To solve the problem, we need to calculate the ratio of the rate constants (k_cat/k_uncat) with and without the presence of a catalyst, given the activation energies.
### Step-by-Step Solution:
1. **Identify Given Values**:
- Activation energy without catalyst (Ea_uncat) = 100 kJ/mol
- The catalyst lowers the activation energy by 75%.
2. **Calculate Activation Energy with Catalyst**:
- The reduction in activation energy due to the catalyst = 75% of 100 kJ/mol.
- Reduction = 0.75 × 100 kJ/mol = 75 kJ/mol.
- Therefore, the activation energy with catalyst (Ea_cat) = 100 kJ/mol - 75 kJ/mol = 25 kJ/mol.
3. **Use the Arrhenius Equation**:
- The Arrhenius equation is given by:
\[
k = A e^{-\frac{E_a}{RT}}
\]
- Where:
- \( k \) = rate constant
- \( A \) = pre-exponential factor
- \( E_a \) = activation energy
- \( R \) = universal gas constant (8.314 J/mol·K)
- \( T \) = temperature in Kelvin (298 K)
4. **Calculate Rate Constants**:
- For the uncatalyzed reaction (k_uncat):
\[
k_{uncat} = A e^{-\frac{100,000 \, J/mol}{(8.314 \, J/mol·K)(298 \, K)}}
\]
- For the catalyzed reaction (k_cat):
\[
k_{cat} = A e^{-\frac{25,000 \, J/mol}{(8.314 \, J/mol·K)(298 \, K)}}
\]
5. **Calculate the Ratio k_cat/k_uncat**:
- The ratio can be expressed as:
\[
\frac{k_{cat}}{k_{uncat}} = \frac{A e^{-\frac{25,000}{(8.314)(298)}}}{A e^{-\frac{100,000}{(8.314)(298)}}}
\]
- The \( A \) cancels out:
\[
\frac{k_{cat}}{k_{uncat}} = e^{-\frac{25,000}{(8.314)(298)}} \cdot e^{\frac{100,000}{(8.314)(298)}}
\]
- This simplifies to:
\[
\frac{k_{cat}}{k_{uncat}} = e^{\left(\frac{100,000 - 25,000}{(8.314)(298)}\right)}
\]
- Calculate the exponent:
\[
= e^{\left(\frac{75,000}{(8.314)(298)}\right)}
\]
6. **Perform the Calculation**:
- Calculate \( \frac{75,000}{(8.314)(298)} \):
\[
\frac{75,000}{(8.314)(298)} \approx 30.2
\]
- Therefore:
\[
\frac{k_{cat}}{k_{uncat}} = e^{30.2} \approx 1.06 \times 10^{13}
\]
### Final Result:
The ratio \( \frac{k_{cat}}{k_{uncat}} \) at T = 298 K is approximately \( 2.34 \times 10^{13} \).
