The Mole. Essential Learning Outcomes: 1. Change can be measured. 2. Changes can occur within a substance that alters its identity.

Transcription

1 The Mole The Mole: Chemical quantities refer to the numbers of different types of particles. These particles are formula units, atoms, and molecules, along with gram formula mass, and liters. Throughout this unit we will look at how these quantities are related to the mole, and how these units are used in the world of chemistry. During this unit there will be a range of topics that will need to be covered. We will review how to make correct conversions using a system called cross cancellation. Shortly after the introduction of cross cancellation we will begin to dissect the Y-diagram and its significance to chemistry. Once we know what the Y-diagram is and how it works, we will begin solving problems related to each leg of the Y-diagram. After the introduction and use of the Y-diagram we will look at percent composition of compounds; what percentage of each element is responsible for the total mass of the compound. Percent composition is the beginning of solving for an empirical or molecular formula. The empirical formula and molecular formula of a substance can be used to determine exactly what type of compound was present this technique is commonly seen on the TV show CSI. An empirical formula is simply the smallest whole number ratio of a compound, and a molecular compound is a simple whole number multiple of an empirical formula. Important vocabulary: Avogadro s hypothesis Avogadro s number Empirical formula Dimensional analysis Molar mass Molar volume Mole (mol) Percent composition Representative particle STP (Standard Temperature and Pressure) All the videos will be posted on YouTube and can be accessed using multiple sources. The videos have been formatted to work on iphones, ipads, Android phones, Kindles, and Nooks. Titles of the videos to search are listed below, and the creator of the videos should be NRHSChemistry. Lesson Target(s) for Each Video: Video 1 Dimensional analysis I can make conversions using the process of dimensional analysis. Video 2 Double conversions I can calculate the density of a gas when given the gram formula mass of that gas. I can make multiple conversions in one problem. Video 3 Avogadro I can explain Avogadro s hypothesis. I can identify various types of representative particles. I can calculate the gram formula mass of a compound. Video 4 Introduction to the Y-diagram I can apply the Y-diagram to making chemistry conversions. Video 5 Applying the Y-Diagram/Gas density I can calculate numeric answers using the Y-diagram. Video 6 Percent composition I can solve for the percent composition of a compound. I can calculate the grams of an element in an unknown sample, when given the percent compositions. Video 7 Empirical formula I can calculate an empirical formula. Video 8 Molecular formula I can solve for a molecular formula of an unknown compound. Essential Learning Outcomes: 1. Change can be measured. 2. Changes can occur within a substance that alters its identity.

2 Required activities: Due dates are in parentheses The Mole P.S. 1-4 (1/24) o Video 1 Dimensional Analysis (1/18) o Video 2 Double Conversions (1/19) The Mole P.S (1/27) o Video 3 Avogadro (1/20) o Video 4 Intro to the Y-Diagram (1/23) o Video 5 Applying the Y-Diagram (1/24) The Mole P.S (1/30) o Video 5 Applying the Y-Diagram (1/24) The Mole P.S (2/3) o Video 5 Applying the Y-Diagram (1/24) o Video 6 Percent Composition (1/26) The Mole P.S (2/7) o Video 7 Empirical Formula (1/30) o Video 8 Molecular Formula (2/1) Molar Relationships Lab (1/25 to 1/27) o Peer Review of Report (1/30) o Lab Report (2/3) Tin Oxide Lab (1/31 to 2/2) o Peer Review of Report (2/6) o Formal Report (2/9) The Mole Unit Quest S.T.P. Additional Resources: Practice Problem Websites: iphone and ipad apps: There are none that I could find.if you find one and feel it is useful, please let me know!!!

