Question

The maximum allowable temperature for a reactor is 800 K. At present our operating set point is 780 K, the 20-K margin of safety to account for fluctuating feed, sluggish controls, etc. Now with a more sophisticated control system we would be able to raise our set point to 792 K with the same margin of safety that we now have. By how much can the reaction rate, hence the production rate be raised by this change if the reaction taking place in the reactor has an activation energy of 175 kJ/mol? (R= 8.3145 J.mol⁻¹.K⁻¹)

Answer 1

The change in rate constant can be calculated using the Arrhenius equation. By rearranging the equation and plugging in the given values, we find that the change in rate constant is approximately 2.44%. Therefore, the production rate can be increased by approximately 2.44%.

Step-by-step explanation:

To calculate the change in production rate, we need to use the Arrhenius equation which relates the reaction rate to the temperature. The Arrhenius equation is given as:

k = Ae^(-Ea/RT)

where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.

By rearranging the equation, we can solve for the change in rate constant:

ln(k2/k1) = Ea/R * (1/T2 - 1/T1)

where k2 is the rate constant at the new temperature, k1 is the rate constant at the original temperature, T2 is the new temperature, and T1 is the original temperature.

Plugging in the given values into the equation, we have:

ln(k2/k1) = (175,000 J/mol) / (8.3145 J/(mol.K)) * (1/800 K - 1/780 K)

Solving for ln(k2/k1), we find:

ln(k2/k1) = 0.024118

Using the exponential function, we can find k2/k1:

e^(ln(k2/k1)) = e^(0.024118)

k2/k1 ≈ 1.0244

Therefore, the change in rate constant is approximately 2.44%. Since the reaction rate is directly proportional to the rate constant, the production rate can be increased by approximately 2.44%.