To solve this problem, we can use the Arrhenius equation, which relates the rate constant (k) to temperature (T) and activation energy (Ea):
k = A * e^(-Ea/RT)
Where: k = rate constant A = pre-exponential factor or frequency factor Ea = activation energy R = gas constant (8.314 J/(mol·K)) T = temperature in Kelvin
We can rearrange the equation to solve for the temperature (T):
T = -Ea / (R * ln(k/A))
Given that the rate constant at 1°C (or 273.15 K) is 4.5 × 10^3 s^(-1), and the activation energy is 58 kJ/mol, we need to find the temperature at which the rate constant is 10^2 s^(-1).
Plugging in the values into the equation:
T = -58,000 J/mol / (8.314 J/(mol·K) * ln(10^2 / 4.5 × 10^3))
Calculating the natural logarithm:
T = -58,000 J/mol / (8.314 J/(mol·K) * ln(0.022222))
T = -58,000 J/mol / (8.314 J/(mol·K) * (-3.806))
T = -58,000 J/mol / (-31.673 K)
T ≈ 1830 K
So, at a temperature of approximately 1830 Kelvin, the rate constant would be 10^2 s^(-1).