Question

The gas-phase decomposition of ethyl bromide is a first-order reaction, occurring with a rate constant that demonstrates the following dependence on temperature: a. Determine the Arrhenius parameters for this reaction. b. Using these parameters, determine \(\Delta H^{\frac{t}{4}}\) and \(\Delta S^{\frac{2}{r}}\) as described by the Eyring equation.

Short answer

To determine the Arrhenius parameters for the gas-phase decomposition of ethyl bromide, first obtain the values of the rate constant \(k\) and temperature \(T\) through experimental data. Then, use the Arrhenius equation \(k = Ae^{\frac{-E_a}{RT}}\) to find the activation energy \(E_a\) and the pre-exponential factor \(A\). Once these values are obtained, use the Eyring equation \(k = \frac{k_BT}{h} e^{\frac{-\Delta H^{\ddagger}}{RT}} e^{\frac{\Delta S^{\ddagger}}{R}}\) to find \(\Delta H^{\ddagger}\) and \(\Delta S^{\ddagger}\), from which \(\Delta H^{\frac{t}{4}}\) and \(\Delta S^{\frac{2}{r}}\) can be calculated using \(\Delta H^{\frac{t}{4}} = \left(\Delta H^{\ddagger}\right)^{\frac{1}{4}}\) and \(\Delta S^{\frac{2}{r}} = \left(\Delta S^{\ddagger}\right)^{\frac{1}{2}}\). These values give insights into the properties of the decomposition of ethyl bromide.

Step-by-step solution

Identify the knowns and unknowns

In this case, we are not given any values for the rate constant k, activation energy E_a or the temperature T. However, we can still provide the general method to determine these parameters from the given data.

Use the Arrhenius equation

To determine the activation energy E_a and the pre-exponential factor A, we first need to obtain the values of k and T through the experimental data. Then we can use the Arrhenius equation: \(k = Ae^{\frac{-E_a}{RT}}\) Where k is the rate constant, A is the pre-exponential factor, E_a is the activation energy, R is the gas constant (8.314 J/(mol K)), and T is the temperature in Kelvin.

Linearize the Arrhenius equation and obtain the parameters

Take the natural logarithm of both sides of the equation and rearrange to obtain: \(\ln k = \ln A - \frac{E_a}{RT}\) This equation can be rewritten in the form of a linear equation, y = mx + b, where y = ln k, x = 1/T, m = -E_a/R, and b = ln A: \(\ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A\) By plotting ln k against 1/T, we can determine the slope, -E_a/R, and intercept, ln A, and thus find the values of E_a and A. b. Use the Eyring equation to determine \(\Delta H^{\frac{t}{4}}\) and \(\Delta S^{\frac{2}{r}}\)

Recall the Eyring equation

The Eyring equation relates the rate constant k of a reaction to its temperature T, the change in enthalpy (\(\Delta H^{\ddagger}\)), and the change in entropy (\(\Delta S^{\ddagger}\)): \(k = \frac{k_BT}{h} e^{\frac{-\Delta H^{\ddagger}}{RT}} e^{\frac{\Delta S^{\ddagger}}{R}}\)

Obtain the Arrhenius parameters and use the Eyring equation

Once we have obtained the values for A and E_a from the Arrhenius equation, we can plug them into the Eyring equation to find \(\Delta H^{\ddagger}\) and \(\Delta S^{\ddagger}\). To do this, we will need to convert E_a and A into appropriate units: \(\Delta H^{\ddagger} = E_a - RT\) \(\Delta S^{\ddagger}\) can be found by dividing the pre-exponential factor A from the Arrhenius equation by the Boltzmann constant (k_B) and the Planck constant (h) and then taking the natural logarithm: \(\Delta S^{\ddagger} = R \ln \left(\frac{A}{k_B \cdot h}\right)\) With these values, we can calculate \(\Delta H^{\frac{t}{4}}\) and \(\Delta S^{\frac{2}{r}}\) as described by the Eyring equation: \(\Delta H^{\frac{t}{4}} = \left(\Delta H^{\ddagger}\right)^{\frac{1}{4}}\) \(\Delta S^{\frac{2}{r}} = \left(\Delta S^{\ddagger}\right)^{\frac{1}{2}}\)

Interpret the results

The obtained values for \(\Delta H^{\frac{t}{4}}\) and \(\Delta S^{\frac{2}{r}}\) represent the fractional changes in enthalpy and entropy during the reaction. These values can help us better understand the properties of the decomposition of ethyl bromide and predict its behavior under different conditions.

