Question

The gas-phase reaction A + B → C + D takes place isothermally at 300 K in a packed-bed reactor in which the feed is equal molar in A and B with CA0 = 0.15 mol/dm3. The reaction is second order in A and zero order in B. Currently, 50% conversion is achieved in a reactor with 100 kg of catalysts for a volumetric flow rate 100 dm3/min. The pressure-drop parameter, α, is α = 0.0099 kg–1. If the activation energy is 10,000 cal/mol, what is the specific reaction rate constant at 450 K?

Answer 1

Step-by-step explanation:To find the specific reaction rate constant at 450 K, we can use the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea) and the temperature (T). The Arrhenius equation is given by:

k = A * exp(-Ea / (R * T))

Where:

k = specific reaction rate constant

A = pre-exponential factor or frequency factor

Ea = activation energy

R = gas constant (8.314 J/(mol·K))

T = temperature in Kelvin

Given:

Initial temperature (T1) = 300 K

Final temperature (T2) = 450 K

To find the specific reaction rate constant at 450 K, we need the value of A. The value of A can be determined using the given data.

Conversion (X) = 50% = 0.5

Feed concentration of A (CA0) = 0.15 mol/dm³

Volumetric flow rate (Q) = 100 dm³/min

Pressure-drop parameter (α) = 0.0099 kg⁻¹

The volume of catalyst (V) can be calculated using the formula:

V = (Q / α) * (1 - X)

Substituting the given values:

V = (100 dm³/min / 0.0099 kg⁻¹) * (1 - 0.5)

V = 10,101 dm³ or 10.101 m³

The amount of catalyst (W) can be calculated using the formula:

W = V * catalyst density

Given that the density of the catalyst is not provided, we cannot determine the exact value of W. However, since we are only interested in the specific reaction rate constant, we can proceed with the calculation.

Now, we can determine the value of A using the equation:

A = k * W * CA0² / X

Substituting the known values:

0.5 = k * W * (0.15 mol/dm³)² / (10.101 m³ * CA0)

Simplifying, we can cancel out the unit of mol/dm³:

0.5 = k * W * 0.15² / (10.101 * 0.15)

0.5 = k * W / 67.34

Now, we can rearrange the equation to solve for A:

A = (0.5 * 67.34) / W

The calculated value of A can be used in the Arrhenius equation to find the specific reaction rate constant at 450 K:

k2 = A * exp(-Ea / (R * T2))

Substituting the given values:

k2 = [(0.5 * 67.34) / W] * exp(-10,000 cal/mol / (8.314 J/(mol·K) * 450 K))

Please note that to obtain consistent units, the activation energy should be converted from calories to joules.

This calculation will yield the specific reaction rate constant (k2) at 450 K, provided that the density of the catalyst (required for calculating W) is given or can be obtained from additional information.