Ch.14 - Chemical Kinetics

Problem 95

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Chapter 14, Problem 95

A certain first-order reaction has a rate constant of 1.0 * 10-3 s-1 at 25 °C. (b) What is the Ea (in kJ/mol) if the same temperature change causes the rate to triple?

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Identify the Arrhenius equation: k = A * e^(-Ea/(RT)), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin.

Recognize that the problem involves a change in the rate constant due to a change in temperature, which can be analyzed using the Arrhenius equation in its logarithmic form: ln(k2/k1) = -Ea/R * (1/T2 - 1/T1).

Since the rate triples, set k2 = 3 * k1. Use the given rate constant k1 = 1.0 * 10^-3 s^-1 and solve for Ea.

Assume the temperature change is negligible for simplification, so T1 = T2 = 25 °C = 298 K. Substitute these values into the equation: ln(3) = -Ea/R * (1/298 - 1/298).

Solve for Ea by rearranging the equation: Ea = -R * ln(3) / (1/298 - 1/298). Note that the temperature difference is zero, so this step is conceptual to understand the relationship.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. The rate law for a first-order reaction can be expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant. Understanding this concept is crucial for analyzing how changes in concentration affect the reaction rate.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to the temperature and activation energy (Ea). It is expressed as k = A * e^(-Ea/RT), where A is the pre-exponential factor, R is the gas constant, and T is the temperature in Kelvin. This equation is essential for determining how temperature changes influence reaction rates and for calculating activation energy.

Effect of Temperature on Reaction Rate

Temperature significantly affects the rate of chemical reactions, typically increasing the rate as temperature rises. This is due to the increased kinetic energy of molecules, leading to more frequent and effective collisions. In this question, understanding how a temperature change can cause the rate to triple is key to calculating the activation energy using the Arrhenius equation.

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Related Practice

Textbook Question

The values of Ea = 248 kJ>mol and ΔE = 41 kJ>mol have been measured for the reaction H21g2 + CO21g2S H2O1g2 + CO1g2 (b) Considering the geometry of the reactants and products, suggest a plausible structure for the transition state.

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Textbook Question

Consider three reactions with different values of E_a and ΔE:

Reaction 1. E_a = 20 kJ>mol; ΔE = -60 kJ/mol

Reaction 2. E_a = 10 kJ>mol; ΔE = -20 kJ/mol

Reaction 3. E_a = 40 kJ>mol; ΔE = +15 kJ/mol

(b) Assuming that all three reactions are carried out at the same temperature and that all three have the same frequency factor A, which reaction is the fastest and which is the slowest?

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Textbook Question

Consider three reactions with different values of E_a and ΔE:

Reaction 1. E_a = 20 kJ>mol; ΔE = -60 kJ/mol

Reaction 2. E_a = 10 kJ>mol; ΔE = -20 kJ/mol

Reaction 3. E_a = 40 kJ>mol; ΔE = +15 kJ/mol

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Textbook Question

If the rate of a reaction increases by a factor of 2.5 when the temperature is raised from 20 °C to 30 °C, what is the value of the activation energy in kJ/mol? By what factor does the rate of this reaction increase when the temperature is raised from 120 °C to 130 °C?

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