The presence of a catalyst provides a reaction pathway in which the activation energy of a certain reaction is reduced from 125 kJ·mol-1 to 75 kJ·mol-1.
(a) By what factor does the rate of the reaction increase at 293 K, all other factors being equal?
(b) By what factor would the rate change if the reaction were carried out at 339 K instead?
To answer both questions, we need to understand the relationship between activation energy and rate of reaction using the Arrhenius equation:
Rate = A * e^(-Ea/RT)
where:
- Rate is the rate of reaction
- A is the pre-exponential factor or frequency factor
- Ea is the activation energy
- R is the ideal gas constant
- T is the temperature in Kelvin
(a) To find the factor by which the rate of reaction increases at 293 K when the activation energy changes from 125 kJ·mol-1 to 75 kJ·mol-1, we can compare the rate equations at each activation energy.
Rate1 = A * e^(-125000 J/mol / (R * 293 K))
Rate2 = A * e^(-75000 J/mol / (R * 293 K))
To find the factor by which the rate increases, we can divide Rate2 by Rate1:
Factor = Rate2 / Rate1
= [A * e^(-75000 J/mol / (R * 293 K))] / [A * e^(-125000 J/mol / (R * 293 K))]
Note that the pre-exponential factor A cancels out.
Factor = e^((-75000 J/mol + 125000 J/mol) / (R * 293 K))
= e^(-50000 J/mol / (R * 293 K))
To evaluate this, we need to know the value of the ideal gas constant R (8.314 J/(mol·K)). Plugging this in, we get:
Factor = e^(-50000 J/mol / (8.314 J/(mol·K) * 293 K))
= e^(-50000 / (8.314 * 293))
Now you can use a calculator or software to calculate the value of this exponential to find the factor by which the rate of reaction increases.
(b) To find the factor by which the rate changes if the reaction is carried out at 339 K instead of 293 K, we can compare the rate equations at each temperature.
Rate1 = A * e^(-75000 J/mol / (R * 339 K))
Rate2 = A * e^(-75000 J/mol / (R * 293 K))
To find the factor by which the rate changes, we divide Rate2 by Rate1:
Factor = Rate2 / Rate1
= [A * e^(-75000 J/mol / (R * 293 K))] / [A * e^(-75000 J/mol / (R * 339 K))]
Once again, the pre-exponential factor A cancels out.
Factor = e^((-75000 J/mol / (R * 293 K)) + (75000 J/mol / (R * 339 K)))
= e^(75000 J/mol * (1 / (R * 339 K - R * 293 K)))
Using the value of the ideal gas constant R (8.314 J/(mol·K)), we can simplify further:
Factor = e^(75000 J/mol / (8.314 J/(mol·K) * (339 K - 293 K)))
= e^(75000 / (8.314 * 46))
Again, you can use a calculator or software to calculate the value of this exponential to find the factor by which the rate changes.
I would use the Arrhenius equation, Ea = 125 kJ/mol, T1 = 273, T2 = 293, k1 = a convenient number such as 10, solve for k2. Then do the same but this time make Ea 125 kJ/mol (remember to change to J/mol in the Arrhenius equation), and re-solve for k2. Then you can find the factor between the two values for k2.
b is done the same way.