91Ó°ÊÓ

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 \(r(\text { crystals/min })=200 D-10 D^{2} \quad(D \text { in } \mathrm{mm})\) (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). (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.)

Step by step solution

01

Identify the units of constants

Since the equation \(r = 200D - 10D^2\) (crystals per min) is dimensionally homogeneous, the units on both sides must be the same. 'r' is in crystals/min and 'D' is in mm, so the units of constants 200 and 10 would be (crystals/min)/mm and (crystals/min)/mm^2 respectively.

02

Calculate the rate in crystals/s for D = 0.050 inch

First, convert 0.050 inch to mm using the conversion 1 inch = 25.4 mm. 0.050 inch = 0.050 * 25.4 = 1.27 mm. Substitute 'D' = 1.27 mm in the equation to find 'r': \(r = 200*1.27 - 10*(1.27)^2 = 192.29\) crystals/min. Then, convert the result to crystals/s using the conversion 1 min = 60 s. Hence, 'r' = 192.29/60 = 3.21 crystals/s.

03

Derive the formula and check using result from step 2

The formula with 'D' in inches and 'r' in crystals/s would be obtained by replacing 200 with \(200/25.4\) (because we are converting mm to inches) and 10 with \(10/(25.4^2)\) (because we are converting mm^2 to inches). Therefore, the formula is \(r = (200/25.4)D - (10/25.4^2)D^2\) crystals/s. Substituting 'D' = 0.050 inch in this equation, it can be seen that it gives the same result as that obtained in step 2.

04

Explain empirical nature of equation

The equation is empirical because it does not reflect any underlying fundamental principles and is based on experimental observations. As 'D' increases, the equation predicts that after reaching a certain size, new crystal formation slows and then actually starts to decrease. In reality, these values cannot decrease as size increases. This discrepancy reveals the empirical nature of the equation, and that this model is not accurate for large values of 'D'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dimensional Analysis

Dimensional analysis is a powerful tool used in physics, engineering, and chemistry to understand the relationships between different physical quantities. It involves the study of the dimensions of physical quantities, which are often represented as powers of the basic physical dimensions, such as mass [M], length [L], time [T], and so forth. By ensuring that all terms in an equation match in terms of their fundamental dimensions, scientists and engineers can verify that equations are dimensionally consistent or homogeneous.

For example, in the nucleation rate problem, dimensional analysis can be employed to determine the units of the constants 200 and 10 in the empirical equation given for crystal growth. Since the nucleation rate, r, is measured in crystals per minute, and the seed crystal diameter, D, is in millimeters, the constants must balance the equation dimensionally. Thus, the units for these constants are derived based on the dimensional form of the rate r and the diameter D, ensuring the equation maintains dimensional homogeneity.

Unit Conversion

Unit conversion is a fundamental concept in many scientific disciplines, as it allows us to translate measurements from one system of units to another. It is especially important when dealing with empirical data or equations that may have been formulated in a different unit system than the one commonly used.

In the context of crystal growth, unit conversion is essential to calculate the nucleation rate in different units, such as converting crystals per minute to crystals per second. This process often involves multiplying or dividing by conversion factors, as seen when converting inches to millimeters, where 1 inch is equivalent to 25.4 millimeters. Accurate unit conversion is crucial for performing meaningful calculations and for validating results obtained from experimental data.

Empirical Equation Analysis

Empirical equation analysis involves studying equations that have been formulated based on experimental data rather than derived from fundamental principles. These equations model the observed behavior under specific conditions and are often used to make predictions or calculate unknown variables based on known quantities.

In our crystal growth problem, we encounter an empirical equation that relates the nucleation rate to the diameter of the seed crystal. Analysis of such an equation involves observing the terms within it and deducing what the equation suggests about the relationship between variables. For instance, the equation suggests that the nucleation rate initially increases with the crystal diameter but then starts to decrease beyond a certain point as the diameter continues to increase. This behavior, which does not necessarily align with theoretical or physical expectations, is a telltale sign of the empirical nature of an equation.

Chemical Engineering Principles

The principles of chemical engineering play a crucial role in understanding and analyzing processes such as nucleation and crystal growth. Chemical engineers often apply these principles to design and optimize systems for the manufacturing of chemicals, materials, and pharmaceuticals.

In the case of nucleation rate, chemical engineering principles can be used to interpret the empirical equation and to consider the physical processes at play. Chemical engineers must also be aware of the limits of empirical models, such as the one provided in the problem, recognizing that these models might not accurately predict behavior outside the range of observed experimental data. Understanding the behavior of nucleation from a chemical engineering standpoint may include considerations of supersaturation levels, temperature effects, agitation, and other factors influencing the rate of crystal formation.