3 Chemistry Name: Problems and Questions The Mole Hour: Directions: Answer the following questions using complete sentences when necessary. On numerical problems, show all work, circle your answers, and follow all rules of significant figures for full credit. 1. List the one conversion factor you would use to convert between these pairs of units. You may need to use chapter 2 in your textbook as a reference. Ex: cubic decimeters and milliliters Answer: 1dm 3 =1000mL a. Seconds and minutes b. Calories and joules c. Grams of water and milliliters of water d. Nanograms and grams e. Joules and kilojoules f. Deciliters and liters 2. Make the following conversions. Use the method of cross-cancellation shown by your instructor. You may need more than one conversion factor in some cases. a. 1.54km to m b. 4.70nm to m c. 4.56cj to joules d. 6.70msec to sec e cm to km f. 5.67cal to joules 3. Make the following conversions using the same method you used in #2. These conversions may take more than one step. a micrograms to centigrams b microliters to milliliters c. 627 nanometers to millimeters d kcal to dcal e. 569cj to kj f. 4.55x10 6 cg to ng 4. Make the following double conversions using the basic methods in #2 and #3. a kg/l to g/cl b km/hour to m/sec c cm/day to m/min d cal/day to joules/sec e. 2.34x10 3 kg/ml to grams/cm 3 f cm/day to mm/sec (express your answer in scientific notation) 5. Name the representative particle (atom, formula unit or molecule) of each of the following substances. Use complete sentences. a. Hydrogen gas d. Carbon monoxide b. Iron (III) oxide e. Silicon dioxide c. Copper f. Aluminum oxide 6. How many hydrogen atoms are in a representative particle of each of these substances? Use complete sentences. a. A formula unit of calcium hydroxide. b. A molecule of C 3H 8O. c. A formula unit of ammonium hydrogen phosphate. d. A molecule of HC 2H 3O. e. A formula unit of manganese (IV) hydrogen sulfate. f. A molecule of C 6H 4(OH) 2.

4 7. What is the total number of ions in a formula unit of each of the following? Use complete sentences. a. Barium chloride b. Ammonium sulfate c. Sodium carbonate d. Iron (III) chloride e. Lithium phosphate f. Aluminum oxide 8. Find the gram formula mass of each of the following elements or compounds. Your answer should have two decimal places. a. Hydrogen gas b. Iron (III) hydroxide c. Copper metal d. Cobalt metal e. Silicon dioxide f. Aluminum oxide 9. Find the gram formula mass of each of the following elements or compounds. Your answer should have two decimal places. a. Calcium carbonate b. Ammonium sulfate c. C 3H 6O d. Iodine liquid e. Hydrogen phosphate f. Dinitrogen pentoxide 10. Find the number of representative particles (formula units, molecules or atoms) in each of these substances. You may need to use Avogadro's number. Use the cross-cancellation method that you used in #2, #3, and #4. Express your answer in proper scientific notation. a mol Fe b mol Cl 2 c mol NaI d. 4.20x10-3 mol Al e mol SO 3 f. 5.25x10-4 mol K 2S 11. How many moles are in each of the following? Use the cross-cancellation method that you used in #2, #3, #4 and #10. a. 6.02x10 22 molecules of Br 2 b. 4.81x10 24 atoms Li c. 1.50x10 23 molecules NH 3 d. 1.00x10 9 molecules O 2 e. 3.61x10 24 formula units Ca(OH) 2 f. 1.00x10 20 molecules propane, C 3H Find the mass of each of the following. a moles sodium hydroxide b. 3.21x10-3 moles nickel (II) nitrate c moles calcium cyanide d moles aluminium oxalate 13. How many moles is each of the following? a g barium chlorate b g chlorine gas c g potassium hydroxide d. 937 g calcium acetate

5 14. Calculate the volume of each of these gases at S.T.P. a. 9.60mol helium gas b. 4.80mol nitrogen gas c. 3.20mol carbon dioxide d. 5.40mol sulfur dioxide 15. How many moles are in each of the following, assuming S.T.P. a L dinitrogen monoxide b L nitrogen gas c L carbon monoxide d L nitrogen trioxide 16. Find each of the following quantities. Use the cross-cancellation method described in #2, #3, #4, and #10 for credit. Show all work. a. The number of molecules in 60.0g of nitrogen dioxide. b. The volume, in liters, of 3.24x10 22 molecules carbon monoxide (at S.T.P.). c. The mass of 18.0L of CH 4 (at S.T.P.). d. The volume, in liters, of 835g of Sulfur trioxide (at S.T.P.). e. The mass of 1.00 molecule of aspirin, C 9H 8O 4. f. The number of atoms in 5.78 moles of NH 4NO 3 (Hint: ammonium nitrate has 9 atoms in 1 formula unit.) 17. Find the gram formula masses of the following compounds using the information provided. Assume S.T.P. and use three significant digits in your answers. a. The density of compound A is 1.96 g/l b. The density of compound B is g/l c. The density of compound C is 2.21 g/l d. The density of compound Z is g/l 18. Calculate the percent composition of each element in each of these compounds. a. Sodium phosphate b. Hydrogen sulfide c. Tin (IV) chloride d. Ammonium oxalate e. Magnesium hydroxide f. Sodium perchlorate 19. Calculate the mass of the element stated in each compound using percent composition. a. Iron in 546 g of iron (II) carbonate b. Calcium in 3.54 g of calcium phosphide c. Oxygen in 256 g of chromium (IV) oxide d. Carbon in 25.0 g of potassium carbonate e. Magnesium in 97.4 g of magnesium chloride 20. Which of the following compounds has the highest iron content? Prove your answer by showing the percentage of iron for each compound. At the end state (in a sentence) which compound has the most iron and explain why. a. Iron (III) oxide b. Iron (III) hydroxide c. Iron (III) chloride d. Iron (II) acetate e. Iron (II) oxide f. Iron (VI) oxide

