Ch.14 - Chemical Kinetics

Problem 87a

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Chapter 14, Problem 87a

The activation energy of an uncatalyzed reaction is 95 kJ/mol. The addition of a catalyst lowers the activation energy to 55 kJ/mol. Assuming that the collision factor remains the same, by what factor will the catalyst increase the rate of the reaction at (a) 25 C

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Identify the Arrhenius equation: \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Convert the temperature from Celsius to Kelvin: \( T = 25 + 273.15 \).

Calculate the rate constant for the uncatalyzed reaction using the activation energy of 95 kJ/mol: \( k_1 = A e^{-\frac{95000}{8.314 \times T}} \).

Calculate the rate constant for the catalyzed reaction using the activation energy of 55 kJ/mol: \( k_2 = A e^{-\frac{55000}{8.314 \times T}} \).

Determine the factor by which the catalyst increases the rate of the reaction by taking the ratio \( \frac{k_2}{k_1} \).

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Key Concepts

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

Activation Energy

Activation energy is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to transform into products. A higher activation energy means that fewer molecules have sufficient energy to react, resulting in a slower reaction rate. Conversely, lowering the activation energy, such as through the use of a catalyst, increases the likelihood of successful collisions between reactant molecules.

Catalysts

Catalysts are substances that increase the rate of a chemical reaction without being consumed in the process. They work by providing an alternative reaction pathway with a lower activation energy. This allows more reactant molecules to have enough energy to undergo the reaction, thus speeding up the reaction rate. Importantly, catalysts do not alter the equilibrium position of the reaction; they only affect the rate at which equilibrium is reached.

Arrhenius Equation

The Arrhenius equation describes the temperature dependence of reaction rates and relates the rate constant of a reaction to the activation energy and temperature. It is expressed as 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 universal gas constant, and T is the temperature in Kelvin. This equation illustrates how a decrease in activation energy, as seen with a catalyst, can significantly increase the reaction rate.

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Arrhenius Equation

Related Practice

(b) Automobile catalytic converters have to work at high temperatures, as hot exhaust gases stream through them. In what ways could this be an advantage? In what ways a disadvantage? (c) Why is the rate of flow of exhaust gases over a catalytic converter important?

Open Question

The enzyme carbonic anhydrase catalyzes the reaction CO2(g) + H2O(l) ↔ HCO3⁻(aq) + H⁺(aq). In water, without the enzyme, the reaction proceeds with a rate constant of 0.039 s⁻¹ at 25 _x001E_C. In the presence of the enzyme in water, the reaction proceeds with a rate constant of 1.0 * 10⁶ s⁻¹ at 25 _x001E_C. Assuming the collision factor is the same for both situations, calculate the difference in activation energies for the uncatalyzed versus enzyme-catalyzed reaction.

Textbook Question

The enzyme urease catalyzes the reaction of urea, 1NH2CONH22, with water to produce carbon dioxide and ammonia. In water, without the enzyme, the reaction proceeds with a first-order rate constant of 4.15 * 10-5 s-1 at 100 C. In the presence of the enzyme in water, the reaction proceeds with a rate constant of 3.4 * 104 s-1 at 21 C. (c) In actuality, what would you expect for the rate of the catalyzed reaction at 100 C as compared to that at 21 C?

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Textbook Question

The activation energy of an uncatalyzed reaction is 95 kJ/mol. The addition of a catalyst lowers the activation energy to 55 kJ/mol. Assuming that the collision factor remains the same, by what factor will the catalyst increase the rate of the reaction at (b) 125 °C?

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Open Question

Suppose that a certain biologically important reaction is quite slow at physiological temperature 137 _x001E_C2 in the absence of a catalyst. Assuming that the collision factor remains the same, by how much must an enzyme lower the activation energy of the reaction to achieve a 1 * 10^5-fold increase in the reaction rate?

Textbook Question

Consider the reaction A + B → C + D. Is each of the following statements true or false? (b) If the reaction is an elementary reaction, the rate law is second order.

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