Question

The degradation of \(\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{~F}\) (an HFC) 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 tropo- spheric 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 \(M / s ?\)

Short answer

The rate of the reaction at this temperature is approximately \(2.77 \times 10^{-35} M/s\).

Step-by-step solution

Convert the concentrations from molecules/cm³ to moles/liter (M)

We need to convert the concentrations from molecules/cm³ to moles/liter (M). To do this, we can use the following conversions: 1 mole = \(6.022 \times 10^{23}\) molecules (Avogadro's number) 1 cm³ = \(1 \times 10^{-3}\) liters. For OH: \(\frac{8.1 \times 10^{5} \, molecules/cm^3}{6.022 \times 10^{23} \, molecules/mol} \cdot \frac{10^3 \, cm^3}{1 \, L} = OH_{M}\) For CF3CH2F: \(\frac{6.3 \times 10^{8} \, molecules/cm^3}{6.022 \times 10^{23} \, molecules/mol} \cdot \frac{10^3 \, cm^3}{1 \, L} = CF_{3}CH_{2}F_{M}\)

Calculate the rate of reaction using the rate law

We have the rate constant k and the concentrations of OH and CF3CH2F in M. Since this is a first-order reaction in both reactants, we can use the rate law equation: Rate = \(k [OH] [CF_{3}CH_{2}F]\) Plug in the given values: Rate = \(1.6 \times 10^{8} M^{-1}s^{-1} \cdot OH_{M} \cdot CF_{3}CH_{2}F_{M}\)

Solve for the rate

Now, plug in the values obtained in Step 1 and solve the equation: Rate = \(1.6 \times 10^{8} M^{-1}s^{-1} \cdot (\frac{8.1 \times 10^{5} \, molecules/cm^3}{6.022 \times 10^{23} \, molecules/mol} \cdot \frac{10^3 \, cm^3}{1 \, L}) \cdot (\frac{6.3 \times 10^{8} \, molecules/cm^3}{6.022 \times 10^{23} \, molecules/mol} \cdot \frac{10^3 \, cm^3}{1 \, L})\) Solve for the rate: Rate ≈ \(2.77 \times 10^{-35} M/s\) So, the rate of reaction at this temperature is approximately \(2.77 \times 10^{-35} M/s\).

Key concepts

Rate Laws

In chemical kinetics, a rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It is a mathematical equation that allows us to calculate the reaction rate given the reactant concentrations and a constant, which varies with temperature and is specific to the reaction.

The general form of a rate law for a reaction with two reactants, A and B, can be expressed as: \[ \text{Rate} = k [A]^x [B]^y \]where \( k \) is the rate constant, \( [A] \) and \( [B] \) represent the concentrations of the reactants, and \( x \) and \( y \) are the reaction orders with respect to each reactant. The sum of \( x \) and \( y \) gives the overall reaction order.

It’s important to note that reaction orders are determined experimentally; they are not necessarily related to the stoichiometry of the reaction. Thus, to solve problems involving rate laws, we often need experimental data to establish the form of the equation before we can apply it to find reaction rates under different conditions.

First-Order Reactions

A first-order reaction is one in which the rate is directly proportional to the concentration of a single reactant. This means that the reaction order with respect to that reactant is one. Mathematically, the rate law for a first-order reaction can be written as:\[ \text{Rate} = k [A] \]In this case, the reaction rate changes linearly with changes in the concentration of the reactant, and the rate constant \( k \) has the units of \( s^{-1} \) (per second). A common example of a first-order reaction is the radioactive decay of isotopes.

For a reaction that is first order in two reactants, like the one between \( \mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{F} \) and OH radicals from our exercise, the rate law would be:\[ \text{Rate} = k [\mathrm{CF}_{3} \mathrm{CH}_{2} \mathrm{F}] [\mathrm{OH}] \]When we say a reaction is first order in each reactant, we mean that the rate of reaction will double if the concentration of either reactant is doubled, assuming the other remains constant. Understanding first-order reactions is essential as they are common in biochemical processes and environmental reactions, such as the degradation of organic compounds in the atmosphere.

Reaction Rates

The rate of a reaction is a measure of how fast a reactant is consumed or a product is formed over time. It's typically expressed in terms of concentration change per unit of time, such as moles per liter per second (\( M/s \)) or molecules per cubic centimeter per second (\( molecules/cm^3/s \)).

The reaction rate can be affected by various factors, including reactant concentrations, temperature, and the presence of a catalyst. In the calculated solution, by knowing the rate law and the reactant concentrations, we derived the reaction rate at a specific temperature for the degradation of an HFC in the presence of OH radicals.

Remember, in practice, reaction rates are often determined by complex interactions between molecules and can be measured by experimental methods such as spectroscopy or chromatography. Being able to calculate and predict these rates is a powerful tool, especially in understanding and controlling chemical processes in industrial, biological, and environmental contexts.