Dimensional Analysis I: Length, Mass, Volume and Area Conversions

Transcription

1 EXPERIMENT Dimensional Analysis I: Length, Mass, Volume and Area Conversions Student will use the Dimensional Analysis Map to introduce / reinforce the process of basic unit conversions in chemistry. OBJECTIVE In laboratories throughout the world, many different measurements are obtained while performing scientific experiments. Most are measurements of mass, volume and length. Early in the history of science, an effort to globalize the language of science led to the adoption of specific units for particular measurements. The SI (Système International) unit for mass is the kilogram and the unit for length is the meter. While most Americans have heard of these units, the majority are much more comfortable using ounces, pounds, and tons for mass; ounces, cups, pints, quarts, and gallons for volume; and inches, feet, yards, and miles for length. Note that ounces are used twice, so the context of the sentence is needed to determine which ounces to use. English-speaking people have found it useful to speak of small (ounces, ounces, inches), medium (pounds, quarts, yards), and large (tons, gallons, miles) quantities by using a variety of units when speaking of mass, volume, or length. The scientific community uses small, medium, and large units, but without changing the base unit. However, before learning more about the Metric system see how much you know about your own system - the English system. Write the correct number in the following blanks to make the equation TRUE. inches = foot cups = quart ounces = gallon inches = yard ounces = pound pounds = ton yards = mile feet = mile quarts = gallon ounces = quart cups = gallon ounces = cup Unlike the English system where many different numbers (numbers like: 3, 4, 8, 2, 6, 32, 28, 760, 2000, 5280) must be memorized to convert between units, the Metric Copyright 2005 Chem2 LLC. No part of this work may be reproduced, transcribed, or used in any form by any means graphic, electronic, or mechanical, including, but not limited to, photocopying, recording, taping, Web distribution, or information storage or retrieval systems without the prior written permission of the publisher. For permission to use material from this work, contact us at support@chem2labs.com. Printed in United States of America.

2 system uses multiples of 0, in a word form, immediately preceding gram, liter, or meter. The origins of these prefixes are truly international: the Greeks supplied the prefixes for very large multipliers (tera, giga, and mega) as well as micro and nano, the French contributed the prefixes for the next smaller multipliers (kilo, hecto, deka, deci, and centi), milli came from the Romans, pico from the Spanish and femto from the Danish. The metric prefixes most commonly used in General Chemistry are: femto (E-5), pico (E- 2), nano (E-9), micro (E-6), milli (E-3), centi (E-2), and kilo (E3). From this information, the following equalities can be derived: Gram Liter Meter fm = E-5 meter fl = E-5 liter fg = E-5 gram pm = E-2 meter pl = E-2 liter pg = E-2 gram nm = E-9 meter nl = E-9 liter ng = E-9 gram μm = E-6 meter μl = E-6 liter μg = E-6 gram mm = E-3 meter ml = E-3 liter mg = E-3 gram cm = E-2 meter cl = E-2 liter cg = E-2 gram km = E3 meter kl = E3 liter kg = E3 gram Table From ANY equality (there are 2 in Table ), two conversion factors can be generated. A conversion factor is a fraction where the numerator and denominator are equal although they have different units. For example, = yard is an equality statement. conversion factors are generated from this equality statement by either dividing both sides by or by dividing both sides by yard: 3 3 feet feet yard or and yard yard yard Two yard or yard It is important to note that all conversion factors are equal to and if a particular measurement is multiplied by a conversion factor, it is the same as multiplying it by. For example, to determine the number of yards in feet one could divide by 3 (in your head perhaps) and arrive at yards. Or, you could set up the following equation: Experiment -2

3 yard feet yards Equation As you can see, units, like numbers, will cancel with one another. Essentially all that has been done to the measurement (000.0 feet) is to multiply it by. The original length has not changed, only the units have been changed (000.0 feet and yards represents the exact same distance). You are probably thinking it would be ridiculous to set up an equation to convert feet to yards when it can be easily done in your head and you would be right. However, we will encounter many units in science that will be so unfamiliar to you that you will likely be confused as to whether to multiply 000 by 3 or divide 000 by 3. Imagine the confusion when you have 3, 4, 5 or more conversion factors in a single equation the number of possible ways to arrive at an incorrect answer will become very large!! To avoid mathematical mistakes that can halt a scientific career, you are expected to learn how to set up equations using Dimensional Analysis (canceling out units). We will begin by converting frequently used English units of mass, volume, length and area into other everyday units of the English system. Then, you will convert Metric units of mass, volume, length and area into other Metric units. Finally you will convert Metric units to English units and vice versa. To assist you in this process, this first lab requires you to learn selected conversion factors using an online Flash Card program called Timed / Repetitive Quizzes (TRQs) and then apply this knowledge using the Dimensional Analysis Map. You are expected to memorize the conversion factors used in this lab; the purpose of this lab is to provide interaction with a process dimensional analysis. The most basic requirement in mastering any process is the possession of essential facts it is assumed that students enrolled in this course have both the ability and the motivation to memorize fundamental relationships between different units. To assist you in this task, TRQs will be assigned. The Dimensional Analysis Map is an interactive map where the various units are clickable. Once clicked, the units are placed in a dimensional analysis problem. Next, numbers are entered in front of the units making a conversion factor. In the Length Box (Figure ), there are 3 arrows and 3 conversion factors. Aligning some of these 3 conversion factors correctly in a dimensional analysis problem allows the conversion between any two length units. Without using dimensional analysis, you would have to know 479,00,600 conversion factors to cover all possible length conversions. Experiment -3

