The rate of bacterial cell decline in the death phase follows:
Decline phase marks period when lethal environmental stresses exceed capacity to repair damage and maintain homeostasis, leading to net loss of viability. Population decay is often well described by exponential decay mathematically analogous to radioactive decay, reflecting stochastic individual death. First-order kinetics states instantaneous rate of viable cell loss is directly proportional to number of viable cells present at that instant expressed as dN/dt equals minus kd times N where kd is first-order death rate constant per time. Integration from time zero yields ln Nt equals ln N0 minus kd t, so plot of log survivors versus time yields straight descending line slope equals minus kd. Biological interpretation is each cell has independent probability of dying per unit time determined by starvation, reactive oxygen accumulation, acidification and protein denaturation. Zero-order would predict constant absolute number dying per time regardless of population size unrealistic for large cultures, second-order would require interaction of two cells to cause death, enzyme-substrate kinetics describes saturation of catalytic rate with substrate not population death. Thus first-order model accurately captures log-linear decline observed in death phase and forms basis for D-value decimal reduction time used to validate sterilization efficacy and predict shelf life of foods and pharmaceuticals.
Ref: Lodish et al., Molecular Cell Biology, 8th ed., Chapter 4: Death Phase and First-Order Kinetics Model.