Problem 118 - Chemistry: The Central Science in SI Units, Global Edition

Question

The reaction between ethyl iodide and hydroxide ion in ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) solution, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}(a l c)+\mathrm{OH}^{-}(a l c) \longrightarrow\) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+\mathrm{I}^{-}(\) alc \()\) has an activation energy of \(86.8 \mathrm{~kJ} / \mathrm{mol}\) and a frequency factor of \(2.1 \times 10^{11} \mathrm{M}^{-1} \mathrm{~s}^{-1}\) (a) Predict the rate constant for the reaction at \(30^{\circ} \mathrm{C}\). (b) A solution of KOH in ethanol is made up by dissolving \(0.500 \mathrm{~g} \mathrm{KOH}\) in ethanol to form \(500 \mathrm{~mL}\) of solution. Similarly, \(1.500 \mathrm{~g}\) of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}\) is dissolved in ethanol to form \(500 \mathrm{~mL}\) of solution. Equal volumes of the two solutions are mixed. Assuming the reaction is first order in each reactant, what is the initial rate at \(30^{\circ} \mathrm{C} ?(\mathbf{c})\) Which reagent in the reaction is limiting, assuming the reaction proceeds to completion? ((d) Assuming the frequency factor and activation energy do not change as a function of temperature, calculate the rate constant for the reaction at \(40^{\circ} \mathrm{C} .\)

Step-by-Step Solution

Answer

The rate constants for the reaction at 30°C and 40°C are approximately 5.56 x 10^7 M^(-1) s^(-1) and 2.09 x 10^8 M^(-1) s^(-1), respectively. The initial rate of the reaction at 30°C is approximately 1.90 x 10^6 M/s. Neither reactant is clearly limiting in this case, as they both have nearly the same number of moles and a stoichiometric ratio of 1:1.

Step 1: Part (a): Rate constant at 30°C

Given: Activation energy (Ea) = 86.8 kJ/mol, Frequency factor (A) = 2.1 x 10^11 M^(-1) s^(-1) Firstly, convert the temperature from Celsius to Kelvin: T = 30 + 273.15 = 303.15 K Now, we will make use of Arrhenius equation to find the rate constant, k: \[k = Ae^{\frac{-Ea}{RT}}\] Here, R is the universal gas constant, R = 8.314 J/(mol.K) Converting Ea to J/mol: 86.8 kJ/mol x 1000 = 86800 J/mol \[k = (2.1 \times 10^{11}) e^{\frac{-86800}{(8.314)(303.15)}}\] Calculate the value of k: k ≈ 5.56 x 10^7 M^(-1) s^(-1) The rate constant for the reaction at 30°C is approximately 5.56 x 10^7 M^(-1) s^(-1).

Step 2: Part (b): Initial rate at 30°C

First, we need to calculate the molar concentration of each reactant. For KOH: Mass = 0.500 g, Molar mass = 56.11 g/mol Moles of KOH = \(\frac{0.500}{56.11}\) = 0.00891 mol Volume of solution = 500 mL = 0.500 L Concentration of OH^(-) = \(\frac{0.00891}{0.500}\) = 0.0178 M For C2H5I: Mass = 1.500 g, Molar mass = 155.97 g/mol Moles of C2H5I = \(\frac{1.500}{155.97}\) = 0.00962 mol Volume of solution = 500 mL = 0.500 L Concentration of C2H5I = \(\frac{0.00962}{0.500}\) = 0.0192 M When equal volumes of the two solutions are mixed, the volume doubles but the concentration remains the same. The rate law for the reaction is: Rate = k [C2H5I][OH^-] Using the rate constant at 30°C (k ≈ 5.56 x 10^7 M^(-1) s^(-1), and concentrations as calculated above: Initial rate ≈ (5.56 x 10^7)(0.0192)(0.0178) ≈ 1.90 x 10^6 M/s The initial rate of the reaction at 30°C is approximately 1.90 x 10^6 M/s.

