Thermal Lethality Kinetics — The relationship between the F- or del-factor (cumulative lethality), temperature, and time often follows an Arrhenius-type form. Which expression reflects this dependence?

Introduction:
Thermal inactivation of microorganisms is often described using Arrhenius-type kinetics. The cumulative lethality (sometimes called the del factor or F-value) increases with both exposure time and temperature, reflecting the temperature dependence of the inactivation rate constant.

Given Data / Assumptions:

• First-order death kinetics with a rate constant that follows k = k0 * e^(-E/(RT)).
• Constant temperature T during a small time interval t (for stepwise integration).
• A represents a proportionality constant incorporating geometry or reference scaling.

Concept / Approach:

Cumulative lethality is proportional to the time integral of the inactivation rate. If the rate constant follows Arrhenius behavior, increasing T accelerates inactivation exponentially. Therefore, Δ grows linearly with t and exponentially with 1/T, matching Δ ∝ t * e^(-E/(RT)). This framework underlies pasteurization and sterilization calculations that sum contributions over variable temperature histories.

Step-by-Step Solution:

Verification / Alternative check:

For variable temperature profiles T(t), lethality is computed as Δ = ∫ A * e^(-E/(RT(t))) dt, which reduces to the given form at constant T.

Why Other Options Are Wrong:

B is an inverse relation and does not capture increasing lethality with time and temperature. C has the wrong sign in the exponential, implying decreasing activation energy effect. D and E lack the exponential temperature dependence central to Arrhenius behavior.

Common Pitfalls:

Confusing Arrhenius temperature dependence with simple linear temperature scaling; always include the exponential term for biochemical reactions and death kinetics.

Final Answer:

Δ = A * t * e^(-E / (R * T))