The rate constants for a reaction are found to be
k(1431 C) = 19.4 x 10 -6 s-1
k(1897 C) = 25.2 x 10 -5 s-1
At what temperature (in degrees C) will the rate constant be 74.7*10-2 s-1?
To find the temperature at which the rate constant is 74.7 x 10^-2 s^-1, we need to use the Arrhenius equation:
k = A * e^(-Ea/RT)
where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin.
First, let's convert the given temperatures from degrees Celsius to Kelvin:
1431°C = 1431 + 273 = 1704 K
1897°C = 1897 + 273 = 2170 K
Next, we can use the two given rate constants to create two separate equations of the Arrhenius equation:
k1 = A * e^(-Ea/RT1)
k2 = A * e^(-Ea/RT2)
We can divide both equations to eliminate A:
k1 / k2 = (A * e^(-Ea/RT1)) / (A * e^(-Ea/RT2))
k1 / k2 = e^(-Ea/RT1 + Ea/RT2)
Now, rearrange the equation to isolate the exponential terms:
ln(k1 / k2) = -Ea/R * (1/RT1 - 1/RT2)
Now, substitute the given values:
ln(19.4 x 10^-6 s^-1 / 25.2 x 10^-5 s^-1) = -Ea/(8.314 J/(mol·K)) * (1/(1704 K) - 1/(2170 K))
Now, solve for Ea:
Ea = -(8.314 J/(mol·K)) * ln(19.4 x 10^-6 s^-1 / 25.2 x 10^-5 s^-1) / (1/(1704 K) - 1/(2170 K))
Once we have Ea, we can rearrange the Arrhenius equation to solve for the temperature T:
T = (-Ea / (8.314 J/(mol·K))) / ln(k / A)
Substituting the given values:
T = (-Ea / (8.314 J/(mol·K))) / ln(74.7 x 10^-2 s^-1 / A)
Finally, substitute Ea into the equation to find the temperature T.