Step 3: Part (c): Limiting reagent

To determine the limiting reagent, compare the mole ratio of reactants: \[\frac{moles\:of\:C_2H_5I}{moles\:of\:OH^-} = \frac{0.00962}{0.00891} = 1.08\] Since the stoichiometric ratio is 1:1, C2H5I and OH^- have nearly the same number of moles, meaning the reaction will consume both reactants almost evenly. Neither reagent is clearly limiting in this case.

Step 4: Part (d): Rate constant at 40°C

Similar to part (a), we will use the Arrhenius equation to find the rate constant at 40°C: T = 40 + 273.15 = 313.15 K \[k = (2.1 \times 10^{11}) e^{\frac{-86800}{(8.314)(313.15)}}\] Calculate the value of k: k ≈ 2.09 x 10^8 M^(-1) s^(-1) The rate constant for the reaction at 40°C is approximately 2.09 x 10^8 M^(-1) s^(-1).

Key Concepts

Rate Constant

The rate constant, often represented by the symbol \( k \), is a crucial part of chemical kinetics, as it quantifies the rate at which a reaction proceeds. It is a unique value for each reaction, influenced by factors such as temperature and the nature of reactants. At a constant temperature, the rate constant remains the same and is used in the rate equation:

• Rate = \( k [ ext{Reactants}]^n \)

where \( n \) is the order of the reaction with respect to each reactant, and the brackets [ ] denote concentration.
The rate constant is calculated using the Arrhenius equation, which connects chemical kinetics with temperature and activation energy.
Understanding the rate constant helps in predicting how fast a reaction can occur under given conditions. It's vital for industries that rely on chemical transformations, such as pharmaceuticals and materials science.

Arrhenius Equation

The Arrhenius Equation provides an essential relationship in chemical kinetics, linking the rate constant to temperature and activation energy. It is expressed as:\[k = A e^{- \frac{E_a}{RT}}\]where:

• \( k \) is the rate constant,
• \( A \) is the frequency factor or pre-exponential factor, indicating the number of times that reactants approach each other with the correct orientation per unit time,
• \( E_a \) is the activation energy, the minimum energy required for a reaction to occur,
• \( R \) is the universal gas constant, and
• \( T \) is the temperature in Kelvin.

The Arrhenius Equation shows that as temperature increases, the rate constant usually increases, speeding up the reaction. This concept is pivotal in understanding how reactions become feasible at higher temperatures, enabling control of reaction speeds in laboratory and industrial settings.

Activation Energy

Activation energy \( (E_a) \) is the energy barrier that reactants must overcome to transform into products during a chemical reaction. It is a critical concept in chemical kinetics, determining whether a reaction will occur spontaneously under given conditions. Activation energy can be visualized as the summit of the energy hill over which reactants must climb during the reaction.

• Higher \( E_a \) means slower reactions at a given temperature, requiring more energy to proceed.
• Lower \( E_a \) signifies faster reactions, as less energy is needed to get over the energy barrier.

Experiments often involve adjusting conditions to lower the activation energy, facilitating more efficient processes. The Arrhenius Equation helps quantify how closely \( E_a \) and temperature affect reaction rates, supporting strategic planning in engineering and environmental controls by reducing energy needs.

Limiting Reagent

The limiting reagent in a chemical reaction determines the extent to which the reaction can proceed. This reagent will be completely consumed first during the reaction, limiting the amount of product formed.
To identify the limiting reagent, compare the mole ratio of the reactants to that present in the balanced chemical equation. This comparison reveals which reactant is in short supply.

• If the mole ratio of one reactant is less than required, it is the limiting reagent.
• If all mole ratios are equivalent, then the mixture is perfectly balanced.

Knowing the limiting reagent is crucial when scaling reactions for industrial purposes, as it dictates the theoretical yield of the reaction and influences cost efficiency. By maximizing the use of the limiting reagent, processes can be optimized for better resource management and waste reduction.