Question

The degradation of \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) (an \(\left.\mathrm{HFC}\right)\) by 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 \(8.1 \times 10^{5}\) and \(6.3 \times 10^{8}\) molecules \(\mathrm{cm}^{-3}\), respectively, what is the rate of reaction at this temperature in \(M /\) s?

Short answer

The rate of the reaction between CF3CH2F and OH radicals in the troposphere at 4°C is \(\boxed{2.24 \times 10^{-13} \mathrm{M/s}}\).

Step-by-step solution

Convert concentrations and constant to appropriate units

Initially, we need to convert the given concentrations and the rate constant to appropriate units. We are given the concentrations of OH and CF3CH2F are \(8.1 \times 10^{5}\) and \(6.3 \times 10^{8}\) molecules/cm³, respectively, and these need to be converted to moles/liter in order to match with the units of rate constant k. Conversion to moles/liter: 1 molecule = \(6.022 \times 10^{23}\) molecules/mol (Avogadro's number) 1 cm³ = 10⁻³ L For OH: \[\frac{8.1 \times 10^{5} \mathrm{\ molecules\ cm}^{-3}}{6.022 \times 10^{23} \mathrm{\ molecules/mol}} \times \frac{1 \mathrm{\ L}}{10^{-3} \mathrm{\ cm}^{3}} = 1.34 \times 10^{-12} \mathrm{M}\] For CF3CH2F: \[\frac{6.3 \times 10^{8} \mathrm{\ molecules\ cm}^{-3}}{6.022 \times 10^{23} \mathrm{\ molecules/mol}} \times \frac{1 \mathrm{\ L}}{10^{-3} \mathrm{\ cm}^{3}} = 1.05 \times 10^{-9} \mathrm{M}\]

Calculate the rate of reaction using the rate equation

Now that we have the concentrations in the appropriate units (M), we can use the first-order rate equation to calculate the rate of reaction: Rate = k[OH][CF3CH2F] Rate \(= 1.6 \times 10^{8} \mathrm{M}^{-1} \mathrm{s}^{-1} \times 1.34 \times 10^{-12} \mathrm{M} \times 1.05 \times 10^{-9} \mathrm{M}\) Rate \(= 2.24 \times 10^{-13} \mathrm{M/s}\)

State the rate of reaction

The rate of the reaction between CF3CH2F and OH radicals in the troposphere at 4°C is \(\boxed{2.24 \times 10^{-13} \mathrm{M/s}}\).

Key concepts

Understanding First-Order Reactions

When we talk about a first-order reaction, we're referring to a chemical reaction where the rate is directly proportional to the concentration of a single reactant. This means as the concentration of the reactant decreases, the rate of the reaction also goes down at a rate that directly corresponds to the concentration at that time. In mathematical terms, the rate law for a first-order reaction is expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Understanding the nature of first-order reactions is crucial because it helps predict how long a reaction will take and how conditions like concentration and temperature will affect the reaction's progress. First-order kinetics are often found in radioactive decay processes and simple enzymatic reactions in biochemistry.

Students often struggle with this concept because it involves exponential decay, a non-intuitive mathematical concept. It's important to visualize how the concentration of the reactant decreases over time in a non-linear fashion. This can be illustrated by plotting concentration vs. time on a graph, which yields a curve that shows a rapid decrease in concentration that slows down over time.

Tropospheric Chemical Reactions

The troposphere is the lowest layer of Earth's atmosphere, where most of our weather occurs and where tropospheric chemical reactions, including the degradation of compounds like Hydrofluorocarbons (HFCs), take place. These reactions are important because they influence air quality and climate change. In the given exercise, we look at the reaction between CF3CH2F, an HFC, and hydroxyl (OH) radicals.

The decomposition of HFCs, initiated by reactions with OH radicals, is a classic example of tropospheric chemistry. OH radicals often act as a 'detergent' of the atmosphere, breaking down potentially harmful compounds.

The conditions in the troposphere, such as temperature, pressure, and the presence of sunlight, can greatly affect the rate of these reactions. When explaining these concepts, it helps to discuss how OH radicals are formed through the interaction with water vapor and sunlight and how they help to 'cleanse' the atmosphere by breaking down pollutants.

The Rate Constant and Its Importance

At the core of chemical kinetics is the rate constant, symbolized as k. This value is a measure of the speed at which a chemical reaction proceeds. It's a fixed value for a given reaction at a certain temperature and does not change with the concentration of reactants.

In a first-order reaction, the rate constant tells us how quickly a reaction will occur when the reactant's concentration is at one molar. It’s essential for calculating the reaction rate and can suggest the sensitivity of the reaction rate to changes in temperature or catalysts. For example, an increase in temperature usually increases the value of the rate constant, speeding up the reaction.

Understanding the rate constant is vital for predicting reaction behavior over time. Teaching this concept might include activities like changing the rate constant in the rate law and observing how the rate of reaction changes, reinforcing the powerful influence of this seemingly simple number in chemical kinetics.