GATE BT 2017 • Q39 NAT • 2 Marks

Section 4: Bioprocess Engineering and Process Biotechnology

Mass transfer in bioprocesses (kLa)

Subtopic: Fundamentals of Mass transfer in bioprocesses (kLa)

Question Statement:

For an E. coli culture in the exponential phase of growth, optical density (OD) at 540 nm is 0.3 at 2 hours and 0.6 at 4 hours. Assuming that the measured OD is linearly proportional to the number of E. coli cells, the growth rate (per hour) for this culture is

Official Answer Key:

0.34 to 0.35

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2017 Q.39)** **1. Governing Concept & Definition**: - **Topic**: Microbial Growth & Specific Growth Rate - In exponential growth: $\ln\left(\frac{OD_2}{OD_1}\right) = \mu \cdot \Delta t \implies \mu = \frac{\ln(OD_2 / OD_1)}{\Delta t}$ **2. Detailed Step-by-Step Calculation**: - **Given**: - $OD_1 = 0.3$ at $t_1 = 0$ - $OD_2 = 0.6$ at $t_2 = 2\text{ hours}$ - **Specific Growth Rate**: $\mu = \frac{\ln(2)}{2\text{ h}} = \frac{0.69315}{2} \approx 0.3465\text{ h}^{-1} \approx 0.35\text{ h}^{-1}$ **3. Final Answer**: - **Calculated Value**: **0.35 h⁻¹** - **Official GATE Accepted Range**: **0.34 to 0.35**

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