Suppose that a catalyst lowers the activation barrier of a reaction from 125 kJ/mol to 55 kJ/mol. By what factor would you expect the reaction rate to increase at 25 °C? (Assume that the frequency factors for the catalyzed and uncatalyzed reactions are identical.)

Verified step by step guidance

1. Identify the Arrhenius equation: $k = A e^{-\frac{E_a}{RT}}$, where $k$ is the rate constant, $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant (8.314 J/mol·K), and $T$ is the temperature in Kelvin.
2. Convert the temperature from Celsius to Kelvin: $T = 25 + 273.15 = 298.15$ K.
3. Calculate the rate constant for the uncatalyzed reaction using the activation energy $E_{a1} = 125$ kJ/mol. Convert $E_{a1}$ to J/mol: $E_{a1} = 125,000$ J/mol.
4. Calculate the rate constant for the catalyzed reaction using the activation energy $E_{a2} = 55$ kJ/mol. Convert $E_{a2}$ to J/mol: $E_{a2} = 55,000$ J/mol.
5. Determine the factor by which the reaction rate increases by taking the ratio of the rate constants: $\frac{k_2}{k_1} = \frac{A e^{-\frac{E_{a2}}{RT}}}{A e^{-\frac{E_{a1}}{RT}}} = e^{-\frac{E_{a2} - E_{a1}}{RT}}$.

Key Concepts

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 form products. Lowering the activation energy, as a catalyst does, increases the number of reactant molecules that can successfully collide and react, thereby increasing the reaction rate.

Arrhenius Equation

The Arrhenius equation describes the temperature dependence of reaction rates and is given by k = A * e^(-Ea/RT), where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation shows that a decrease in activation energy (Ea) leads to an exponential increase in the rate constant (k), which directly correlates to an increase in reaction rate.

Catalysis

Catalysis is the process of increasing the rate of a chemical reaction by adding a substance called a catalyst, which is not consumed in the reaction. Catalysts work by providing an alternative reaction pathway with a lower activation energy, allowing more reactant molecules to participate in the reaction at a given temperature, thus enhancing the overall reaction rate.