mathrm{M}^{-1} \mathrm{~s}{ }^{-1}\) at \(4^{\circ} \mathrm{C}\). If the tropospheric concentrations of \(\mathrm{OH}\) and \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) are \(8.1 \times 10^{3}\) and \(6.3 \times 10^{3}\) molecules/ \(/ \mathrm{cm}^{3}\), respectively, what is the rate of reaction at this temperature in \(M / s\) ?

Short answer

The rate of reaction at the given temperature and concentrations of the reactants is approximately \(2.25E-37 \, M/s\).

Step-by-step solution

Conversion of concentrations to mol/L (Molarity)

The given concentrations of reactants are in molecules/cm³. To convert them to mol/L (Molarity), we need to use Avogadro's number (6.022 x 10²³ molecules/mol) and the conversion factor 1L = 10⁶ cm³: \(C(OH) = \frac{8.1E3 \,molecules/cm^3}{(6.022E23 \, molecules/mol) (\frac{1}{1E6} \, L/cm^3)} \approx 1.346E-14 \, M\) \(C(CF_3CH_2F) = \frac{6.3E3 \,molecules/cm^3}{(6.022E23 \, molecules/mol) (\frac{1}{1E6} \, L/cm^3)} \approx 1.046E-14 \, M\)

Applying the rate law for first-order reaction

The rate law for the first-order reaction with two reactants is given by: Rate = \(k [OH][CF_{3}CH_{2}F]\) Plug in the values of k and the concentrations of OH and CF₃CH₂F: Rate = \((1.6E8 \, M^{-1}s^{-1}) (1.346E-14 \, M)(1.046E-14 \, M) \approx 2.25E-37 \, M/s\)

Reporting the rate of reaction

The rate of reaction at the given temperature and concentrations of the reactants is: Rate = \(2.25E-37 \, M/s\)

Key concepts

Rate Law

Understanding the rate law is crucial for predicting how the concentration of reactants will affect the speed of a chemical reaction. In essence, the rate law is an equation that relates the reaction rate to the concentrations of reactants. It takes the general form \begin{align*}\text{Rate} = k [A]^m[B]^n...\text{Rate} = k [A]^m[B]^n...\end{align*}where \(k\) is the rate constant, a measure of the inherent reactivity of the reaction, and \([A]\), \([B]\), ... represent the molar concentrations of the reactants raised to the powers of \(m, n, ...\), which are the reaction orders with respect to each reactant. These exponents can be either integers or fractions and are determined experimentally.
Reaction order is essential in determining the relationship between reactant concentration and reaction rate. For example, if a reaction is first order with respect to a certain reactant, doubling the concentration of that reactant will double the reaction rate. If it's second order, the rate would be quadrupled, and so on. For zero-order reactions, the rate is independent of the concentration of the reactant.
When expression \begin{align*}\text{Rate} = k [A]^m[B]^n...\end{align*}is applied with known concentrations and the rate constant, we can calculate the reaction rate, as seen in the original exercise. The conversion of reactants' concentrations into the appropriate units is key to correctly using the rate law.

First-Order Reaction

A first-order reaction is one where the rate of reaction is directly proportional to the concentration of a single reactant. This means that if the concentration of the reactant is doubled, the reaction rate also doubles. For a reaction that is first-order in reactant A, the rate law can be expressed as\begin{align*}\text{Rate} = k [A]\end{align*}In this kind of reaction, the unit of the rate constant \(k\) (as seen from the exercise) is \(s^{-1}\) or \(M^{-1}s^{-1}\), depending on whether the reaction involves one or more species. This reveals not just how swiftly the reaction proceeds but also provides insight into the reaction mechanism. The rate constant is affected by factors like temperature and the presence of a catalyst. In the given exercise, the reaction is first-order with respect to both \(OH\) radicals and \(CF_3CH_2F\) molecules, indicating that the rate is dependent on the concentration of each reactant separately, not their combined or overall concentration.

Reaction Rates

Reaction rate is a measure of how quickly the concentration of reactants decreases or the concentration of products increases over time. It can be described as the change in concentration of a reactant or product per unit time. Typically, it's measured in mol/L/s (Molarity per second). A higher reaction rate signifies a faster reaction. Factors that influence reaction rates include reactant concentration, temperature, surface area, and the presence of a catalyst.
In kinetic studies, the initial rates method often employs the initial concentrations of reactants to determine the order of reaction and the rate constant. The real-time monitoring of concentration changes requires sophisticated methods such as spectrophotometry or gas chromatography, depending on the nature of the reactants and products.
Understanding reaction rates is not only important for academic purposes but also for industrial applications, where the speed of reactions can impact the efficiency and cost-effectiveness of the production process. The calculation of reaction rates as seen in the solution provided, illustrates this importance and gives a glimpse into the practical aspects of studying chemical kinetics.