181. In a CSTR, first order reaction takes place converting A to B. If at a dilution rate (D)= 0.5 h-1 , 50% of A gets
converted to B, then the rate constant ‘k’ of the reaction is:
1. 0.5 h-1
2. 1.0 h-1
3. 0.25 h-1
4. 2.0 h-1

Introduction:

Continuous Stirred Tank Reactors (CSTRs) are commonly used in chemical and biochemical processes where reactions occur at steady state. In this scenario, a first-order reaction is converting reactant A into product B. The relationship between the dilution rate (D) and the rate constant (k) plays a crucial role in determining how effectively the reaction proceeds.

This article explains how to calculate the rate constant for a first-order reaction in a CSTR when the dilution rate and the conversion of the reactant are provided.

Problem Overview:

Given a CSTR with a first-order reaction where A is converted to B, the following data is provided:

• Dilution rate (D): 0.5 h⁻¹

• Conversion of A to B: 50%

We need to calculate the rate constant k for the reaction.

Key Concept:

For a first-order reaction, the rate of reaction is proportional to the concentration of reactant A. The general rate equation for a first-order reaction is:

d[A]dt=−k[A]\frac{d[A]}{dt} = -k[A]dtd[A]​=−k[A]

Where:

• k is the rate constant (in h⁻¹),

• [A] is the concentration of reactant A.

In a CSTR, the dilution rate (D) is defined as the volumetric flow rate divided by the volume of the reactor. At steady state, the rate of reaction is balanced by the inflow and outflow of the reactants, leading to the following relationship for a first-order reaction:

$$D=k(1-X)$$

Where:

• D is the dilution rate (h⁻¹),

• k is the rate constant (h⁻¹),

• X is the conversion (fraction of A converted to B).

Step-by-Step Calculation:

1. Given:

   • Dilution rate (D) = 0.5 h⁻¹

   • Conversion of A to B (X) = 50% = 0.5

2. Substitute the given values into the equation:

$$0.5=k(1-0.5)$$ $$0.5=k(0.5)$$

3. Solve for k:

k=0.50.5=1.0 h−1k = \frac{0.5}{0.5} = 1.0 \, \text{h}^{-1}k=0.50.5​=1.0h−1

Conclusion:

The rate constant k for the first-order reaction in the CSTR is 1.0 h⁻¹.

Answer:

The correct answer is 2. 1.0 h⁻¹.