GATE BT 2019 • Q31 NAT • 1 Mark

Section 4: Bioprocess Engineering and Process Biotechnology

Mass transfer in bioprocesses (kLa)

Subtopic: Fundamentals of Mass transfer in bioprocesses (kLa)

Question Statement:

A pure culture of bacteria with an initial concentration of 1.0 x 10^3 cells/mL doubles every 30 minutes in exponential growth. The bacterial cell concentration after 3 hours of incubation is ________ x 10^3 cells/mL.

Official Answer Key:

16 to 16

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2019 Q.31)** **1. Governing Concept & Definition**: - **Topic**: Microbial Kinetics & Exponential Cell Proliferation - In exponential growth, the cell concentration after $n$ doublings is: $N(t) = N_0 \cdot 2^n = N_0 \cdot 2^{t / t_d}$ **2. Detailed Step-by-Step Calculation**: - **Given**: - Initial concentration $N_0 = 1.0 \times 10^3\text{ cells/mL}$ - Doubling time $t_d = 30\text{ min} = 0.5\text{ h}$ - Elapsed time $t = 2\text{ hours} = 120\text{ min}$ - **Number of Doublings ($n$)**: $n = \frac{120\text{ min}}{30\text{ min}} = 4\text{ doublings}$ - **Final Cell Concentration**: $N = 1.0 \times 10^3 \times 2^4 = 1.0 \times 10^3 \times 16 = 16 \times 10^3\text{ cells/mL}$ - Expressed in units of $10^3\text{ cells/mL}$: $\text{Value} = 16$ **3. Final Answer**: - **Calculated Value**: **16** - **Official GATE Accepted Range**: **16 to 16**

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