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
think it has something to do with the Arrhenius equation but i have no idea how to do this?
Look up the Arrhenius equation, substitute the numbers and solve. It's that simple.
I get the answer 5 times. Is that correct?
I got 2 times
To determine how much faster a reaction is at a certain temperature compared to another temperature, you can use the Arrhenius equation. The Arrhenius equation relates the rate constant of a reaction to the activation energy and temperature. The equation is as follows:
k = Ae^(-Ea/RT)
Where:
- k is the rate constant of the reaction
- A is the pre-exponential factor or frequency factor, which represents the likelihood of successful collisions between reactant molecules
- Ea is the activation energy of the reaction in joules/mol
- R is the gas constant (8.314 J/(mol⋅K))
- T is the absolute temperature in Kelvin
First, convert the activation energy from kilojoules to joules by multiplying it by 1000: 51.6 kJ/mol = 51,600 J/mol.
Now, let's calculate the rate constant at 50 degrees C (323 Kelvin) and 25 degrees C (298 Kelvin).
For 50 degrees C (323 K):
k1 = Ae^(-Ea/RT1)
For 25 degrees C (298 K):
k2 = Ae^(-Ea/RT2)
To find the ratio of the rates (k1/k2), divide equation 1 by equation 2:
k1/k2 = (Ae^(-Ea/RT1))/(Ae^(-Ea/RT2))
The pre-exponential factor (A) and the activation energy (Ea) are the same for both reactions, so they cancel out:
k1/k2 = e^(-Ea/RT1)/e^(-Ea/RT2)
Simplifying further:
k1/k2 = e^(-Ea/RT1 + Ea/RT2)
Now, substitute the given temperatures into the equation and simplify:
k1/k2 = e^(-Ea(1/T1 - 1/T2)/R)
k1/k2 = e^(-51,600(1/323 - 1/298)/(8.314))
Using a calculator, evaluate e^(-51,600(1/323 - 1/298)/(8.314)) = 2.26
The ratio of the rates is approximately 2.26.
Since the question asks for how much faster the reaction is, we can round 2.26 to the nearest whole number, which is 2.
Therefore, the answer to the question is a. 2 times.