Im struggling with part b on 229 I cant figure out the unit

2.29. A seed crystal of diameter D (mm) is placed in a solution of dissoived 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) 3200D 10D2 (D in 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 0.050 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 ofknown 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. density of a fluid is given by the empirical equation The 70.5 exp(8.27 x 10-7P) where p is density (bm/fts) and P is pressure (bvin2). (a) What are the units of 70.5 and 8.27 x 10-7? N/m2. (b) Calculate the density in glcm for a pressure of 9.00 x 106 2.6-1)Check your result (c) for cm3) as a function of P(N/m2). (See Example using the solution of Part (b)

Solution

a) If we call the units of 200 and 10 (same unit) is A then we have:

r [crystals/min] = 200[A].D[mm] - 10[A].D^2[mm^2]

then we can rewrite:

[crystals/min] = [A][mm] - [A][mm^2] = [A][mm].(1 - [mm])

so [A] = [crystals/min] / { [mm].(1 - [mm]) }

c) 1 min = 60 sec and 1 inch = 25.4 mm, so we have equation in term of sec and inch :

r/60 = (200/25.4)D - (10/25.4)D^2

b) so if D = 0.050 inch => r [crystals/sec] =
={(200/25.4)D - (10/25.4)D^2}*60
={(200/25.4)*0.050 -(10/25.4)*0.050^2}*60
= 23.56
The answer is r (crystals/s)=23.56.