6 21. Calculate the empirical formula of each of these compounds. a. 42.9% C, and 57.1% O b % Cl, 16.16% O, and 12.12% C c % C, 42.66% O, 18.67% N, and 6.670% H d % C, 1.15% H, 16.09% N, 55.17% O e. 17.6% Na, 39.7% Cr, 42.7% O f. 67.6% Hg, 10.8% S, 21.6% O 22. Write the formulas of the names given below and classify them as either empirical or molecular formulas. a. Dinitrogen pentoxide b. Tin (II) oxide c. Potassium sulfate d. Disulfur dichloride e. Ammonium carbonate f. Sucrose (I'll give you this one: C 6H 12O 6) 23. Determine the molecular formula for each of these compounds. a. 40.4% C, 53.4% O, and 6.60% H; gfm = 120.0g/mole b. 94.1% O and 5.90% H; gfm = 34.0g/mole c. 54.5% C, 13.6% H, and 31.8% N; gfm = 88.0g/mole d % N, 69.54% O; gfm = 92.0g/mole 24. Using the mass spectrometer data below, calculate the molecular formula of the unknown compound. 47.9% O Gram formula mass = 100.1g/mol 39.0% K 12.0% C 1.01% H

7 Lab 14: Molar Relationships: What Are the Identities of the Unknown Compounds? Introduction The concept of the mole is important for understanding chemistry. The mole provides a measure of the number of atoms present in a sample of a compound. One mole of an element or compound contains 6.022x10 23 atoms or molecules. This quantity is referred to as the Avogadro constant. Knowing the amounts of particles allows chemists to understand how different chemicals behave during chemical reactions and predict the outcomes of reactions. Moles provide a standardized way of comparing elements. Using the Avogadro constant, chemists can use other measures, such as mass or volume, to determine the amount of particles a sample has. To use mass to determine the number of moles of an element or molecule in a sample, you must also know the molar mass of that element or molecule. The molar mass refers to the total mass of an element present in one mole of that element. The unit for these masses is grams per mole (g/mol). The molar mass of an element is easily identified on most periodic tables, where it is typically listed in the box provided for a particular element. Examples of molar mass include carbon (C), g/mol; oxygen (O), g/mol; and gold (Au), g/mol. To determine the molar mass for a compound made of larger molecules, you must add up the molar masses of all the atoms present in the molecular formula. For example, the molar mass of CO 2 is g/mol, which is calculated by g/mol (C) g/mol (O) g/mol (O). Remember that you have to include the total number of atoms in the molecular formula when calculating molar mass, so be mindful of the subscripts in those formulas. By knowing the molar mass of a compound and the mass of a sample of that compound, you can determine the number of moles in the compound. Continuing from the example above, if you have a sample of CO 2 whose mass is g, then you can determine the number of moles in that sample by dividing the actual mass by the molar mass (e.g., g / g/mol = moles of CO 2 ). You will now use your understanding of the relationships between moles, molar mass, and mass of a sample to identify some unknown compounds. Remember, moles provide a standardize unit of measure (based on the Avogadro constant) so that chemists can compare a wide variety of substances, including the amount of substances needed and produced by a chemical reaction. Your Task You will be given seven sealed bags. Each bag will be filled with a different powder and will be labeled with the number of moles of powder that is inside the bag. Your task will be to identify the powder in each bag. The unidentified powders could be any of the following compounds: Calcium acetate, Ca(C 2 H 3 O 2 ) 2 Sodium carbonate, Na 2 CO 3 Calcium oxide, CaO Sodium chloride, NaCl Potassium sulfate, K 2 SO 4 Zinc oxide, ZnO Sodium acetate, NaC 2 H 3 O 2 The guiding question of this investigation is, What are the identities of the unknown compounds? Materials You may use any of the following materials during your investigation: Consumables Sealed plastic bags of unknown compounds Empty plastic bags Equipment Electronic balance Periodic table

