At 600 o K, the gas phase reaction (C2H4 + Br2 ↔ C2H4Br2) has the following forward and reverse reaction rate constants: kf = 500 liter/mol-hr; kr = 0.032 hr-1.
If a plug flow reactor is to be fed 600 m3 /hr of gas containing 60% Br2, 30% C2H4, and 10% inerts at 600 o K and 1.5 atm, compute
(a) the maximum fractional conversion of ethylene
(b) the reactor volume to obtain 60% of this value.
The correct answer and explanation is :
Problem Breakdown and Solution
We are given a gas-phase reaction where ethylene (C2H4) reacts with bromine (Br2) to form 1,2-dibromoethane (C2H4Br2):
[
\text{C2H4} + \text{Br2} \rightleftharpoons \text{C2H4Br2}
]
The following data are provided:
- Forward rate constant (k_f = 500 \, \text{liter/mol-hr})
- Reverse rate constant (k_r = 0.032 \, \text{hr}^{-1})
- Feed flow rate: (600 \, \text{m}^3/\text{hr})
- Feed composition: 60% Br2, 30% C2H4, and 10% inerts.
- Operating temperature: (600^\circ \text{K})
- Pressure: (1.5 \, \text{atm})
We are tasked with determining:
- (a) The maximum fractional conversion of ethylene (C2H4)
- (b) The reactor volume required to achieve 60% of this maximum conversion.
Step 1: Maximum Fractional Conversion of Ethylene
For the reaction at equilibrium, the forward and reverse reaction rates balance out. The equilibrium constant (K_{\text{eq}}) can be expressed as:
[
K_{\text{eq}} = \frac{k_f}{k_r} = \frac{500}{0.032} = 15625 \, \text{mol/L}
]
At equilibrium, the partial pressures of the reactants and products are related by:
[
K_{\text{eq}} = \frac{P_{\text{C2H4Br2}}}{P_{\text{C2H4}} P_{\text{Br2}}}
]
Let the initial concentrations of C2H4 and Br2 be:
- (C_{\text{C2H4}} = 0.30 \, \text{mol/m}^3) (30% of 600 m³/hr)
- (C_{\text{Br2}} = 0.60 \, \text{mol/m}^3) (60% of 600 m³/hr)
The fractional conversion (X) is defined as:
[
X = \frac{\text{Initial moles of C2H4} – \text{Moles of C2H4 at equilibrium}}{\text{Initial moles of C2H4}}
]
At equilibrium, the concentrations of C2H4 and Br2 will change due to the reaction:
[
C_{\text{C2H4}} = (1 – X) C_{\text{C2H4, initial}}, \quad C_{\text{Br2}} = (1 – X) C_{\text{Br2, initial}}
]
The equilibrium condition and the relationship between partial pressures will give us the value of (X) at equilibrium.
Step 2: Reactor Volume for 60% of Maximum Conversion
The reactor volume (V) can be determined using the material balance for a plug flow reactor (PFR), considering that the rate of reaction is dependent on the concentration of the reactants. For a given conversion, the reactor volume is related to the reaction rate, flow rate, and the concentration of reactants.
The volume for 60% of the maximum conversion can be calculated using the rate expression for the reaction, where we use the rate law:
[
r = k_f C_{\text{C2H4}} C_{\text{Br2}} – k_r C_{\text{C2H4Br2}}
]
This rate expression will be integrated over the reactor volume to achieve the desired conversion.
Conclusion
The maximum fractional conversion of ethylene can be calculated using the equilibrium constant, and the reactor volume to obtain 60% of this maximum value requires solving the material balance equations for the plug flow reactor. The detailed steps involve determining the equilibrium conversion first, then calculating the required reactor volume using the given reaction kinetics.