Law vs. Theory A law summarizes what happens A theory (model) is an attempt to explain why it happens. Unit 2: (Chapter 5) Measurements and Calculations Cartoon courtesy of NearingZero.net Steps in the Scientific Method 1. Observations - quantitative - qualitative 2. Formulating hypotheses - possible explanation for the observation 3. Performing experiments - gathering new information to decide whether the hypothesis is valid YOU WILL NEED YOUR CALCULATOR TO PARTICIPATE IN CLASS AND PERFORM MANY OF THE PROBLEMS THAT WE WILL BE DOING. BRING YOUR CALCULATOR!! Outcomes Over the Long-Term Theory (Model) - A set of tested hypotheses that give an overall explanation of some natural phenomenon. Natural Law - The same observation applies to many different systems - Example - Law of Conservation of Mass Measuring Matter in Two Ways Last week we looked at many Qualitative Measurements: which are usually descriptive like observations. Now it is important to start making Quantitative Measurements: measurements in the form of numbers and units.
Numbers in Science Picture a microscopic cell. Scientific Notation Scientists need a way to express extremely LARGE and extremely SMALL numbers in their quantitative measurement. Scientific notation shows the product of two numbers: Picture the galaxy. A coefficient X 10 to some exponent 5.1 Scientific Notation In science, we deal with some very LARGE numbers: 1 mole = 602000000000000000000000 In science, we deal with some very SMALL numbers: Mass of an electron = 0.000000000000000000000000000000091 kg Scientific Notation: A method of representing very large or very small numbers in the form: M x 10 n M is a number between 1 and 10 n is an integer Imagine the difficulty of calculating the mass of 1 mole of electrons! 0.000000000000000000000000000000091 kg x 602000000000000000000000??????????????????????????????????? 2 500 000 000 9 8 7. Step #1: Insert an understood decimal point Step #2: Decide where the decimal must end up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form M x 10 n 6 5 4 3 2 1
2.5 x 10 9 The exponent is the number of places we moved the decimal. PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION ADDITION AND SUBTRACTION 0.0000579 1 2 3 4 5 Step #2: Decide where the decimal must end up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form M x 10 n Review: Scientific notation expresses a number in the form: 1 M < 10 M x 10 n n is an integer 5.79 x 10-5 The exponent is negative because the number we started with was less than 1. 4 x 10 6 IF the exponents are + 3 x 10 6 the same, we simply add or subtract the 7 x 10 6 numbers in front and bring the exponent down unchanged.
4 x 10 6 The same holds true - 3 x 10 6 for subtraction in scientific notation. 1 x 10 6 A Problem for you 2.37 x 10-6 + 3.48 x 10-4 4 x 10 6 If the exponents are + 3 x 10 5 NOT the same, we must move a decimal to make them the same. Solution 002.37 x 10-6 + 3.48 x 10-4 4.00 x 10 6 4.00 x 10 6 + 3.00 x 10 5 +.30 x 10 6 Move the decimal on the smaller number! 4.30 x 10 6 Solution 0.0237 x 10-4 + 3.48 x 10-4 3.5037 x 10-4
Scientific Notation and your Calculator Using Scientific Notation: Use an "e" to replace the "x10". Don't skip any spaces. For example: use 3.25e-6 to represent 3.25 x 10-6 *Make sure that when you enter a negative exponent that you use the (-) button instead of the button for subtraction. S.I. Units The International System of Units is used ALMOST exclusively worldwide (the U.S. is one the of the exceptions). ALL science is done using S.I. units. Most calculators (TI-89 and higher) you will need to hit the 2 nd button and the comma button. 5.2 Units Measurement - quantitative observation consisting of 2 parts Part 1 - number Part 2 - scale (unit) Examples: 20 grams 6.63 x 10-34 Joule seconds You must always write down your units in chemistry! The United States System of Measurement In 1975, the U.S. government attempted to adopt the metric system with little success. The U.S. currently uses the English System of Measurement. A Short History of Standard Units INTERNATIONAL SYSTEM (le Système International, SI) Humans did not always have standards by which to measure temperature, time, distance, etc. It wasn t until 1790 that France established the first metric system.
SI Prefixes Common to Chemistry Prefix Unit Abbr. Exponent Mega M 10 6 Try to avoid parallax errors. Parallax errors arise when a meniscus or needle is viewed from an angle rather than from straight-on at eye level. Kilo k 10 3 Deci d 10-1 Centi c 10-2 Milli m 10-3 Micro µ 10-6 Nano n 10-9 Incorrect: viewing the meniscus from an angle Correct: Viewing the meniscus at eye level 5.3 Measuring Volume Temperature Mass The glass cylinder has etched marks to indicate volumes, a pouring lip, and quite often, a plastic bumper to prevent breakage. Graduated Cylinders Reading the Meniscus Always read volume from the bottom of the meniscus. The meniscus is the curved surface of a liquid in a narrow cylindrical container. Measuring Volume Determine the volume contained in a graduated cylinder by reading the bottom of the meniscus at eye level. Read the volume using all certain digits and one uncertain digit. Certain digits are determined from the calibration marks on the cylinder. The uncertain digit (the last digit of the reading) is estimated.