8 Safety Precautions Follow all normal lab safety rules. Your teacher will explain relevant and important information about working with the chemicals associated with this investigation. In addition, take the following safety precautions: Wear indirectly vented chemical-splash goggles Wash your hands with soap and water before leaving the laboratory. Investigation Proposal Required? Yes No Getting Started To answer the guiding question, you will need to design and conduct an investigation. To accomplish this task, you must first determine what type of data you need to collect, how you will collect the data, and how you will analyze the data. To determine what type of data you need to collect, think about what type of measurements you will need to make during your investigation. To determine how you will collect the data, think about the following questions: o How will you make sure that your data are of high quality (i.e., how will you reduce error)? o How will you keep track of the data you collect and how will you organize it? To determine how you will analyze the data, think about the following questions: o What type of table or graph could you create to help make sense of your data? o What types of calculations will you need to make? Connections to Crosscutting Concepts As you work through your investigation, be sure to think about the importance of identifying patterns, which proportional relationships are critical to the understanding of this investigation, how scientific knowledge changes over time in light of new evidence, and the difference between data and evidence. Initial Argument Once your group has finished collecting and analyzing your data, you will need to develop an initial argument. Your argument must include a claim, which is your answer to the guiding question. Your argument must also include evidence in support of your claim. The evidence is your analysis of the data and your interpretation of what the analysis means. Finally, you must include a justification of the evidence in your argument. You will therefore need to use a scientific concept or principle to explain why the evidence that you decided to use is relevant and important. You will create your initial argument on a whiteboard. Your whiteboard must include all the information shown in Figure L14.1. FIGURE L14.1 Argument presentation on a whiteboard

9 Argumentation Session The argumentation session allows all of the groups to share their arguments. One member of each group stays at the lab station to share that group s argument, while the other members of the group go to the other lab stations one at a time to listen to and critique the arguments developed by their classmates. The goal of the argumentation session is not to convince others that your argument is the best one; rather, the goal is to identify errors or instances of faulty reasoning in the initial arguments so these mistakes can be fixed. You will therefore need to evaluate the content of the claim, the quality of the evidence used to support the claim, and the strength of the justification of the evidence included in each argument that you see. To critique an argument, you might need more information that what is included on the whiteboard. You might, therefore, need to ask the presenter one or more of the following questions, such as: How did your group collect the data? Why did you use that method? What did your group do to make sure the data you collected are reliable? What did you do to decrease measurement error? What did your group do to analyze the data? Did you check your calculations? Is that the only way to interpret the results of your group s analysis? How do you know that your interpretation of the analysis is appropriate? Why did your group decide to present your evidence in that manner? What other claims did your group discuss before deciding on that one? Why did you abandon those alternative ideas? How confident are you that your group s claim is valid? What could you do to increase your confidence? Once the argumentation session is complete, you will have a chance to meet with your group and revise your original argument. Your group might need to gather more data or design a way to test one or more alternative claims as part of this process. Remember, your goal at this stage of the investigation is to develop the most valid or acceptable answer to the research/guiding question! Report Once you have completed your research, you will need to prepare an investigation report that consists of three sections that provide answers to the following questions: 1. What question were you trying to answer and why? 2. What did you do during your investigation and why did you conduct your investigation in this way? 3. What is your argument? Your report should answer these questions in four pages or less. The report must be typed and any diagrams, figures, or tables should be embedded into the document. Be sure to write in a persuasive style; you are trying to convince others that your claim is acceptable and valid!

