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(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.

Explanation:

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)