Question

Hydrogen abstraction from hydrocarbons by atomic chlorine is a mechanism for \(\mathrm{Cl} \cdot\) loss in the atmosphere. Consider the reaction of \(\mathrm{Cl} \cdot\) with ethane: $$\mathrm{C}_{2} \mathrm{H}_{6}(g)+\mathrm{Cl} \cdot(g) \rightarrow \mathrm{C}_{2} \mathrm{H}_{5} \cdot(g)+\mathrm{HCl}(g)$$ This reaction was studied in the laboratory, and the following data were obtained: $$\begin{array}{cc}\mathbf{T}(\boldsymbol{K}) & \mathbf{k}\left(\times \mathbf{1 0}^{-\mathbf{1 0}} \mathbf{M}^{-\mathbf{2}} \mathbf{s}^{-\mathbf{1}}\right) \\\\\hline 270 & 3.43 \\\370 & 3.77 \\ 470 & 3.99 \\\570 & 4.13 \\\670 & 4.23\end{array}$$ a. Determine the Arrhenius parameters for this reaction. b. At the tropopause (the boundary between the troposphere and stratosphere located approximately \(11 \mathrm{km}\) above the surface of Earth \(),\) the temperature is roughly \(220 \mathrm{K}\). What do you expect the rate constant to be at this temperature? c. Using the Arrhenius parameters obtained in part (a), determine the Eyring parameters \(\Delta H^{\dagger}\) and \(\Delta S^{\frac{1}{r}}\) for this reaction at \(220 \mathrm{K}\)

Short answer

Question: Using the given temperature and rate constant data, determine the Arrhenius parameters, calculate the rate constant at 220 K, and find the Eyring parameters for the given reaction.

Step-by-step solution

Determine the Arrhenius parameters

We are given a set of data points for the temperature in Kelvin (T) and the rate constant (k). To determine the Arrhenius parameters, we need to fit the data using the Arrhenius equation: $$k = Ae^{-\frac{E_a}{RT}}$$ Where: - A is the pre-exponential factor - E_a is the activation energy - R is the gas constant, given by \(8.314 \mathrm{J \cdot mol^{-1} \cdot K^{-1}}\) - T is the temperature in Kelvin Taking the natural logarithm of both sides of the Arrhenius equation and rearranging the terms, we get the linear equation: $$\ln k = \ln A - \frac{E_a}{R}\cdot\frac{1}{T} $$ Now, using the given data, we can perform a linear regression to obtain the slope, which is \(-\frac{E_a}{R}\), and the y-intercept, which is \(\ln A\).

Calculate the rate constant at 220 K

With the Arrhenius parameters (A and E_a) determined from the linear regression, we can now calculate the rate constant at 220 K using the Arrhenius equation: $$ k = Ae^{-\frac{E_a}{RT}} $$

Determine the Eyring parameters

The Eyring parameters, \(\Delta H^{\dagger}\) and \(\Delta S^{\dagger}\), are related to the Arrhenius parameters through the Eyring equation: $$k = \frac{k_b T}{h}e^{-\frac{\Delta H^{\dagger}}{RT}(1-\frac{\Delta S^{\dagger}}{R})}$$ Where: - k_b is Boltzmann's constant, given by \(1.381\times10^{-23}\mathrm{J \cdot K^{-1}}\) - h is Planck's constant, given by \(6.626\times10^{-34}\mathrm{J \cdot s}\) We can obtain the desired Eyring parameters by comparing the Eyring equation to the Arrhenius equation, and using the values previously obtained for A and E_a: $$\Delta H^{\dagger} = E_a$$ $$\Delta S^{\dagger} = R\left(1-\frac{\ln A - \ln (k_b T /h)}{\ln A}\right)$$ By using these equations, we can calculate the Eyring parameters \(\Delta H^{\dagger}\) and \(\Delta S^{\frac{1}{r}}\) for this reaction at 220 K.

