Bacteria can serve as catalysts for the conversion of low-cost chemicals, such as glucose, into higher value compounds, including commodity chemicals (with large production rates) and high-value specialty chemicals such as pharmaceuticals, dyes, and cosmetics. Commodity chemicals are produced from bacteria in very large bioreactors. For example, cultures up to 130,000 gallons are used to produce antibiotics and other therapeutics, industrial enzymes, and polymer intermediates. When a healthy bacteria culture is placed in a suitable environment with abundant nutrients, the bacteria experience balanced growth, meaning that they continue to double in number in the same fixed period of time. The doubling time of mesophilic bacteria (bacteria that live comfortably at temperatures between 35 degree C and 40 degree C) ranges anywhere from 20 minutes to a few hours. During balanced growth, the rate of growth of the bacteria is given by the expression dCdt = mu C where C(g/L) is the concentration of bacteria in the culture and mu is called the specific growth rate of the bacteria (also described in Problem 2.33). The balanced growth phase eventually comes to an end, due either to the presence of a toxic byproduct or the lack of a key nutrient. The following data were measured for the growth of a particular species of mesophilic bacteria at a constant temperature: (a) If bacteria are used in the production of a commodity chemical, would a low or high value of p be desirable? Explain.
(b) In the rate expression, separate the variables and integrate to derive an expression of the form f(C, C_0) = mu t, where C_0 is the bacteria concentration that would be measured at t = 0 if balanced growth extended back that far. (It might not.) What would you plot versus what on what kind of coordinates (rectangular, semilog, or log) to get a straight line if growth is balanced, and how would you determine mu and C_0 from the plot? (Review Section 2.7 if necessary.)
(c) From the given data, determine whether balanced growth was maintained between t = 1 h and t = 8 h. If it was, calculate the specific growth rate. (Give both its numerical value and its units.)
(d) Derive an expression for the doubling time of a bacterial species in balanced growth in terms of mu. (You may make use of your calculations in Part (b). Calculate the doubling time of the species for which the data are given.