= KN 2. The growth of a bacterial colony can be modeled as a first-order process in which the rate of growth of the number of bacteria cells is proportional to the number of cells dN dt (a) Use the first-order rate law expression above to show that the number of cells in the colony at any time is given by N = NOM where N, is the initial number of cells. (b) The generation time is the amount of time it takes for a given number of cells to double. Determine an expression for the generation time. (C) If the generation time for the bacteria is 75 minutes at 37C, determine its rate constant. (d) The change in rate constants as a function of temperature is very important for the food industry as perishables need to be transported such that they have not spoiled by the time they reach consumers. Ratkowsky et. al. measured the rate constants of several bacteria strains. According to their data (shown in Figure 1, is it appropriate to apply an Arrhenius relationship to determine how the rate constant changes with temperature? Why? 50 45 401 0 OVO 35) 0 a Sport carne 301 doc CIFE O Escherichia coll Logarithm of growth rate constant 35 o 20 cherche 15| ale Aeroter Geropened 10 05 55 50 45 00030 00031 Degrees Celsius 40 35 30 25 20 00032 00033 00034 00035 Reciprocal of absolute temperature 00036 00037 Figure 1: Arrhenius plot relating the natural logarithm of the rate constant with the inverse temperature for 6 bacteria strains. Image taken from Ratkowsky, D. A., et al. "Relationship between temperature and growth rate of bacterial cultures." Journal of Bacteriology 149.1 (1982): 1-5