A seed crystal of diameter D (mm) is placed in a solution of dissolved salt, and

Question

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 - 10D^2 (D in mm) What are the units of the constants 200 and 10? (Assume the given equation is valid and therefore dimensionally homogeneous.) Calculate the crystal nucleation rate in crystals/s corresponding to a crystal diameter of 0.050 inch. 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 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.

Explanation / Answer

a.

the general dimensions of the equation are:

Crystals/min

for the left side

therefore the right side must have the same units (must be homogenoeus)

then

200*D = x Crystals/min

-10D^2 = x Crystals/min

Since D = mm

then

200(mm) = x Crystals/min

200/x = Crystals/min-mm

Units of 200 must be Crystals / (min-mm) or Crystals per minute and unit mm

-10(mm^2) = x Crystals/min

-10/x = (Crystals)/(min) / (mm^2)

Units of 200 must be Crystals / (min-mm) or Crystals per minute and unit mm to the second power.

b.

if d = 0.05 inch

1 in = 25.4 mm

d = 0.05 in * 25.4 mm = 1.27 mm

R(crystal/min) = 200*(1.27) - 10*(1.27^2) = 237.871 Crystals per minute

c)

Solve for D

R = 200D - 10 D^2

10D^2 - 200D + R = 0

this is a quadratic formula so:

divide by 10

D^2 - 20D + R/10 = 0

a = 1, b = -20, c = R/10

D = (-b +/- sqrt(b^2 -4ac) ) /2a

D = (20 +/- sqrt((-20^2) -4*1*R/10) ) /2*1

D = 10 +/- sqrt(400-2R/5) ) /2

d.

The empirical nature are the constants 200 and 10, since they MUST be used in "mm"

If it was a mathematical model, they should be able to be calculated with any units.

Also, if D increases, it states that there is a max of D = 25 mm

In theory, there should be NO limits for the size of diameters, since crystal diameter is a continuos (i.e. it can be up to 100 mm or even 10000 mm if needed)

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