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)?
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ACn = Cn0 * exp[μ * (t - t0)]
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BCn0 = Cn * exp[μ * (t - t0)]
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CCn / Cn0 = (1/μ) * exp(t - t0)
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DCn0 / Cn = (1/μ) * exp(t - t0)
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ECn = Cn0 + μ * (t - t0)
Answer
Correct Answer: Cn = Cn0 * exp[μ * (t - t0)]
Explanation
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:
1) Write rate law: dCn/dt = μ * Cn.2) Separate and integrate: ∫ dCn/Cn = ∫ μ dt.3) Apply bounds: ln(Cn) − ln(Cn0) = μ (t − t0).4) Exponentiate: Cn = Cn0 * exp[μ (t − t0)].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)]