If a temperature increase from 19.0 degrees C to 36.0 degrees C 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:
k = A * exp(-Ea / (R * T))
where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant (8.314 J/(mol·K))
T = absolute temperature (in Kelvin)
Given that the rate constant triples when the temperature increases from 19.0 degrees Celsius to 36.0 degrees Celsius, we can use this information to find the activation energy (Ea).
Let's follow these steps to calculate Ea:
Step 1: Convert the given temperatures from Celsius to Kelvin.
T1 = 19.0 + 273.15 = 292.15 K
T2 = 36.0 + 273.15 = 309.15 K
Step 2: Set up a ratio of the rate constants at two temperature points.
k1 / k2 = exp((Ea / R) * ((1 / T2) - (1 / T1)))
Step 3: Substitute the known values into the equation.
3 / 1 = exp((Ea / (8.314 J/(mol·K))) * ((1 / 309.15 K) - (1 / 292.15 K)))
Step 4: Solve for Ea.
Taking the natural logarithm (ln) of both sides of the equation, we can isolate Ea:
ln(3) = (Ea / (8.314 J/(mol·K))) * ((1 / 309.15 K) - (1 / 292.15 K))
Now, rearranging the equation, we find:
Ea = (ln(3) * 8.314 J/(mol·K)) / ((1 / 309.15 K) - (1 / 292.15 K))
Evaluating this expression will give us the value of the activation barrier (Ea).