i'm stumped on how to write this in mathematical form. it says the rate increased by a factor of 2000, I am supposed to determine by what percent the Energy of activation will be reduced. The answer is 25% in the book. I don't know how they get the answer using this formula: k1/k2=Ae^-Ea/RT. A is constant for both reactions. T = 298 K, R = 8.3145
I am not sure, and someone can come along after I am done and check what I am telling you, but I believe you are missing some information like the temperature change. The question seems to be incomplete.
yeah I'm not sure...an example in the book just equates them, its for a catalyzed reaction by saying k1/k2=(Ae^-Ea/RT)/(Ae^-Ea/Rt)=e^-Ea/RT but I'm not sure about that way.
Is this a multipart problem; i.e., is there another part of the problem in which Ea is calculated? or given? If so, what is the value of Ea in that part of the problem?
no they didn't give anything else
To determine the percent reduction in energy of activation, let's break down the steps using the given formula:
1. Start by considering two reactions, let's call them Reaction 1 and Reaction 2.
2. The given information states that the rate of Reaction 2 increased by a factor of 2000 compared to Reaction 1. Mathematically, this can be expressed as:
Rate of Reaction 2 = 2000 * Rate of Reaction 1
3. Now, let's apply the formula k1/k2 = Ae^(-Ea/RT), where k1 and k2 represent the rate constants of Reaction 1 and Reaction 2, respectively. A is a constant for both reactions, T is the temperature (298 K), R is the gas constant (8.3145 J/(mol*K)), and Ea is the energy of activation.
4. Substituting the given information into the formula, we have:
Rate of Reaction 1 / Rate of Reaction 2 = Ae^(-Ea/RT) / (2000 * Ae^(-Ea/RT))
5. We can simplify the expression by canceling out the A's, resulting in:
1 / 2000 = e^(-Ea/RT) / e^(-Ea/RT)
6. Since the exponential functions on both sides have the same base (e), the exponents must be equal. Therefore:
-Ea/RT = -Ea/RT
7. From here, we can solve for the energy of activation Ea. Divide both sides of the equation by -RT:
Ea = Ea / 2000
8. Now, to determine the reduction in the energy of activation, we calculate the difference:
Reduction in energy of activation = Ea - Ea / 2000
9. Simplifying the equation, we have:
Reduction in energy of activation = Ea(1 - 1/2000)
10. To express the reduction as a percentage, we divide the reduction in energy of activation by the original energy of activation and multiply by 100:
Percent reduction = Reduction in energy of activation / Ea * 100
11. Substituting the simplified equation from step 9, we get:
Percent reduction = (Ea(1 - 1/2000)) / Ea * 100
12. Canceling out the Ea's, we obtain:
Percent reduction = (1 - 1/2000) * 100
13. Evaluating the expression, we find that the percent reduction is 99.95%.
Therefore, it seems there might be an error in the book. The correct percent reduction in the energy of activation should be approximately 99.95% instead of 25%.