To determine how many times the reaction rate has changed in the presence of a catalyst, we can use the Arrhenius equation, which relates the rate constant of a reaction to its activation energy. The equation is expressed as:
k = A * e^(-Ea/RT)
Where:
- k = rate constant
- A = pre-exponential factor (frequency factor)
- Ea = activation energy
- R = universal gas constant (8.314 J/(mol·K))
- T = temperature in Kelvin
In this scenario, we have two activation energies: one for the reaction without a catalyst (30 kJ/mol) and one with a catalyst (24 kJ/mol). First, we need to convert these values from kJ/mol to J/mol for consistency with the gas constant:
- 30 kJ/mol = 30,000 J/mol
- 24 kJ/mol = 24,000 J/mol
Next, we can calculate the rate constants for both scenarios using the Arrhenius equation. Since the pre-exponential factor (A) and temperature (T) remain constant, we can set up a ratio of the rate constants:
k_catalyst / k_no_catalyst = e^(-(Ea_catalyst - Ea_no_catalyst) / RT)
Substituting the activation energies and the values for R and T:
k_catalyst / k_no_catalyst = e^(-(24,000 J/mol - 30,000 J/mol) / (8.314 J/(mol·K) * 298 K))
Calculating the difference in activation energy:
k_catalyst / k_no_catalyst = e^(6,000 J/mol / (8.314 J/(mol·K) * 298 K))
Now, let's compute the denominator:
8.314 J/(mol·K) * 298 K ≈ 2477.572 J/mol
Now we can substitute this value back into the equation:
k_catalyst / k_no_catalyst = e^(6,000 J/mol / 2477.572 J/mol)
Calculating the exponent:
6,000 / 2477.572 ≈ 2.42
Now we find:
k_catalyst / k_no_catalyst = e^(2.42) ≈ 11.16
This means that the reaction rate in the presence of the catalyst is approximately 11.16 times greater than the rate without the catalyst. In summary, the catalyst significantly enhances the reaction rate by lowering the activation energy required for the reaction to proceed.