Key concepts

Eyring equation

The Eyring equation is a fundamental equation in chemical kinetics that provides a deep insight into the transition state theory. It relates the rate constant (\( k \)) of a chemical reaction to temperature (\( T \)), and changes in enthalpy (\( \Delta H^{\ddagger} \)) and entropy (\( \Delta S^{\ddagger} \)) of the reaction. The equation is:\[k = \frac{k_B T}{h} e^{\frac{-\Delta H^{\ddagger}}{RT}} e^{\frac{\Delta S^{\ddagger}}{R}}\]- Here, \( k_B \) is the Boltzmann constant and \( h \) is the Planck constant.- The term \( \frac{k_B T}{h} \) signifies the ratio of thermal energy to the quantum of action.Compared to the Arrhenius equation, the Eyring equation provides additional insights by considering entropy changes. Understanding this equation helps in analyzing how temperature and molecular changes influence reaction rates. For chemical reactions, a positive \( \Delta \)S (entropy change) usually suggests disorder increase, often speeding up the reaction. Conversely, changes in enthalpy \( \Delta \)H, often reflect the energy required to reach the transition state. This equation, therefore, is very insightful for observing how reactions transpire at a molecular level.

Activation energy

Activation energy (\( E_a \)) is a crucial concept in chemical reactions that denotes the minimum energy required for reactants to transform into products. Essentially, it is an energy barrier that must be overcome for a reaction to proceed. In the context of the Arrhenius equation, it is represented by the parameter:\[E_a = RT \ln \left( \frac{A}{k} \right)\]- \( E_a \) is typically expressed in units of Joules per mole (J/mol).- The higher the \( E_a \), the slower the reaction since fewer molecules possess the necessary energy to surpass this barrier.The concept of activation energy explains why some reactions occur more quickly than others due to their lower energy thresholds. It can vary with different factors:

• Temperature: As temperature increases, more molecules can overcome the energy barrier, speeding up the reaction.
• Catalysts: These lower the activation energy, allowing reactions to proceed at lower temperatures.

Understanding \( E_a \) is essential for controlling and optimizing chemical reactions in various fields such as pharmaceuticals and materials science.

Rate constant

The rate constant (\( k \)) is an essential parameter in the kinetics of chemical reactions that indicates how rapidly a reaction proceeds. It is a part of the rate equation, showing the relationship between reaction rate and concentration of reactants. For a first-order reaction, the rate constant is given by:\[k = Ae^{-\frac{E_a}{RT}}\]- - Depending on the order of the reaction, the units of \( k \) vary. For instance, first-order reactions have units of s\(^{-1}\).- It is independent of reactant concentrations and only changes with temperature.The rate constant provides valuable information on how factors such as:

• Temperature: An increase in temperature generally increases \( k \), leading to a faster reaction.
• Presence of a catalyst: Catalysts can enhance \( k \), allowing for a quicker reaction process without altering the equilibrium.

Understanding \( k \) is vital for predicting how long a reaction will take under specific conditions and is integral in fields like chemistry and chemical engineering.

Enthalpy and entropy changes

Enthalpy and entropy changes are important concepts in the study of chemical reactions. They help us understand the energy and disorder aspects of reactions.- **Enthalpy Change (\( \Delta H \)):** This is the amount of energy absorbed or released during a reaction. It is indicative of the heat change in a reaction at constant pressure: - A negative \( \Delta H \) signifies an exothermic reaction (releases heat). - A positive \( \Delta H \) denotes an endothermic reaction (absorbs heat).In the context of the Eyring equation, \( \Delta H^{\ddagger} \) relates to the activation energy, adjusting for the system's heat requirement to reach the transition state.- **Entropy Change (\( \Delta S \)):** This describes the degree of disorder or randomness in a system. An increase in entropy (\( \Delta S > 0 \)) corresponds to increased disorder: - Reactions with increased randomness (\( \Delta S > 0 \)) are often favored and can progress more readily.Overall, understanding variations in enthalpy and entropy provides critical insights into the spontaneity and energy dynamics of reactions. These concepts are useful for calculating Gibbs free energy changes and determining reaction feasibility.