10 Investigation Timeline Option D Stage 1: Identify the task and the guiding question. Hold a tool talk Day 1 Small groups of students then 50 Minutes Stage 2: Design a method and collect data Groups then Stage 3: Analyze data and develop a tentative argument Day 2 Each group then shares its argument during an Stage 4: Argumentation session If needed, groups can Collect additional data or reanalyze the collected data 50 Minutes The teacher then leads an The teacher then leads an Stage 5: Explicit and reflective discussion Homework Individual students then Stage 6: Write and investigation report Day 3 The report then goes through a Stage 7: Double-blind group peer review 30 Minutes Homework Each student then Stage 8: Revises and submits his or her report

11 Lab 26: Composition of Chemical Compounds: What Is the Empirical Formula of Magnesium Oxide? Introduction Chemists can describe the composition of a chemical compound in at least three different ways. The first way is to define the percent composition; this is the percent of each element found in the compound by mass. For example, a chemical compound called acetylene is composed of 92.25% carbon and 7.75% hydrogen by mass. The second way to describe the composition of a compound is to provide its empirical formula, which indicates the lowest whole-number ratio of the atoms found within one unit of that compound. The empirical formula of acetylene, for example, is CH. The empirical formula indicates the ratio of atoms in a compound but does not always represent the actual number of each kind of atom found in one unit of that compound. The compound benzene, for example, has the same empirical formula (CH) as acetylene because both acetylene and benzene contain one carbon atom for every atom of hydrogen. The third way to define the composition of a compound is to use a molecular formula, which indicates the actual number of atoms that are found in a single unit of that compound. The molecular formulas of acetylene and benzene are different even though they share the same empirical formula because each compound contains a different number of carbon and hydrogen atoms. The molecular formula of acetylene is C 2 H 2, whereas the molecular formula for benzene is C 6 H 6. Chemists rely on two important principles when they attempt to determine the composition of an unknown compound. The first principle is the law of definite proportions, which indicates that a compound is always made up of the exact same proportion of elements by mass. The percent composition of a compound is therefore a constant and does not depend on the amount of a sample. The second principle is the law of conservation of mass, which states that mass is neither created nor destroyed during a chemical reaction. These two principles enable chemists to determine the percent composition of an unknown compound. Chemists can then use the percent composition of the compound and some simple mathematics to determine its empirical formula. In this investigation, you will have an opportunity to use these two principles to determine the empirical formula of a compound that you create inside the lab by heating magnesium in the presence of oxygen. Your Task Magnesium oxide is a compound that consists of magnesium and oxygen. It is produced when magnesium metal is heated. The heat causes the magnesium to combine with molecules of oxygen found in the air. The magnesium and oxygen may combine in a number of different ratios during this reaction. These ratios include, but are not limited to, MgO, Mg2O, Mg3O2, and Mg5O4. Your goal is to determine the percent composition of magnesium oxide and then use this information to calculate its empirical formula. The guiding question for this lab is, What is the empirical formula of magnesium oxide? Materials You may use any of the following materials during your investigation: Consumables Magnesium ribbon (3-4 cm long) Equipment Bunsen burner Lighter Crucible with lid Clay triangle Crucible tongs Ring stand with metal ring Wire gauze square Electronic balance Periodic table

12 Safety Precautions Follow all normal lab safety rules. Magnesium is a flammable solid, and hydrochloric acid is a corrosive liquid. Your teacher will explain relevant and important information about working with the chemicals associated with this investigation. In addition, take the following safety precautions: Wear indirectly vented chemical-splash goggles Use caution when working with Bunsen burners. They can burn skin, and combustibles and flammables must be kept away from the open flame. If you have long hair, tie it back behind you head. Inspect crucible for cracks. Exchange cracked or broken crucibles for a new one. Wipe crucible and lid clean before using them. Be careful with a crucible after removing it from a flame it will be extremely hot. Handle all glassware with care. Wash your hands with soap and water before leaving the laboratory. Investigation Proposal Required? Yes No Getting Started The first step in your investigation is to determine the percent composition of magnesium oxide. As noted earlier, when magnesium is heated, it combines with oxygen to form magnesium oxide. The law of conservation of mass suggests that the total mass of the products of a chemical reaction must equal the mass of the reactants. The mass of the oxygen is a sample of magnesium oxide will therefore equal the mass of the magnesium oxide minus the mass of the original piece of magnesium that was used in the reaction. You can use this information to determine the percent of oxygen by mass found in magnesium oxide. You will, however, need to first determine what type of data you need to collect and how you will collect the data to be able to make this calculation. To determine what type of data you need to collect, think about the following questions: o What type of measurements or observations will you need to record during your investigation? o When will you need to make these measurements or observations? To determine how you will collect the data, think about the following questions: o What equipment will you need to use to make magnesium oxide? o How much magnesium will use to make magnesium oxide? o How will you know when all the magnesium has been converted to magnesium oxide? o How will you make sure that your data are of high quality (i.e., how will you reduce error)? o How will you keep track of the data you collect and how will you organize is? The second step in your investigation is to calculate the empirical formal for magnesium oxide from its percent composition. This calculation requires two steps. First, you will need to calculate the number of moles of magnesium and oxygen in your sample of magnesium oxide. Next, you will need to use the ratio between the number of moles of magnesium and the number of moles of oxygen to calculate the empirical formula of magnesium oxide. Keep in mind that fractions of atoms do not exist. Connections to Crosscutting Concepts As you work through your investigation, be sure to think about how scale, proportion, and quantity play a role in science; the flow of energy and matter within systems; how scientific knowledge can change over times in light of new evidence, and the different methods used in scientific investigations.

