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) = 200Dâˆ'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.

a) the units of the constants 200 and 10 are as follows:

unit of 200 = unit of r / unit of D

= crystals/min * mm

= crystals / (min*mm)

unit of 10 = unit of r / unit of D^2

= crystals/min * mm²

= crystals / (min*mm²)

b) The objective here is to determine the crystal nucleation rate in crystals/s corresponding to a crystal diameter of 0.050 inch; T o do that; let's first convert the inch to mm

We all know that

1 inch = 25.4 mm

0.050 inch = 0.050 * 25.4 mm

= 1.27 mm

nucleation rate = 200*D - 10*D²

= 200*1.27 - 10 * (1.27) ²

=237.9 Crystals/min

=237.9/60 crystals/sec

= 3.96 crystals/sec

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.

r (crystals/sec) = A D-B D² (D in inch)

unit of 200 = crystals / (min*mm)

unit of 10=crystals / (min*mm²)

A = 200 crystals / (min*mm) * 1/60 min/sec * 25.4 mm/inch

= 84.7 crystals / (sec-inch)

B = 10 crystals / (min*mm²) * 1/60 min/sec * 25.4 mm/inch*25.4 mm/inch

=107.5 crystals / (sec-inch)