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.
To determine the activation barrier for the reaction, we can use the Arrhenius equation, which relates the rate constant of a reaction with the temperature and the activation energy. The equation is as follows:
k = A * exp(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant (8.314 J/mol·K)
T = temperature in Kelvin
In this case, we are given two sets of rate constant values at different temperatures, 28°C (301 K) and 79°C (352 K). We can set up two equations with the given data:
(1) 1.23 x 10^(-4) /s = A * exp(-Ea / (8.314 J/mol·K * 301 K))
(2) 0.235 /s = A * exp(-Ea / (8.314 J/mol·K * 352 K))
To solve for Ea, we can take the ratio between these two equations:
(1.23 x 10^(-4) /s) / (0.235 /s) = exp(-Ea / (8.314 J/mol·K) * (301 K - 352 K))
Simplifying:
5.23404 x 10^(-4) = exp(Ea / 8.314 * (-51 K))
Take the natural logarithm (ln) of both sides:
ln(5.23404 x 10^(-4)) = Ea / 8.314 * (-51 K)
Rearranging the equation, we get:
Ea = -8.314 * 51 K * ln(5.23404 x 10^(-4))
Evaluating the expression:
Ea ≈ 34,065.64 J/mol
So the correct answer for the activation barrier is approximately 34,065.64 J/mol or 34.07 kJ/mol.