Question

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

The catalyzed reaction takes approximately 218 seconds to occur, assuming the same initial concentrations.

Step-by-step solution

Recall the Arrhenius equation

$k=A{e}^{-\frac{E_a}{RT}}$, where $k$ is the rate constant of the reaction, $A$ is the frequency factor (which is assumed constant in this problem), $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the temperature in Kelvin.

Calculate catalyzed and uncatalyzed rate constants

Using the Arrhenius equation, we can calculate the rate constant for both the uncatalyzed and catalyzed reactions. Let $k_1$ be the rate constant for the uncatalyzed reaction with activation energy $E_{a1}=184\mathrm{kJ}/\mathrm{mol}$ and $k_2$ be the rate constant for the catalyzed reaction with activation energy $E_{a2}=59.0\mathrm{kJ}/\mathrm{mol}$. First, let's convert the activation energies to $\mathrm{J}/\mathrm{mol}$: $E_{a1}=184\times {10}^3\mathrm{J}/\mathrm{mol}$ $E_{a2}=59.0\times {10}^3\mathrm{J}/\mathrm{mol}$ Since the temperature and frequency factor are the same for both reactions, we can calculate the ratio of rate constants: $\frac{k_2}{k_1}=\frac{A{e}^{-\frac{E_{a2}}{RT}}}{A{e}^{-\frac{E_{a1}}{RT}}}=e^{\frac{E_{a1}-E_{a2}}{RT}}$

Calculate the ratio of rate constants

Now, we know that $R=8.314\mathrm{J}/(\mathrm{mol}\cdot \mathrm{K})$, $T=600\mathrm{K}$, $E_{a1}=184\times {10}^3\mathrm{J}/\mathrm{mol}$, and $E_{a2}=59.0\times {10}^3\mathrm{J}/\mathrm{mol}$. Plug these values into the equation: $\frac{k_2}{k_1}=e^{\frac{(184-59.0)\times {10}^3}{(8.314)(600)}}\approx 3.467\times {10}^8$

Calculate the time for the catalyzed reaction

We know that the time for the uncatalyzed reaction is $t_1=2400\mathrm{years}$. We assume that the initial concentrations of the reactants are the same for the uncatalyzed and catalyzed reactions. So, the time taken for the catalyzed reaction, $t_2$, can be related to the time for the uncatalyzed reaction by the ratio of the rate constants: $t_2=\frac{k_1}{k_2}\times t_1=\frac{1}{\frac{k_2}{k_1}}\times t_1=\frac{1}{3.467\times {10}^8}\times t_1$ Now find the time for the catalyzed reaction: $t_2=\frac{1}{3.467\times {10}^8}\times 2400\mathrm{years}\approx 6.92\times {10}^{-6}\mathrm{years}$ Converting to seconds: $t_2\approx 6.92\times {10}^{-6}\mathrm{years}\times \frac{365\cdot 24\cdot 60\cdot 60}{1\mathrm{year}}\approx 218\mathrm{s}$

Present the final answer

The catalyzed reaction takes approximately 218 seconds to occur assuming the same initial concentrations.

Key concepts

Activation Energy

Activation energy is a fundamental concept in chemical kinetics that represents the minimum energy required for a chemical reaction to occur. It acts as an energy barrier that reactants need to overcome to form products.
If reactants do not possess enough energy to pass this barrier, the reaction will not proceed. The higher the activation energy, the slower the reaction because fewer molecules have the necessary energy to react at a given temperature.

• Activation energy is usually expressed in units of kilojoules per mole (kJ/mol).
• In the Arrhenius equation, it is represented as $E_a$.

In the given exercise, the activation energy of the uncatalyzed reaction was very high (184 kJ/mol), indicating a slow reaction requiring 2400 years. By introducing a catalyst, the activation energy was significantly reduced to 59 kJ/mol, facilitating a much faster reaction.

Catalyst

Catalysts are substances that increase the rate of a chemical reaction without being consumed in the process. They function by providing an alternative pathway for the reaction with a lower activation energy.

• This reduction in activation energy allows more reactant molecules to have enough energy to overcome the energy barrier, increasing the reaction rate.
• Although catalysts do not change the thermodynamics of a reaction, such as the enthalpy change, they have a significant effect on the kinetics, or speed, of the reaction.

In our specific problem, the introduction of a catalyst reduced the activation energy from 184 kJ/mol to 59 kJ/mol, drastically reducing the time required for the reaction from centuries to a mere 218 seconds.

Reaction Rate Constant

The reaction rate constant, denoted as $k$ in the Arrhenius equation, is a critical factor in determining how quickly a reaction proceeds. For a given reaction, a higher rate constant signifies a faster reaction.
It's determined by factors like activation energy and temperature and is specific to each reaction.

• In the Arrhenius equation: $k=A{e}^{-\frac{E_a}{RT}}$, $k$ changes with temperature $T$ and the activation energy $E_a$.
• The frequency factor $A$ is a constant that reflects the number of collisions resulting in a reaction.

Our exercise used the rate constants to show the impact of reducing activation energy with a catalyst. This reduction led to a vast increase in the rate constant by over $3.467\times {10}^8$ times, allowing the catalyzed reaction to complete in much less time.

Temperature Dependence

Temperature plays a pivotal role in chemical reactions; it often determines the reaction rate. Generally, increasing the temperature increases the reaction rate by providing more energy to the reactant molecules.

• The Arrhenius equation clearly shows this temperature dependence: $k=A{e}^{-\frac{E_a}{RT}}$. An increase in temperature $T$ decreases the exponent $-\frac{E_a}{RT}$, thus raising $k$.
• This means more molecules have sufficient energy to surpass the activation energy barrier, speeding up the reaction.

In the exercise, the reaction temperature was held constant at 600 K to isolate the effect of the catalyst on activation energy. This allowed for a direct comparison of reaction rates with and without a catalyst.