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1–28, find a general solution to the given | Ch 4.10 - 11RP

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider ISBN: 9780321747730 43

Solution for problem 11RP Chapter 4.10

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 11RP

In 1–28, find a general solution to the given differential equation. , t > 0

Step-by-Step Solution:
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Week 3 Notes Recrystalization- Single Solvent Goal: Purify an impure or crude experimentally synthesized product Basic Principle: Solubility of a compound usually increases with increasing temperature of the solvent (solubility depends on concentration, polarity, and temperature) Major Difficulty: Most impurities and reaction by-products have similar solubility properties as the desired product. 1. Dissolve the impure product in a minimum amount of a soluble hot solvent. 2. Filter the hot solution to remove any insoluble impuriies. 3. Let the solution cool slowly to allow the pure crystal form without trapping impuities. 4. Filter the purified product Solubility in Recrystalization Example 2: Suppose we take 25 g of sugar and dissolve it in 10 mL of boiling water. We then allow the solution to slowly cool down to room temperature. Will we observe any sugar crystals in the cooled solution Example 3: Suppose we take 25 g of sugar and dissolve it in 12.5 mL of boiling water. We then allow the solution to slowly cool down to room temperature. Will we observe any sugar crystals in the cooled solution Solubility of Sugar in Water Temperature 1.80 g/mL 25 C Using excess solvent in a recrystallization will reduce 2.50 g/mL 100 C the amount of recovered purified product (lower yield!) Example 2: At 100 C: 2.50 gml×10ml = 25g [Can dissolve all sugar] o 1.80 ×10ml=18g [Cannot dissolve all sugar] At25 C: ml 25g initially dissolved - 18 g remaining in solution = 7 g or sugar cyrstals formed Example 3: At 100 C: 2.50 gml×12.5ml = 31.25g [Can dissolve all sugar] At 25 C: 1.80 g ×12.5ml = 22.5g [Cannot dissolve all sugar] ml 25g initially dissolved - 22.5 g remaining in solution = ONLY 2.5 g of sugar crystals formed Using Recrystalization to Remove Impurities Consider the example system where we have 10 g of desired product (A) and 2 g of impurity (B) Temp = 20°C Temp =100°C solubility (0.1 g/mL 1 g/mL solubility (B0.1 g/mL 1 g/mL Dissolve all crude product (10 gAand 2 g B) in 12 mL of solvent at 100oC. Cooling to 20oC, 1.2 g ofAand and 1.2 g of B remain in solution while 8.8 g ofAand 0.8 g of B crystalize out. Before recrystallization 10g / 12g ×100% = 83.3% 12g After recrystallization 8.8g / 9.6g ×100% = 91.7% Choosing a Good Crystallization Solvent A good crystallization solvent should: Be soluble for (dissolve) a large quantity of the product at high temperature Be insoluble for (not dissolve) the product at low temperature Be soluble for (dissolve) the impurities at all temperatures [Optional, becuase very difficult to achieve] Not react with the product Considering Crystalization of Glycine Example 4: Is water or ethanol a better solvent for glycine Solubility of Glycine in Water Temperature Solubility of Glycine in Ethanol Temperature 14.0 g/100 mL 0 C 0.01 g/ 100 mL 0 C 25 g/ 100 mL 25 C 39 g/ 100 mL 50 C 45 g/ 100 mL 25 C 54 g/ 100 mL 75 C o o 65 g/ 100 mL 65 C 67 g/ 100 mL 100 C Considering Crystalization of Glycine Example 4: Is water or ethanol a better solvent for glycine Use percent recovery to decide (amount dissolved at high T - amount at low T / amount at high T) x 100% = Percent Recovery Using water as the solvent: (67 g -14 g) / 67 g % recovery = ×100% = 79% Using ethanol as the solvent: [Preferred Solution] % recovery =(65 g - 0.01g) / 6×100% = 99.98% Synthesis of Aspirn Lab salicylic acid (limiting reagent) + acetic anhydride ——> aspirin + acetic acid Example 5: When performing the aspirin lab, you weigh out exactly 3 g of salicylic acid (SA). Given that you made 3.21 g of aspirin (A) after recrystallization, what is the % yield 3gSA 1mol SA / 138.1g SA) 1mol A / 1 mol SA)180.2g A / 1 mol A=3.91g % yield = actual amount / theoretical amount ×100% = 3.21 g / 3.91 g×100% = 82.1% Recrystalization of Aspirin What solvent do we use to recrystalize aspirin Water: not very soluble Ethanol: way too soluble Solubility ofAspirin in Water Temperature Solubility of Aspirin in Temperature 0.025 g / 100 mL of wat0 C Ethanol 0.30 g / 100 mL of wate25 C 20.0 g / 100 mL 25 C of ethanol 1.0 g / 100 mL of water37 C 35.0 g / 100 mL o of ethanol 70 C Decomposes at high T o 100 C Mixed Solvent System for Recrystalization • Made with two mutually miscible (mixable) liquids • One of the two solvents will dissolve the product more readily than the other solvent • Usually works better than single solvent system We will use a mixed solvent recrystallization for purifying aspirin – use a mixture of water and ethanol! Procedurte for Recrystalization with Mixed Solvent System • Dissolve the impure product in the high-solubility solvent (ethanol) at high temperature. • Add the low-solubility solvent (water) drop-wise until solution turns cloudy. • Heat solution to re-dissolve all crystals. Add a few drops of high- solubility solvent (ethanol) if required to force all the crystals back into solution • Allow solution to cool slowly to allow the pure products to form without trapping impurities. Graphs •Online in powerpoint provided and in notes

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Chapter 4.10, Problem 11RP is Solved
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Textbook: Fundamentals of Differential Equations
Edition: 8
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
ISBN: 9780321747730

The full step-by-step solution to problem: 11RP from chapter: 4.10 was answered by , our top Calculus solution expert on 07/11/17, 04:37AM. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations , edition: 8. Fundamentals of Differential Equations was written by and is associated to the ISBN: 9780321747730. This full solution covers the following key subjects: Differential, equation, Find, general, given. This expansive textbook survival guide covers 67 chapters, and 2118 solutions. The answer to “In 1–28, find a general solution to the given differential equation. , t > 0” is broken down into a number of easy to follow steps, and 15 words. Since the solution to 11RP from 4.10 chapter was answered, more than 275 students have viewed the full step-by-step answer.

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1–28, find a general solution to the given | Ch 4.10 - 11RP