Section 4: Bioprocess Engineering and Process Biotechnology
Mass transfer in bioprocesses (kLa)
Subtopic: Fundamentals of Mass transfer in bioprocesses (kLa)
Question Statement:
A substrate is consumed in a zero order reaction such that its concentration falls
from 42 g L-1 to 14 g L-1 in 4 hours. The total time taken for complete utilization of
substrate will be ________ hours. (answer in integer)
Official Answer Key:
6 to 6
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.60)**
**1. Governing Concept & Definition**:
- **Topic**: Chemical Reaction Kinetics & Zero-Order Reactions
- For a zero-order reaction, the rate of consumption is constant:
$-\frac{d[S]}{dt} = k \implies [S]_0 - [S]_t = k t \implies k = \frac{[S]_0 - [S]_t}{t}$
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- Initial substrate concentration $[S]_0 = 42\text{ g L}^{-1}$
- Substrate concentration after 4 hours $[S]_4 = 14\text{ g L}^{-1}$
- Elapsed time $t = 4\text{ hours}$
- **Step 1: Calculate reaction rate constant $k$**:
$k = \frac{42 - 14}{4} = \frac{28}{4} = 7\text{ g L}^{-1}\text{ h}^{-1}$
- **Step 2: Calculate time for complete substrate depletion ($[S] = 0$)**:
$t_{\text{total}} = \frac{[S]_0}{k} = \frac{42}{7} = 6\text{ hours}$
**3. Final Answer**:
- **Calculated Value**: **6 hours**
- **Official GATE Accepted Range**: **6 to 6**
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