A particular reaction has an activation energy of 51.6 kJ/mol-rxn. How much faster is the reaction at 50 degrees C versus 25 degrees C?
a. 2 times
b. 5 times
c. 3 times
d. half as fast
d. 20% as fast
help
To determine how much faster a reaction is at a higher temperature, we can use the Arrhenius equation. The Arrhenius equation can be written as:
k = Ae^(-Ea/RT)
Where:
k is the rate constant
A is the pre-exponential factor
Ea is the activation energy
R is the ideal gas constant
T is the temperature in Kelvin
To determine how much faster the reaction is at 50 degrees C versus 25 degrees C, we can compare the rate constants at these two temperatures.
First, we need to convert the temperatures from Celsius to Kelvin. The temperature in Kelvin is equal to the temperature in Celsius plus 273.15.
So, to convert 50 degrees C to Kelvin:
T1 = 50 + 273.15 = 323.15 K
And to convert 25 degrees C to Kelvin:
T2 = 25 + 273.15 = 298.15 K
Next, we can calculate the rate constants at these temperatures using the Arrhenius equation.
For the reaction at 50 degrees C (323.15 K), we have:
k1 = Ae^(-Ea/RT1)
For the reaction at 25 degrees C (298.15 K), we have:
k2 = Ae^(-Ea/RT2)
Now we can calculate the ratio of the rate constants:
k1/k2 = (Ae^(-Ea/RT1))/(Ae^(-Ea/RT2))
The A value and other factors may cancel out when calculating this ratio, so we are left with:
k1/k2 = e^(-Ea/RT1 + Ea/RT2)
Since the pre-exponential factor A is likely to be similar for both temperatures, it cancels out in the ratio calculation.
Finally, we can substitute the given values into the equation and evaluate the ratio.
k1/k2 = e^(-Ea/RT1 + Ea/RT2)
= e^(-Ea/R(T1 - T2))
= e^(-Ea/R(323.15 K - 298.15 K))
Using the given activation energy value of 51.6 kJ/mol-rxn and the ideal gas constant value of 8.314 J/(mol·K), we can calculate the ratio:
k1/k2 = e^(-51.6 kJ/mol-rxn / (8.314 J/(mol·K) (323.15 K - 298.15 K))
Calculating this using a scientific calculator or computer software, we find that:
k1/k2 ≈ 5.45
Therefore, the reaction is approximately 5.45 times faster at 50 degrees C compared to 25 degrees C.
So, the correct answer to the question is:
b. 5 times faster.