Key concepts

Eyring Equation

The Eyring equation is a fundamental equation in chemical kinetics used to understand the rate of chemical reactions. It is similar to the Arrhenius equation, providing insights into the energy changes occurring during a reaction. The Eyring equation is represented as:\[k = \frac{k_b T}{h} e^{-\frac{\Delta H^{\dagger}}{RT} \left(1-\frac{\Delta S^{\dagger}}{R}\right)}\]In this equation:

• \(k\) is the rate constant
• \(k_b\) is Boltzmann's constant
• \(T\) is the temperature in Kelvin
• \(h\) is Planck's constant
• \(\Delta H^{\dagger}\) is the change in enthalpy (activation enthalpy)
• \(\Delta S^{\dagger}\) is the change in entropy (activation entropy).

Unlike the Arrhenius equation which provides the activation energy, the Eyring equation provides both enthalpic and entropic contributions to the activation barrier. This makes it highly useful in detailed studies of reaction mechanisms, particularly in smaller, less obvious mechanisms or those involving transition states.

Activation Energy

Activation energy is a crucial concept in understanding chemical kinetics. It refers to the minimum amount of energy required for a chemical reaction to occur. In the context of the Arrhenius equation:\[k = Ae^{-\frac{E_a}{RT}} \]\(E_a\) denotes the activation energy. The role of activation energy can be illustrated as the barrier that reactants must overcome to convert into products. If the activation energy is low, reactions tend to proceed faster since fewer energy inputs are needed. Conversely, high activation energy implies slower reactions as reactants need more energy to proceed.Factors affecting activation energy include:

• The nature of the reactants: Stronger bonds mean higher activation energy.
• The temperature: Higher temperatures can provide the kinetic energy required to overcome activation energy barriers.

Understanding activation energy is pivotal in designing reactions that are efficient, safer, or more selective in industrial processes.

Rate Constant

The rate constant, often denoted by the symbol \(k\), is a fundamental element in the study of chemical kinetics. It is part of both the Arrhenius and Eyring equations and plays a central role in the speed at which reactions proceed.The basic role of the rate constant can be seen in the rate equations:

• Rate = \(k[A]^m [B]^n\) for a reaction with reactants \(A\) and \(B\), where \(m\) and \(n\) are the reaction orders.

For a particular reaction at a specific temperature, \(k\) remains constant. Its value is influenced by:

• Temperature: As temperature increases, particle energy increases, facilitating more successful collisions.
• Catalysts: These can lower the activation energy, thereby changing the rate constant value.

A high rate constant indicates a fast reaction under given conditions, while a low rate constant suggests a slow reaction. Understanding and calculating the rate constant is essential for both predicting reaction behavior and optimizing conditions to achieve desired reaction speeds.

Hydrogen Abstraction

Hydrogen abstraction is a type of chemical reaction where a hydrogen atom is removed from a molecule, often involving radicals. This reaction mechanism is significant in atmospheric chemistry, particularly in the behavior of compounds like hydrocarbons.In the given example, we have:\[\text{C}_2\text{H}_6(g) + \text{Cl} \cdot (g) \rightarrow \text{C}_2\text{H}_5 \cdot (g) + \text{HCl}(g)\]Here, the chlorine radical (\(\text{Cl} \cdot\)) abstracts a hydrogen atom from ethane \((\text{C}_2\text{H}_6)\), producing a new radical \((\text{C}_2\text{H}_5 \cdot)\) and \(\text{HCl}\). Because radicals are highly reactive, hydrogen abstraction reactions can occur rapidly, making them important in atmospheric reactions involving chlorine or other radical-forming species.These reactions can:

• Transform stable molecules into reactive ones, allowing further reactions to occur.
• Influence chemical pathways in the atmosphere, contributing to processes like ozone depletion.

Understanding hydrogen abstraction is critical for both environmental studies and industrial applications, where managing radical behavior is necessary.