If the activation energy for a given compound is found to be 42.0 kJ/mol, with a frequency factor of 8.0 × 1010 s-1, what is the rate constant for this reaction at 298 K?
Use the Arrhenius equation.
k = Ae^-[Ea/(RT)]
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To find the rate constant for a reaction, you can use the Arrhenius equation:
k = A * e^(-Ea/RT)
Where:
- k is the rate constant for the reaction
- A is the frequency factor
- Ea is the activation energy
- R is the gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
First, convert the activation energy from kJ/mol to J/mol:
Ea = 42.0 kJ/mol * 1000 J/kJ = 42,000 J/mol
Next, convert the frequency factor from s^-1 to s^-1/mol:
A = 8.0 × 10^10 s^-1 * (1 mol/6.022 × 10^23 molecules) = 1.33 x 10^13 s^-1/mol
Now, we can substitute the values into the equation and solve for the rate constant (k) at a specific temperature (298 K):
k = A * e^(-Ea/RT)
R = 8.314 J/(mol·K)
T = 298 K
k = (1.33 x 10^13 s^-1/mol) * e^(-42,000 J/mol / (8.314 J/(mol·K) * 298 K))
Now, let's solve this equation step by step:
1. Calculate the value inside the exponent:
-42,000 J/mol / (8.314 J/(mol·K) * 298 K) ≈ -50.9 mol^-1
2. Calculate the exponent using e (the base of natural logarithm):
e^(-50.9 mol^-1) ≈ 1.4 x 10^-22
3. Now, substitute this value into the equation:
k ≈ (1.33 x 10^13 s^-1/mol) * (1.4 x 10^-22) ≈ 1.9 x 10^-9 s^-1/mol
The rate constant (k) for this reaction at 298 K is approximately 1.9 x 10^-9 s^-1/mol.