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 \(k=1.6 \times 10^{8} M^{-1} \mathrm{s}^{-1}\) 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/cm\(^{3},\) respectively, what is the rate of reaction at this temperature in \(M / \mathrm{s} ?\)

Short answer

The rate of reaction at \(4^{\circ} \mathrm{C}\) is approximately \(2.269 \times 10^{-36} M/s\).

Step-by-step solution

Convert concentrations from molecules/cm³ to Molarity (M)

To do this, we first need to use Avogadro's number, which is \(6.022 \times 10^{23}\) particles/mole. We will divide the given concentrations by Avogadro's number and then convert from cm³ to L (1 L = 1000 cm³): \[ [\mathrm{OH}] = \dfrac{8.1 \times 10^{5}}{\ (6.022 \times 10^{23})} \times \dfrac{1}{1000} M = 1.35 \times 10^{-15} M\] \[ [\mathrm{CF}_{3}\mathrm{CH}_{2}\mathrm{F}] = \dfrac{6.3 \times 10^{8}}{\ (6.022 \times 10^{23})} \times \dfrac{1}{1000} M = 1.05 \times 10^{-12} M\]

Use the rate law for the first-order reaction

The rate law for this first-order reaction involving \(\mathrm{OH}\) and \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{F}\) can be written as: \[ Rate = k \times [\mathrm{OH}] \times [\mathrm{CF}_{3}\mathrm{CH}_{2}\mathrm{F}]\] Substitute the given rate constant \(k = 1.6 \times 10^{8} M^{-1} \mathrm{s}^{-1}\), and the calculated concentrations in the rate equation: \[ Rate = \ (1.6 \times 10^{8} M^{-1} \mathrm{s}^{-1}) \times (1.35 \times 10^{-15} M) \times (1.05 \times 10^{-12} M) \]

Calculate the rate of reaction

Now, just multiply the values to get the rate of reaction: \[ Rate = 2.269 \times 10^{-36} M^2/s \] Since the units cancel each other out (\(M^{-1} \times M \times M = M^0\)), the units of the rate of reaction are in M/s: \[ Rate = 2.269 \times 10^{-36} M/s \] The rate of reaction at \(4^{\circ} \mathrm{C}\) is approximately \(2.269 \times 10^{-36} M/s\).

Key concepts

Understanding Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates of chemical reactions and the mechanisms by which they occur. In other words, it delves into how fast a reaction goes from start to finish, and what happens at a molecular level during the reaction.

One of the key concepts in kinetics is the reaction rate, which is a measure of how quickly the concentration of a reactant or product changes over time. A rate can either be expressed as an increase in the concentration of a product per unit of time, or a decrease in the concentration of a reactant per unit of time. Several factors can affect the reaction rate, including the concentration of the reactants, temperature, presence of a catalyst, and the surface area of solid reactants or catalysts.

In the exercise, we examine a reaction that is 'first order' in each reactant, which means the rate of reaction is directly proportional to the concentration of that one reactant. If we had multiple reactants, a different order (such as second-order) would mean the rate is proportional to the square of the concentrational change of one reactant or the product of the concentrations of two different reactants.

Rate Law and Its Role in Kinetics

Determining Reaction Order

Rate laws are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. For a first-order reaction, the rate law typically has the form:\[ Rate = k \times [A] \]where \(k\) is the rate constant, and \([A]\) represents the concentration of reactant \(A\). As seen in the solution, both \(\mathrm{OH}\) and \(\mathrm{CF}_3\mathrm{CH}_2\mathrm{F}\) are reactants in our rate law equation.

The rate constant \(k\) is a crucial factor in this equation because it can change with temperature and provides information about the reaction's speed under certain conditions; the higher the value of \(k\), the faster the reaction proceeds. It's also specific to each reaction and must be determined experimentally.

In our problem, after determining the concentrations of our reactants, we plug those values into the rate law to calculate the reaction rate. For students, understanding how to construct and utilize a rate law is critical for predicting the behavior of chemical reactions under varying conditions.

The Significance of Avogadro's Number in Calculations

Avogadro's number, denoted as \(6.022 \times 10^{23}\), is a fundamental constant in chemistry, representing the number of particles (such as atoms or molecules) in one mole of a substance. This number is essential when working with chemical quantities at the molecular level, as it allows the conversion between the number of particles and the amount of substance in moles.

For instance, in the given exercise, we use Avogadro's number to convert the concentrations of the reactants from molecules per cubic centimeter to molarity, which is moles per liter. This conversion is vital for calculating the rate of the reaction using the rate law, as the rate constant \(k\) is given in terms of molarity.

Converting Units with Avogadro's Number

To illustrate, consider the concentration of \(\mathrm{OH}\) radicals. The number of \(\mathrm{OH}\) molecules per cm³ is divided by Avogadro's number to find the number of moles per cm³, and then converted to moles per liter (Molarity, M) since there are 1000 cm³ in a liter. The precise understanding and application of Avogadro's number are fundamental to both stoichiometry and reaction kinetics, allowing chemists to predict how much of a substance is involved in a reaction or produced by it.