The rate constant for decomposition of compound X was determined to be 1.0x10^-4 s-1 at 298 K and 1.0x10^-2 at 365 K. The activation energy for reaction is:
I don't know if this is correct 8.314 x 10^-3 x 298x365 divided by 365 - 298 x In 1.0 x10^-2 divided by 1.0-4. This gives a different answer.
What equation is used and all the maths steps as I really need help with this problem.
The answer should be 6.2 x 10^4 J mol -1
Thank you
You should use the Arrhenius equation. Substitute the numbers and solve Ea.
To determine the activation energy for a reaction, you can use the Arrhenius equation. The Arrhenius equation relates the rate constant (k) to the temperature (T) and activation energy (Ea) of a reaction:
k = A * e^(-Ea / (R * T))
Where:
- k is the rate constant
- A is the pre-exponential factor
- e is the base of the natural logarithm
- Ea is the activation energy
- R is the ideal gas constant (8.314 J/(mol*K))
- T is the temperature in Kelvin
In this case, you have two sets of temperature and rate constant data:
T1 = 298 K, k1 = 1.0x10^-4 s^-1
T2 = 365 K, k2 = 1.0x10^-2 s^-1
To find the activation energy, we need to compare the two equations at different temperatures:
k1 = A * e^(-Ea / (R * T1))
k2 = A * e^(-Ea / (R * T2))
Dividing the second equation by the first equation, we get:
k2 / k1 = e^(-Ea / (R * T2)) / e^(-Ea / (R * T1))
Since e^x / e^y = e^(x - y), we can simplify it as:
k2 / k1 = e^(-Ea / (R * T2) + Ea / (R * T1))
To solve for the activation energy (Ea), we can take the natural logarithm of both sides:
ln(k2 / k1) = -Ea / (R * T2) + Ea / (R * T1)
Rearranging the equation to isolate Ea, we have:
ln(k2 / k1) = Ea / (R * T1 * T2) - Ea / (R * T2^2)
Now we can substitute the given values:
ln(1.0x10^-2 / 1.0x10^-4) = Ea / (8.314 J/(mol*K) * 298 K * 365 K) - Ea / (8.314 J/(mol*K) * (365 K)^2)
Simplifying further:
ln(1.0x10^2) = Ea / (8.314 J/(mol*K) * 298 K * 365 K) - Ea / (8.314 J/(mol*K) * (365 K)^2)
ln(1.0x10^2) = Ea / (8.314 J/(mol*K) * 119,654,760) - Ea / (8.314 J/(mol*K) * 133,225,225)
Solving for Ea:
Ea = (ln(1.0x10^2) * (8.314 J/(mol*K) * 133,225,225) - ln(1.0x10^2) * (8.314 J/(mol*K) * 119,654,760)) / (8.314 J/(mol*K) * 119,654,760)
Calculating this expression gives the activation energy Ea = 6.2 x 10^4 J/mol.
Therefore, the correct answer is indeed 6.2 x 10^4 J/mol.