The activation energy for a particular reaction is Ea = 14.80 kJ. What percentage of the molecules are at or above the activation energy at 500.0 K?
To determine the percentage of molecules at or above the activation energy, you need to utilize the Boltzmann distribution equation.
The equation is given by:
P(E) = e^(-Ea / (k * T))
Where:
P(E) is the probability of a molecule having energy E or higher,
Ea is the activation energy,
k is the Boltzmann constant (8.617333262145 x 10^(-5) eV/K),
T is the temperature in Kelvin.
To find the percentage of molecules, you will need to calculate the integral of the Boltzmann distribution equation from the activation energy to infinity and divide it by the total integral from zero to infinity.
So, to solve the problem:
1. Convert the activation energy from kJ to eV (electron volts), as the Boltzmann constant is in eV:
Ea_eV = Ea * 1000 / 6.242 x 10^18
2. Convert the temperature from Celsius to Kelvin if necessary:
T_K = 500 + 273.15
3. Calculate the integral of the Boltzmann distribution equation from the activation energy to infinity using:
integral formula = ∫(e^(-E / (k * T)), (Ea_eV, ∞))
4. Calculate the total integral from zero to infinity using:
total integral formula = ∫(e^(-E / (k * T)), (0, ∞))
5. Obtain the percentage by dividing the integral calculated in step 3 by the total integral calculated in step 4 and multiplying by 100.
Please note that step 3 and step 4 require solving definite integrals, which might involve complex calculations. You can use software programs, numerical methods, or built-in integral calculators to perform the calculations.
Finally, the obtained value will represent the percentage of molecules at or above the activation energy at the given temperature.