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Chapter 12: Problem 74

The activation energy for a reaction is changed from \(184 \mathrm{~kJ} / \mathrm{mol}\) to \(59.0 \mathrm{~kJ} / \mathrm{mol}\) at \(600 . \mathrm{K}\) by the introduction of a catalyst. If the uncatalyzed reaction takes about 2400 years to occur, about how long will the catalyzed reaction take? Assume the frequency factor \(A\) is constant and assume the initial concentrations are the same.

Short Answer

Expert verified
The time it would take for the catalyzed reaction to occur is approximately 0.00493 years or about 1.8 days.

Step by step solution

01

Write down the Arrhenius equation for both reactions

\(k_1 = Ae^{\frac{-E_1}{RT}}\), where \(k_1\) is the rate constant for the uncatalyzed reaction, \(A\) is the frequency factor, \(E_1 =184\, kJ/mol \) is the activation energy of the uncatalyzed reaction, R = 8.314 J/mol K is the gas constant, and T = 600 K is the temperature. \(k_2 = Ae^{\frac{-E_2}{RT}}\), where \(k_2\) is the rate constant for the catalyzed reaction and \(E_2 = 59.0\, kJ/mol \) is the activation energy of the catalyzed reaction.
02

Divide both equations to eliminate A from the equations

Dividing both equations, we have \(\frac{k_2}{k_1} = \frac{Ae^{\frac{-E_2}{RT}}}{Ae^{\frac{-E_1}{RT}}}\) After simplification, we have \(\frac{k_2}{k_1} = e^{\frac{E_1-E_2}{RT}}\)
03

Calculate ratio of rate constants, k2/k1

From the above equation, we can calculate the ratio of the rate constants, k2/k1. Convert the activation energy from kJ/mol to J/mol. \(E_1 = 184 * 10^3\, J/mol\) \(E_2 = 59.0 * 10^3 \, J/mol\) \(\frac{k_2}{k_1} = e^{\frac{184*10^3 - 59.0*10^3}{8.314*600}}\) \(\frac{k_2}{k_1} \approx e^{19.99}\) \(\frac{k_2}{k_1} \approx 486880.20\)
04

Estimate the time for the catalyzed reaction

Since the initial concentrations are the same, we can assume the time required for complete reaction is inversely proportional to the rate constant (t ∝ 1/k). Thus, \(\frac{t_1}{t_2} = \frac{k_2}{k_1}\), where t1 is the time for the uncatalyzed reaction and t2 is the time for the catalyzed reaction. Given the time required for the uncatalyzed reaction, \(t_1 = 2400 \,years\). So, we can solve for t2: \(t_2 = \frac{k_1}{k_2} * t_1\) \(t_2 = \frac{1}{486880.20} * 2400 \,years\) \(t_2 \approx 0.00493 \,years\) The time it would take for the catalyzed reaction to occur is approximately 0.00493 years or about 1.8 days.

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Most popular questions from this chapter

Sulfuryl chloride undergoes first-order decomposition at \(320 .{ }^{\circ} \mathrm{C}\) with a half-life of \(8.75 \mathrm{~h}\). $$ \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) $$ What is the value of the rate constant, \(k\), in \(\mathrm{s}^{-1} ?\) If the initial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is 791 torr and the decomposition occurs in a \(1.25\) -L container, how many molecules of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) remain after \(12.5 \mathrm{~h}\) ?

Cobra venom helps the snake secure food by binding to acetylcholine receptors on the diaphragm of a bite victim, leading to the loss of function of the diaphragm muscle tissue and eventually death. In order to develop more potent antivenoms, scientists have studied what happens to the toxin once it has bound the acetylcholine receptors. They have found that the toxin is released from the receptor in a process that can be described by the rate law Rate \(=k[\) acetylcholine receptor-toxin complex \(]\) If the activation energy of this reaction at \(37.0^{\circ} \mathrm{C}\) is \(26.2 \mathrm{~kJ} / \mathrm{mol}\) and \(A=0.850 \mathrm{~s}^{-1}\), what is the rate of reaction if you have a \(0.200 M\) solution of receptor-toxin complex at \(37.0^{\circ} \mathrm{C}\) ?

Make a graph of [A] versus time for zero-, first-, and second-order reactions. From these graphs, compare successive half-lives.

Chemists commonly use a rule of thumb that an increase of \(10 \mathrm{~K}\) in temperature doubles the rate of a reaction. What must the activation energy be for this statement to be true for a temperature increase from 25 to \(35^{\circ} \mathrm{C}\) ?

The activation energy for the reaction $$ \mathrm{NO}_{2}(g)+\mathrm{CO}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g) $$ is \(125 \mathrm{~kJ} / \mathrm{mol}\), and \(\Delta E\) for the reaction is \(-216 \mathrm{~kJ} / \mathrm{mol}\). What is the activation energy for the reverse reaction \(\left[\mathrm{NO}(g)+\mathrm{CO}_{2}(g) \longrightarrow\right.\) \(\left.\mathrm{NO}_{2}(g)+\mathrm{CO}(g)\right] ?\)
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