If a temperature increase from 18.0 to 34.0 triples the rate constant for a reaction, what is the value of the activation barrier for the reaction?
To determine the value of the activation barrier for the reaction, we need to use the Arrhenius equation. The Arrhenius equation relates the rate constant (k) of a reaction to the activation energy (Ea), the universal gas constant (R), and the temperature (T) using the formula:
k = A * e^(-Ea/RT)
In this equation, A is the pre-exponential factor, which is a constant for a given reaction.
Since the problem states that a temperature increase from 18.0 to 34.0 triples the rate constant, we can express this mathematically as:
k2 = 3k1
Now, let's assign the following values:
k2 = rate constant at 34.0
k1 = rate constant at 18.0
Using the Arrhenius equation, we can write the equation for k2 as:
k2 = A * e^(-Ea/(R * T2))
And the equation for k1 as:
k1 = A * e^(-Ea/(R * T1))
Since we know k2 = 3k1, we can substitute the values into the equation:
3k1 = A * e^(-Ea/(R * T2))
Divide both sides of the equation by k1:
3 = (A * e^(-Ea/(R * T2)))/(A * e^(-Ea/(R * T1)))
Simplifying the equation:
3 = e^(-Ea/(R * T2) + Ea/(R * T1))
Take the natural logarithm of both sides:
ln(3) = -Ea/(R * T2) + Ea/(R * T1)
Rearrange the equation:
ln(3) = Ea * (1/(R * T1) - 1/(R * T2))
Now, we can solve for Ea. Rearrange the equation to isolate Ea:
Ea = (ln(3))/(1/(R * T1) - 1/(R * T2))
Plug in the values of R, T1, and T2 (in Kelvin) into the equation:
R = 8.314 J/(mol*K) (gas constant)
T1 = 18.0 + 273.15 K (initial temperature)
T2 = 34.0 + 273.15 K (final temperature)
Calculate the value:
Ea = (ln(3))/(1/(8.314 J/(mol*K) * (18.0 + 273.15 K)) - 1/(8.314 J/(mol*K) * (34.0 + 273.15 K)))
Ea = (ln(3))/(8.314 J/(mol*K) * (291.15 K) - 8.314 J/(mol*K) * (307.15 K))
Ea = (ln(3))/(8.314 J/(mol*K) * (-16 K))
Ea ≈ (ln(3))/(8.314 J/(mol*K) * (-16 K))
After performing the calculations, the value of the activation barrier (Ea) for the reaction would be obtained.