13 Initial Argument Once your group has finished collecting and analyzing your data, you will need to develop an initial argument. Your argument must include a claim, which is your answer to the guiding question. Your argument must also include evidence in support of your claim. The evidence is your analysis of the data and your interpretation of what the analysis means. Finally, you must include a justification of the evidence in your argument. You will therefore need to use a scientific concept or principle to explain why the evidence that you decided to use is relevant and important. You will create your initial argument on a whiteboard. Your whiteboard must include all the information shown in Figure L26.1. FIGURE L26.1 Argument presentation on a whiteboard Argumentation Session The argumentation session allows all of the groups to share their arguments. One member of each group stays at the lab station to share that group s argument, while the other members of the group go to the other lab stations one at a time to listen to and critique the arguments developed by their classmates. The goal of the argumentation session is not to convince others that your argument is the best one; rather, the goal is to identify errors or instances of faulty reasoning in the initial arguments so these mistakes can be fixed. You will therefore need to evaluate the content of the claim, the quality of the evidence used to support the claim, and the strength of the justification of the evidence included in each argument that you see. To critique an argument, you might need more information that what is included on the whiteboard. You might, therefore, need to ask the presenter one or more of the following questions, such as: How did your group collect the data? Why did you use that method? Is that the only way to interpret the results of your group s analysis? How do you know that your interpretation of the analysis is appropriate? Why did your group decide to present your evidence in that manner? What other claims did your group discuss before deciding on that one? Why did you abandon those alternative ideas? How confident are you that your group s claim is valid? What could you do to increase your confidence? Once the argumentation session is complete, you will have a chance to meet with your group and revise your original argument. Your group might need to gather more data or design a way to test one or more alternative claims as part of this process. Remember, your goal at this stage of the investigation is to develop the most valid or acceptable answer to the research/guiding question! Report Once you have completed your research, you will need to prepare an investigation report that consists of three sections that provide answers to the following questions: 1. What question were you trying to answer and why? 2. What did you do during your investigation and why did you conduct your investigation in this way? 3. What is your argument? Your report should answer these questions in four pages or less. The report must be typed and any diagrams, figures, or tables should be embedded into the document. Be sure to write in a persuasive style; you are trying to convince others that your claim is acceptable and valid!

14 Investigation Timeline Option D Stage 1: Identify the task and the guiding question. Hold a tool talk Day 1 Small groups of students then 50 Minutes Stage 2: Design a method and collect data Groups then Stage 3: Analyze data and develop a tentative argument Day 2 Each group then shares its argument during an Stage 4: Argumentation session If needed, groups can Collect additional data or reanalyze the collected data 50 Minutes The teacher then leads an The teacher then leads an Stage 5: Explicit and reflective discussion Homework Individual students then Stage 6: Write and investigation report Day 3 The report then goes through a Stage 7: Double-blind group peer review 30 Minutes Homework Each student then Stage 8: Revises and submits his or her report

15 Investigation Proposal A The guiding question Hypothesis 1 Hypothesis 2 IF IF AND Procedure The Test What data will you collect? How will you analyze the data? What safety precautions will you follow? Predicted Result if hypothesis 1 is valid THEN Predicted result if hypothesis 2 is valid THEN AND The Actual Results I approve of this investigation. Instructor s Signature Date