Bacterial exponential growth law: if the specific growth rate μ is constant during the exponential phase, how is cell number concentration Cn related to time (with initial value Cn0 at t0)?

Introduction / Context:
During exponential growth, cell populations increase at a rate proportional to their current size. This yields an exponential law widely used to estimate doubling times and to model bioreactor start-up phases. The question asks for the correct mathematical form for cell number concentration Cn over time at constant μ.

Given Data / Assumptions:

• Specific growth rate μ is constant.
• Initial condition: Cn = Cn0 at t = t0.
• No limitations (substrate, oxygen) during the interval.

Concept / Approach:
The differential equation is dCn/dt = μ * Cn. Separating variables and integrating between t0 and t yields ln(Cn/Cn0) = μ * (t − t0), hence Cn = Cn0 * exp[μ * (t − t0)].

Step-by-Step Solution:

Verification / Alternative check:
Doubling time td satisfies Cn/Cn0 = 2 = exp(μ td) → td = ln(2)/μ, consistent with standard microbiology.

Why Other Options Are Wrong:

• Options with 1/μ multipliers: dimensionally inconsistent and not solutions of the differential equation.
• Linear form Cn = Cn0 + μ (t − t0): describes zero-order growth, not exponential.
• Cn0 = Cn * exp[…]: rearranged incorrectly.

Common Pitfalls:
Using base-10 instead of natural exponent; forgetting to apply initial conditions leading to missing Cn0 term.

Final Answer:
Cn = Cn0 * exp[μ * (t - t0)]