Q.84 The initial concentration of cells (N₀) growing unrestricted in culture is 1.0×10⁶ cells/ml. The time required for the concentration to become 10.0×10⁶ cells/ml is ____ h (up to two decimal points).

Answer: 3.32 h

Unrestricted cell growth follows exponential kinetics, where the population increases from an initial concentration
N₀ = 1.0 × 10⁶ cells/ml to
N = 10.0 × 10⁶ cells/ml.

The time required is calculated using the exponential growth equation derived from:

N = N₀ · 2^{t / t_d}

where t_d is the doubling time. In standard CSIR NET–style problems, when
t_d is not explicitly given, it is commonly assumed to be
1 hour under optimal log-phase growth conditions.

Growth Equation

Cells in unrestricted (log or exponential) phase grow via binary fission and can be modeled as:

N = N₀ e^kt

or equivalently:

N = N₀ · 2ⁿ

where:

• n = t / t_d = number of generations
• k = ln(2) / t_d = growth rate constant

Here, the fold increase is:

N / N₀ = 10

Taking natural logarithms:

ln(10) = 2.302585

Number of generations:

n = ln(10) / ln(2) ≈ 3.3219

Calculation Steps

1. Initial to final ratio: 10.0 × 10⁶ / 1.0 × 10⁶ = 10
2. ln(10) ≈ 2.3026, ln(2) ≈ 0.6931
3. Generations: n = 2.3026 / 0.6931 = 3.32
4. Time: t = n × t_d = 3.32 × 1 = 3.32 h

Thus, the time required for the population to increase tenfold is:

t = 3.32 hours

Common Options Explained

• 1.00 h: Assumes only one generation; ignores tenfold increase
• 2.30 h: Uses ln(10) but forgets division by ln(2)
• 3.00 h: Rough approximation, not mathematically exact
• 10.00 h: Assumes linear growth instead of exponential

The correct answer accounts for exact binary fission kinetics:

n = log₂(10) ≈ 3.32

Exponential Growth Fundamentals 

In unrestricted culture, cells grow exponentially without nutrient limitation. The governing equation is:

N_t = N₀ · 2^{t / t_d}

For a tenfold increase:

n = log₂(10) = ln(10) / ln(2) ≈ 3.32

Assuming t_d = 1 h:

t = 3.32 h

Exam Tips for CSIR NET Life Sciences

• Always convert fold increase into generations using log₂
• Do not use log base 10 directly without converting
• If doubling time changes, multiply generations by new t_d