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

Step-by-step solution

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.

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 \(\frac{k_2}{k_1} = e^{\frac{E_1-E_2}{RT}}\)

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. \(E_1 = 184 * 10^3\, J/mol\) \(E_2 = 59.0 * 10^3 \, J/mol\) \(\frac{k_2}{k_1} = e^{\frac{184*10^3 - 59.0*10^3}{8.314*600}}\) \(\frac{k_2}{k_1} \approx e^{19.99}\) \(\frac{k_2}{k_1} \approx 486880.20\)

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.

Key concepts

Arrhenius Equation

The Arrhenius Equation is a formula that describes how the rate constant (\(k\)) of a chemical reaction is affected by temperature and activation energy. This equation is given as:\[k = Ae^{-E_a/RT}\]where:

• \(k\) is the rate constant that indicates how fast or slow a reaction occurs.
• \(A\) is the frequency factor, representing the number of times that reactants approach the activation barrier per unit time.
• \(E_a\) is the activation energy, which is the minimum energy required to initiate a reaction.
• \(R\) is the universal gas constant, approximately equal to 8.314 J/mol·K.
• \(T\) is the temperature in Kelvin.

The Arrhenius equation shows that as the activation energy decreases or the temperature increases, the rate constant \(k\) increases. Thus, reactions happen faster when they require less energy to overcome the activation barrier or occur at higher temperatures. The exponential part of the equation highlights how sensitively the reaction rate increases with these changes.

Catalyst Effect

Catalysts play a crucial role in chemical reactions by providing an alternative pathway for the reaction to occur, which has a lower activation energy compared to the reaction without a catalyst. This dramatic reduction in activation energy is why reactions tend to happen more quickly when a catalyst is present.In the given exercise, the introduction of a catalyst reduces the activation energy from \(184 \, \mathrm{kJ/mol}\) to \(59.0 \, \mathrm{kJ/mol}\). This significant decrease in energy requirement means that more molecules can overcome the energy barrier and react, thus speeding up the reaction.It's important to note that while catalysts speed up the rate of a reaction, they do not affect the equilibrium state of the reaction. This means that the relative concentrations of reactants and products at equilibrium remain unchanged. Instead, catalysts help systems reach equilibrium more rapidly. Additionally, catalysts themselves are not consumed in the reaction, which allows them to be used repeatedly.
This can be extremely useful in industrial processes where reaction speed is a critical factor for efficiency and productivity.

Rate Constant

The rate constant (symbolized as \(k\)) is a critical factor in determining the speed of a chemical reaction. It varies with temperature and can be affected by the presence of a catalyst. The rate constant is part of the rate law expression, which connects the rate of a chemical reaction to the concentration of the reactants.In our example, we calculate the change in the rate constant due to the introduction of a catalyst. By using the Arrhenius equation, we can compute the ratio of the rate constants for the catalyzed and uncatalyzed reactions. Here, the catalyzed reaction's rate constant \(k_2\) was approximately 486,880 times that of the uncatalyzed reaction's rate constant \(k_1\).This enormous increase in \(k\) results from the lowered activation energy, proving how powerful and beneficial catalysts can be. As mentioned earlier, the rate constant is inversely related to the time it takes for a reaction to complete, which explains why the catalyzed reaction occurs much faster — moving from 2400 years to just about 1.8 days.
Understanding the rate constant and its dependence on activation energy and temperature is vital for controlling and optimizing chemical reactions in both laboratory and industrial settings.