This chemistry question involves key chemical concepts and calculations. The detailed solution below walks through each step, from identifying the reaction type to computing the final answer.
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Here's the solution for Question 2, covering both parts. 2.1. Set up a stoichiometric table and express concentrations. The reactions are: 1. A → B (desired) 2. A → C (undesired) Given that the reactions are elementary and the rate constant k_B = 2 dm^3/mol · min (units for a second-order reaction) and k_C = 1 min^-1 (units for a first-order reaction), we assume the reaction orders are dictated by these units. Therefore: Rate of formation of B: r_B = k_B C_A^2 Rate of formation of C: r_C = k_C C_A The total rate of disappearance of A is: -r_A = -r_A1 + (-r_A2) = k_B C_A^2 + k_C C_A The reactor operates isothermally and with constant density ( = _0). Let X be the overall conversion of A. Let X_B be the fractional conversion of A to B, and X_C be the fractional conversion of A to C. The total conversion of A is X = X_B + X_C. Stoichiometric Table (in terms of concentrations): | Component | Initial Concentration (C_i0) | Change in Concentration | Final Concentration (C_i) | | :-------- | :------------------------------- | :---------------------- | :-------------------------- | | A | C_A0 | -C_A0X | C_A = C_A0(1-X) | | B | 0 | +C_A0X_B | C_B = C_A0X_B | | C | 0 | +C_A0X_C | C_C = C_A0X_C | The concentrations as a function of overall conversion X and initial concentration C_A0 are: C_A = C_A0(1-X) C_B = C_A0X_B C_C = C_A0X_C 2.2. Evaluate whether a CSTR or PFR will provide a higher yield of B. To determine which reactor provides a higher yield of B, we need to calculate the yield for each reactor type at 90% conversion of A (X=0.9). The yield of B (Y_B) is defined as the moles of B formed per mole of A fed, which is Y_B = (C_B)/(C_A0) = X_B. Given values: C_A0 = 4 mol/dm^3 X = 0.9 k_B = 2 dm^3/mol · min k_C = 1 min^-1 The instantaneous selectivity of B with respect to A reacted is: S_BA = (r_B)/(-r_A) = (k_B C_A^2)/(k_B C_A^2 + k_C C_A) = (k_B C_A)/(k_B C_A + k_C) Since S_BA depends on C_A, it is not constant. To maximize S_BA, we need to maximize C_A. For a Plug Flow Reactor (PFR): Step 1: Calculate the conversion of A to B (X_B) for a PFR. For a PFR, the differential conversion of B is related to the instantaneous selectivity: dX_B = S_BA dX. X_B = _0^X S_BA dX = _0^X (k_B C_A)/(k_B C_A + k_C) dX Substitute C_A = C_A0(1-X): X_B = _0^X k_B C_A0(1-X)k_B C_A0(1-X) + k_C dX Plug in the given values: k_B = 2, k_C = 1, C_A0 = 4, and X = 0.9. X_B = _0^0.9 (2 × 4 (1-X))/(2 × 4 (1-X) + 1) dX = _0^0.9 (8(1-X))/(8(1-X) + 1) dX Let u = 1-X, so du = -dX. The limits change from X=0 u=1 to X=0.9 u=0.1. X_B = _1^0.1 (8u)/(8u+1) (-du) = _0.1^1 ( 1 - (1)/(8u+1) ) du X_B = [ u - (1)/(8) (8u+1) ]_0.1^1 X_B = ( 1 - (1)/(8) (8(1)+1) ) - ( 0.1 - (1)/(8) (8(0.1)+1) ) X_B = ( 1 - (1)/(8) (9) ) - ( 0.1 - (1)/(8) (1.8) ) X_B = 0.9 - (1)/(8) ( 9 - 1.8) = 0.9 - (1)/(8) ( (9)/(1.8) ) X_B = 0.9 - (1)/(8) (5) X_B = 0.9 - (1.6094379)/(8) = 0.9 - 0.2011797 = 0.69882 The yield of B for a PFR is Y_B^PFR = 0.6988. For a Continuous Stirred-Tank Reactor (CSTR): Step 2: Calculate the conversion of A to B (X_B) for a CSTR. In a CSTR, the concentration of A is uniform throughout the reactor and equal to the outlet concentration. C_A = C_A0(1-X) = 4 mol/dm^3 (1-0.9) = 4 × 0.1 = 0.4 mol/dm^3 The instantaneous selectivity S_BA is constant throughout the CSTR, evaluated at this outlet concentration: S_BA^CSTR = (k_B C_A)/(k_B C_A + k_C) = (2 × 0.4)/(2 × 0.4 + 1) = (0.8)/(0.8 + 1) = (0.8)/(1.8) = (4)/(9) For a CSTR, the overall conversion of A to B is X_B = X × S_BA^CSTR. X_B = 0.9 × (4)/(9) = 0.4 The yield of B for a CSTR is Y_B^CSTR = 0.4000. Conclusion: Step 3: Compare the yields. Comparing the yields: Y_B^PFR = 0.6988 Y_B^CSTR = 0.4000 Since 0.6988 > 0.4000, the Plug Flow Reactor (PFR) will provide a higher yield of B. This is because the selectivity to B increases with higher concentrations of A, and a PFR maintains a higher average concentration of A throughout the reactor compared to a CSTR for the same overall conversion. Last free one today — make it count tomorrow, or type /upgrade for unlimited.