2.29. WP A seed crystal of diameter D (mm) is placed in a solution of dissolved salt, and new crystals are observed to nucleate (form) at a constant rate r (crystals/min). Experiments with seed crystals of different sizes show that the rate of nucleation varies with the seed crystal diameter as r(crystals/min) 200D – 10D² (Din mm) a. What are the units of the constants 200 and 10? (Assume the given equation is valid and therefore dimensionally homogeneous.) b. Calculate the crystal nucleation rate in crystals/s corresponding to a crystal diameter of o.050 inch. c. Derive a formula for r(crystals/s) in terms of D(inches). (See Example 2.6-1.) Check the formula using the result of Part (b). Exploratory Exercise-Research and Discover d. The given equation is empirical; that is, instead of being developed from first principles, it was obtained simply by fitting an equation to experimental data. In the experiment, seed crystals of known size were immersed in a well-mixed supersaturated solution. After a fixed run time, agitation was ceased and the crystals formed during the experiment were allowed to settle to the bottom of the apparatus, where they could be counted. Explain what it is about the equation that gives away its empirical nature. (Hint: Consider what the equation predicts as D continues to increase.) 2.30. WP The density of a fluid is given by the empirical equation p = 70.5 exp(8.27 × 10-7P) where p is density (lbm/ft³) and P is pressure (lbf/in²). a. What are the units of 70.5 and 8.27 × 10-7? b. Calculate the density in g/cm3 for a pressure of 9.00 × 10° N/m². Answer c. Derive a formula for p(g/cm³) as a function of P(N/m²). (See Example 2.6-1.) Check your result using the solution of Part (b). BIOENGINEERING 2.31. WP The volume of a microbial culture is observed to increase according to the formula V(cm³) = aebt where t is time in seconds. a. Calculate the expression for V(in°) in terms of t(h). b. Since both the exponential function and its argument must be dimensionless, what must be the units of a and b? 2.33. WP The following table is a summary of data taken on the growth of yeast cells in a bioreactor: Time, t(h) Yeast Concentration, X(g/L) 0.010 0.048 8 0.152 12 0.733 16 2.457 The data can be fit with the function X — Хо ехp (ut) where X is the concentration of cells at any time t, X, is the starting concentration of cells, and u is the specific growth rate. a. Based on the data in the table, what are the units of the specific growth rate? b. Give two ways to plot the data so as to obtain a straight line. Each of your responses should be of the form "plot versus on axes." c. Plot the data in one of the ways suggested in Part (b) and determine u from the plot. d. How much time is required for the yeast population to double? BIOENGINEERING -2.34. WP You arrive at your lab at 8 A.M. and add an indeterminate quantity of bacterial cells to a flask. At 11 A.M. you measure the number of cells using a spectrophotometer (the absorbance of light is directly related to the number of cells) and determine from a previous calibration that the flask contains 3850 cells, and at 5 P.M. the cell count has reached 36,530. a. Fit each of the following formulas to the two given data points (that is, determine the values of the two constants in each formula): linear growth, C = C, + kt; exponential growth, C = Coekt; power-law growth, C = kt'. In these expressions, Co is the initial cell concentration and k and b are constants. b. Select the most reasonable of the three formulas and justify your selection. c. Estimate the initial number of cells present at 8 A.M. (t = 0). State any assumptions you make. Answer d. The culture needs to be split into two equal parts once the number of cells reaches 2 million. Estimate the time at which you would have to come back to perform this task. State any assumptions you make. If this is a routine operation that you must perform often, what does your result suggest about the scheduling of the experiment? BIOENGINEERING