Explain what it is about the equation that gives away its empirical nature.

{ “cells”: [ { “cell_type”: “markdown”, “metadata”: {}, “source”: [ “\n”, “<center><h2> ChE 113: Chemical Process Analysis <br> HW #2 <br> Due: 9-16-2018 </h2> Problems adapted from *Elementary Principles of Chemical Processes*, 4th Edition, by Felder, Rousseau, and Bullard) </center>” ] }, { “cell_type”: “markdown”, “metadata”: {}, “source”: [ “### Problem 1 ###\n”, “A published study of a chemical reaction, $ A \\to P $, indicates that if the reactor initially contains $ A $ at a concentration *C<sub>A0</sub>* and the reaction temperature, *T*, is kept constant, then the concentration of $ P $ in the reactor increases with time according to the formula \n”, “\n”, “$$ C_P(g/L)= C_{A0}(1-e^{-kt}) $$\n”, “\n”, “The *rate constant*, *k*(s<sup>-1</sup>), is reportedly a function only of the reaction temperature. To test this finding, the reaction is run in three different laboratories. The reported experimental results (values of $ C_P $ in g/L), are given below. \n”, “\n”, “\n”, “\n”, “| | **Lab 1** <br> *T* = 275°C <br> *C*<sub>A0</sub> = 4.83| **Lab 2** <br> *T* = 275°C <br> *C*<sub>A0</sub> = 5.14 | **Lab 3** <br> *T* = 275°C <br> C<sub>A0</sub> = 3.69 | \n”, “|—:|:–:|:—:|:—:|\n”, “|t (s) <td colspan=3> <center>*C*<sub>P</sub> (g/L)</center>|\n”, “| 0 | 0.0 | 0.0 | 0.0 |\n”, “| 10 | 0.287 | 0.310 | 0.245 |\n”, “| 20 | 0.594 | 0.614 | 0.456 |\n”, “| 30 | 0.871 | 0.885 | 0.670 |\n”, “| 60 | 1.51 | 1.64 | 1.20 |\n”, “|120 | 2.62 | 2.66 | 2.06 |\n”, “|240 | 3.91 | 3.87 | 3.03 |\n”, “|360 | 4.30 | 4.61 | 3.32 |\n”, “|480 | 4.62 | 4.89 | 3.54 |\n”, “|600 | 4.68 | 5.03 | 3.59 |\n”, “\n”, “\n”, “**(a)** What plot would yield a straight line if the given equation is correct? \n”, “**(b)** Use Python to generate a plot of the data in Part(a) and determine the corresponding value of *k* using least squares regression. \n”, “**(c)** Use the result in Part(b) to come up with a good estimate to the value of *k* at 275°C. Explain how you did it.\n”, “\n”, “\n”, “\n” ] }, { “cell_type”: “markdown”, “metadata”: {}, “source”: [ “### Problem 2 ###\n”, “\n”, “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\n”, “\n”, “$$ r = 200D – 10D^2,$$\n”, “where $D$ is in mm.\n”, “<br>\n”, “\n”, “**(a)** What are the units of the constants 200 and 10? (Assume the given equation is valid and therefore dimensionally homogeneous.)\n”, “<br>\n”, “**(b)** Calculate the crystal nucleation rate in crystals/s corresponding to a crystal diameter of 0.050 inches.\n”, “<br>\n”, “**(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).\n”, “<br>\n”, “**(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.)” ] } ], “metadata”: { “kernelspec”: { “display_name”: “Python 3”, “language”: “python”, “name”: “python3” }, “language_info”: { “codemirror_mode”: { “name”: “ipython”, “version”: 3 }, “file_extension”: “.py”, “mimetype”: “text/x-python”, “name”: “python”, “nbconvert_exporter”: “python”, “pygments_lexer”: “ipython3”, “version”: “3.6.5” } }, “nbformat”: 4, “nbformat_minor”: 2 }

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