At 425°C, a reaction has a rate constant of 7.7 x 10–4 s–1 and has an activation energy of 275 kJ mol–1. What is the frequency factor?
The answer is 2.9 x 1017 s–1 , I need help with working out I used arrhenius equation and rearranged for A = K /e^(-Ea/RT). My answer comes out too big.
As I understand it, the ‘frequency factor’ does not have units. What you are showing in the posted question is ‘rate constant’ calculated from the Arrhenius Equation. The Arrhenius Equation is derived from the frequency factor. From the KM Theory f = e¯∆E/RT and represents the fraction of molecular collisions having energy values greater than the activation energy. In the posted problem, if ∆E = 275 Kj/mole = 2.75 x 10ˢ J/mole, T = 275⁰C = 698 K and R = 8.314 J/mol·K => ∆E/RT = [(2.75 x 10ˢ J/mole)/(8.314 J/ mol·K)(698 K)] = 47.39. All units cancel out. Substituting into f = e¯∆E/RT => f = e^-(47.39) = 2.63 x 10¯²¹. Also, this type problem usually comes with a couple of temperature values which serve to illustrate – after calculation – the significant increase in number of collisions having energy values greater than the activation energy. Compare frequency at 698 K to frequency at 708 K, a 10⁰ increase in temperature. Frequency at 708 K = e^-(46.72) = 5.13 x 10¯²¹. This represents about a 2-times increase in reaction rate. Hence, the generalization ‘For every 10⁰C increase in temperature, the reaction rate doubles … b/c frequency of collisions greater than the activation energy doubles.
To find the frequency factor (A), you correctly used the Arrhenius equation:
A = k / e^(-Ea/RT)
where:
- A is the frequency factor (in s^(-1))
- k is the rate constant at a specific temperature
- Ea is the activation energy (in J/mol)
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the absolute temperature (in K)
In this case, you have the rate constant (k) and the activation energy (Ea), but you obtained a value for A that seems too big. Let's review the calculations step by step to identify the mistake.
Given:
- k = 7.7 x 10^(-4) s^(-1)
- Ea = 275 kJ/mol
- R = 8.314 J/(mol·K)
First, convert the activation energy from kJ/mol to J/mol:
Ea = 275 kJ/mol x 1000 J/1 kJ
Ea = 275,000 J/mol
Next, convert the activation energy from J/mol to J:
Ea = 275,000 J/mol
Now, let's calculate the frequency factor (A) using the rearranged Arrhenius equation:
A = k / e^(-Ea/RT)
Substituting the known values:
A = (7.7 x 10^(-4) s^(-1)) / e^(-(275,000 J/mol) / (8.314 J/(mol·K) x 425°C + 273.15 K)
Notice that the temperature needs to be converted from Celsius to Kelvin by adding 273.15.
Now we can simplify the equation:
A = (7.7 x 10^(-4) s^(-1)) / e^(-647.054)
Calculating the exponential term:
A = (7.7 x 10^(-4) s^(-1)) / e^(-647.054)
A = (7.7 x 10^(-4) s^(-1)) / 5.827 x 10^(-282)
Finally, divide the rate constant by the exponential term:
A = (7.7 x 10^(-4) s^(-1)) / 5.827 x 10^(-282)
A = 2.9 x 10^(-17) s^(-1)
The correct value for the frequency factor (A) is 2.9 x 10^(-17) s^(-1). Make sure you have entered all the values correctly into your calculations and check for any potential errors in using scientific notation or performing calculations.