4 Figure In addition to the conversions previously mentioned, you will be introduced to conversion factors that span the English and Metric systems, and conversion factors specific to chemistry. One such conversion factor is the Atomic Mass Unit (amu). Atomic Mass Units are chemical units of mass that are appropriate when referring to the mass of a single molecule. Chemists use the unit grams when referring to a large mass of atoms or molecules. The conversion between amu and grams is given by the following equality: x 0 23 amu = gram. The Periodic Table contains the amus of individual atoms of an element. For example, molecule of H 2 SO 4 has 2 Hydrogen atoms, Sulfur atom, and 4 Oxygen atoms so, the mass of molecule of H 2 SO 4 is amu ((2 x.0079 amu) amu + (4 x amu)). If you know the number of amu s for one molecule, you can use this conversion factor to determine the number of molecules present in a collection that has a certain mass given in amu. It should be noted that while chemists speak of amus, there are no balances that can weigh a single molecule or even a small collection of molecules. A complete listing of the conversion factors needed for this laboratory is given in Table 2. Experiment -4

5 Length Volume Mass 2 inches = foot 8 ounces = cup 6 ounces = pound = yard 2 cups = pint 2000 pounds = ton ENGLISH 760 yards = mile 2 pints = quart 5280 feet = mile 4 quarts = gallon 32 ounces = quart 8 pints = gal 4 cups = quart fm = E-5 meter fl = E-5 liter fg = E-5 gram pm = E-2 meter pl = E-2 liter pg = E-2 gram METRIC nm = E-9 meter nl = E-9 liter ng = E-9 gram μm = E-6 meter μl = E-6 liter μg = E-6 gram mm = E-3 meter ml = E-3 liter mg = E-3 gram cm = E-2 meter cl = E-2 liter cg = E-2 gram km = E3 meter kl = E3 liter kg = E3 gram CHEMISTRY Å = E-0 meter 6.022E23 amu = gram 2.54 cm = in.057 qt = L g = lb Table 2 Table 2 introduces a unit of length specific to Chemistry - the Angstrom (Å = E-0 m). The Angstrom was named after Anders Jöns Ångström, a Swedish physicist, who studied the wavelengths of light in the electromagnetic spectrum. inch While not specifically represented in Table 2, area conversions can be derived from the length conversion factors. However, the units will be squared and the numbers associated with the units must be squared. For example, since inch = 2.54 cm, it must also be true that inch 2 = (2.54) 2 cm 2. This becomes obvious when one considers the area of the following square: 2.54cm 2.54cm inch inch x inch = 2.54 cm x 2.54 cm or (2.54) 2 cm 2 Experiment -5

6 Log in to Click on the Lab : Problems assignment. PROCEDURE 2. Click the first Work Problem link. 3. Read the problem... Convert 5.23E6 in 2 Substance A [SubA] to mi 2. Find the Starting number and unit (5.23E6 in 2 ). 4. Mouse over the Instructions link on the Dimensional Analysis Map. 5. Click the Starting unit for the example above, place your mouse over the Inch 2 text (the mouse arrow changes to a hand) and left-click on it. A yellow circle will appear behind the text and in 2 SubA will appear at the beginning and end of the problem that is centered on the Dimensional Analysis Map. 6. Locate a path between the Starting (in 2 ) and Ending (mi 2 ) units. Click any unit on the path (Foot 2 ) and click the Ending unit (Mile 2 ). 7. If you have successfully connected the Starting and Ending units, your units will cancel out leaving only the Ending units (in 2 cancel and ft 2 cancel leaving only mi 2 ). 8. Click the Next Button to display the Dimensional Analysis problem. 9. Fill in all the text fields and make sure Sub A is selected in each dropdown box. For every conversion factor, the computer will check the units, the numbers in front of the units, and a label for each unit. For this first lab, the label for each unit will be SubA (Substance A). In most dimensional analysis calculations, the unit label will change, Experiment -6

7 sometimes several times. For example, if a hamburger costs $5 and contains 3 pickles, how many pickles would you get if you have $5? To answer this question, you must convert $ to hamburgers and then hamburgers to pickles. 0. If the number in front of a unit is, enter the. For squared numbers, you can square the number in a calculator and enter that value, or use parentheses to bookend the number and the ^ to denote an exponent. For example, mm 2 can be entered as E-6 m 2 or (E-3)^2 m 2.. When you have entered the information required for each conversion factor, enter the answer (with the correct number of significant figures) in the appropriate space and press the Submit button. For this lab, the number of significant figures is based on the sigfigs of the number in front of the Staring unit. 2. The computer program will check to see if you have entered the correct information for each conversion factor and if your answer is correct (with the correct number of significant figures). If there is an error, this will be denoted in red. You will be given additional submissions (the number and penalty to be determined by your instructor) to correct the answer. 3. In the example shown, the 5280 was not squared. This caused that part of the problem to be incorrect as well as the answer. 4. Once the 5280 is squared and the answer recalculated (and entered with 3 SigFigs), the problem was graded as correct. 5. One final item it is critical that you enter numbers like 5.23E6 in your calculator using Scientific Notation. First enter 5.23, then click the EE or EXP button, finally enter 6. Do not enter 5.23E6 as 5.23 x 0^6 or x 6 or some other way!! Experiment -7