Q: A reaction rate has a constant of 1.23 x 10-4/s at 28 degrees C and 0.235 at 79 degrees C. Determine the activation barrier for the reaction.
A: My work thus far:
ln(1.23 x 10-4/0.235)=(Ea/8.314)(1/79-1/28)
-7.551=(Ea/8.314)(0.01265-0.03571)
-7.551=(Ea/8.314)(-0.02305)
327.53=(Ea/8.314)
Ea=2,723.08 J/mol
2,723 J/mol = 0.002723 kJ/mol
My answer is incorrect, and I would like to know where I went wrong and what the correct answer is.
Thanks for showing your work.
I didn't work it out but your error is you didn't convert T to kelvin.
28 C = 301.15
79 C = 352.15
To determine the activation barrier for the reaction, you can use the Arrhenius equation:
k = A * exp(-Ea / (R * T))
In this equation, k is the reaction 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.
Given the reaction rate constants at two different temperatures (T1 = 28°C and T2 = 79°C), we can write two equations using the Arrhenius equation:
k1 = A * exp(-Ea / (R * T1))
k2 = A * exp(-Ea / (R * T2))
In this case, k1 = 1.23 x 10^(-4)/s at 28°C, and k2 = 0.235/s at 79°C. We need to solve these two equations to find the activation energy (Ea).
First, let's convert the temperatures from Celsius to Kelvin:
T1 = 28 + 273.15 = 301.15 K
T2 = 79 + 273.15 = 352.15 K
Now, let's divide the two equations:
k1 / k2 = (A * exp(-Ea / (R * T1))) / (A * exp(-Ea / (R * T2)))
k1 / k2 = exp((-Ea / (R * T1)) + (Ea / (R * T2)))
Take the natural logarithm (ln) on both sides of the equation:
ln(k1 / k2) = (-Ea / (R * T1)) + (Ea / (R * T2))
Rearrange the equation:
ln(k1 / k2) = Ea / (R * T2) - Ea / (R * T1)
ln(k1 / k2) = Ea * (1 / (R * T2) - 1 / (R * T1))
ln(k1 / k2) = (Ea / R) * (1 / T2 - 1 / T1)
Now, substitute the values:
ln(1.23 x 10^(-4) / 0.235) = (Ea / (8.314 J/(mol*K))) * (1 / (352.15 K) - 1 / (301.15 K))
Evaluate the expression on the right-hand side of the equation:
ln(1.23 x 10^(-4) / 0.235) = (Ea / 8.314) * (0.002834 - 0.003322)
ln(1.23 x 10^(-4) / 0.235) = (Ea / 8.314) * (-0.000488)
Now, let's solve for Ea:
ln(1.23 x 10^(-4) / 0.235) / (-0.000488) = Ea / 8.314
Ea = (ln(1.23 x 10^(-4) / 0.235) / (-0.000488)) * 8.314
Calculating this will give you the correct activation energy (Ea).