Final answer:
To determine how much faster the reaction occurs at 343K versus 314K, the rate constants at both temperatures are calculated using the Arrhenius equation and their ratio provides the rate enhancement factor.
Step-by-step explanation:
To determine how much faster a reaction progresses at 343K compared to 314K, we can use the Arrhenius equation which describes the temperature dependence of the reaction rate. The equation is given by k = Ae-Ea/RT, where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the ideal gas constant, and T is the temperature in Kelvin.
First, we must convert the activation energy given in kJ/mol to J/mol by multiplying it with 1000, since 1 kJ = 1000 J. For the specified reaction, Ea = 42.2 kJ/mol or 42200 J/mol. Using the Arrhenius equation, we can calculate the rate constants k1 and k2 at temperatures 314K and 343K respectively.
For T1 = 314K: k1 = Ae-Ea/(R*T1)
For T2 = 343K: k2 = Ae-Ea/(R*T2)
By taking the ratio of k2 to k1, we can find out how many times faster the reaction is at the higher temperature:
- Rate enhancement = k2/k1 = (Ae-Ea/(R*T2))/(Ae-Ea/(R*T1))
- Rate enhancement = e(Ea/R)(1/T1 - 1/T2)
Finally, plug in the values of Ea, R, T1, and T2 to calculate the rate enhancement:
- Rate enhancement = e(42200 J/mol)/(8.314 J/mol*K)(1/314K - 1/343K)
- Calculate the value of the exponent and then the value of e to the power of that exponent to find the rate enhancement factor.