A certain reaction has an activation energy of 27.34 kJ/mol. At what Kelvin temperature will the reaction proceed 3.50 times faster than it did at 335 K?
Use the Arrhenius equation and set k1 = k1(T1 = 335K) and k2 = 3.5k1 (T2 = ?)
what is the arrhenius equation?
To find the Kelvin temperature at which the reaction proceeds 3.50 times faster than it did at 335 K, we can use the Arrhenius equation, which relates the rate constant of a reaction with the activation energy and temperature.
The Arrhenius equation is given by:
k = A * e^(-Ea/RT)
where:
k is the rate constant of the reaction
A is the pre-exponential factor (the frequency factor)
Ea is the activation energy of the reaction
R is the gas constant (8.314 J/(mol*K))
T is the temperature in Kelvin
In this case, we want to compare the rate constant at two different temperatures. Let's call the rate constant at 335 K as k1, and the rate constant at the unknown temperature (T) as k2.
We know that the reaction is 3.50 times faster at T compared to 335 K. Mathematically, this can be written as:
k2 = 3.50 * k1
Now, let's substitute the equation for k1 and k2 into the Arrhenius equation:
3.50 * k1 = A * e^(-Ea/(R * 335 K))
k2 = A * e^(-Ea/(R * T))
Since both sides of the equation have A in common, we can cancel it out:
3.50 = e^(-Ea/(R * 335 K)) / e^(-Ea/(R * T))
Next, we can take the natural logarithm (ln) of both sides of the equation:
ln(3.50) = ln(e^(-Ea/(R * 335 K)) / e^(-Ea/(R * T)))
Using the properties of logarithms, we can rewrite the equation:
ln(3.50) = (-Ea/(R * 335 K)) - (-Ea/(R * T))
Simplifying further:
ln(3.50) = Ea/(R * 335 K) - Ea/(R * T)
Now, we can rearrange the equation to solve for the unknown temperature, T:
ln(3.50) = Ea/R * (1/335 K - 1/T)
To find T, we need to rearrange the equation as follows:
1/(335 K) - 1/T = ln(3.50) * R / Ea
Now, plug in the known values:
R = 8.314 J/(mol*K)
Ea = 27.34 kJ/mol (convert it to Joules by multiplying with 1000)
1/(335 K) - 1/T = ln(3.50) * 8.314 J/(mol*K) / (27.34 kJ/mol * 1000)
Simplify and solve for 1/T:
1/(335 K) - 1/T = ln(3.50) * 8.314 J / (k * mol) / (27.34 * 10^3 J / mol)
Now, you can plug in the values to calculate the 1/T term. Take the reciprocal of this value to find T in Kelvin.