Use the graduations to find all certain digits There are two unlabeled graduations below the meniscus, and each graduation represents 1 ml, 52 so ml. the certain digits of the reading are 25mL graduated cylinder What is the volume of liquid in the graduate? 1 1. _ 5 0_ ml Estimate the uncertain digit and take a reading The meniscus is about eight tenths of the way to the next graduation, so the final digit in the reading is 0.8 ml. The volume in the graduated cylinder is 52.8 ml. 100mL graduated cylinder What is the volume of liquid in the graduate? 5 2. _ 7 ml 10 ml Graduate What is the volume of liquid in the graduate? 6_. _ 6 _ 2 ml Self Test Examine the meniscus below and determine the volume of liquid contained in the graduated cylinder. The cylinder contains: 7 6. _ 0 ml
The Thermometer o Determine the temperature by reading the scale on the thermometer at eye level. o Read the temperature by using all certain digits and one uncertain digit. Measuring Mass - The Beam Balance o Certain digits are determined from the calibration marks on the thermometer. o The uncertain digit (the last digit of the reading) is estimated. o On most thermometers encountered in a general chemistry lab, the tenths place is the uncertain digit. Our balances have 4 beams the uncertain digit is the thousandths place ( _. X) Do not allow the tip to touch the walls or the bottom of the flask. If the thermometer bulb touches the flask, the temperature of the glass will be measured instead of the temperature of the solution. Readings may be incorrect, particularly if the flask is on a hotplate or in an ice bath. Balance Rules In order to protect the balances and ensure accurate results, a number of rules should be followed: Always check that the balance is level and zeroed before using it. Never weigh directly on the balance pan. Always use a piece of weighing paper to protect it. Do not weigh hot or cold objects. Clean up any spills around the balance immediately. Reading the Thermometer Determine the readings as shown below on Celsius thermometers: _ 8 _ 7. _ 4 C 3 5. _ 0 C Mass and Significant Figures o Determine the mass by reading the riders on the beams at eye level. o Read the mass by using all certain digits and one uncertain digit. othe uncertain digit (the last digit of the reading) is estimated. o On our balances, the thousandths place is uncertain.
Determining Mass 1. Place object on pan 2. Move riders along beam, starting with the largest, until the pointer is at the zero mark 1 1 _ 4. _ 4_ 9_ 7 Read Mass More Closely Check to see that the balance scale is at zero Measuring Length on a meter stick What is the value in cm? in mm? 41.45 cm 414.5 mm 1 1 _ 4.??_? 5.4 Uncertainty in Measurement A digit that must be estimated is called uncertain. Read Mass A measurement always has some degree of uncertainty due to the limitations of the measuring device.
Why Is there Uncertainty? Measurements are performed with instruments No instrument can read to an infinite number of decimal places Which of these balances has the greatest uncertainty in measurement? Pg 124 of text Precision and Accuracy Accuracy refers to the agreement of a particular value with the true value. Precision refers to the degree of agreement among several measurements made in the same manner. Rules for Counting Significant Figures - Details Nonzero integers always count as significant figures. 3456 has 4 sig figs. Neither accurate nor precise Precise but not accurate Precise AND accurate 5.5 Significant Figures Significant Figures» The digits in a measured quantity that are known exactly, plus one digit that is inexact. For Example» If you divided 1.20 g by 0.07023 ml, your calculator would tell you 17.08671507903 g/ml.» You should probably round the number off, but where? Rules for Counting Significant Figures - Details Zeros - Leading zeros do not count as significant figures. 0.0486 has 3 sig figs.» They are a set of rules to use that will tell you how many significant figures there are and where to round off after a calculation.
Rules for Counting Significant Figures - Details Zeros - Captive zeros always count as significant figures. 16.07 has 4 sig figs. Sig Fig Practice #1 How many significant figures in each of the following? 1.0070 m 5 sig figs 17.10 kg 4 sig figs 100,890 L 5 sig figs 3.29 x 10 3 s 3 sig figs 0.0054 cm 2 sig figs 3,200,000 2 sig figs Rules for Counting Significant Figures - Details Zeros Trailing zeros are significant only if the number contains a decimal point. 9.300 has 4 sig figs. Rules for Significant Figures in Mathematical Operations Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation. 6.38 x 2.0 = 12.76 13 (2 sig figs) Rules for Counting Significant Figures - Details Exact numbers have an infinite number of significant figures. 1 inch = 2.54 cm, exactly Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m x 7.0 m 22.68 m 2 23 m 2 100.0 g 23.7 cm 3 4.219409283 g/cm 3 4.22 g/cm 3 0.02 cm x 2.371 cm 0.04742 cm 2 0.05 cm 2 710 m 3.0 s 236.6666667 m/s 240 m/s 1818.2 lb x 3.23 ft 5872.786 lb ft 5870 lb ft 1.030 g 2.87 ml.3588850174g/ml.359 g/ml
Rules for Significant Figures in Mathematical Operations Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. 6.8 + 11.934 = 18.734 18.7 (3 sig figs) US Conversion Steps (Dimensional Analysis) 1. Read the question to figure out what you have/know for information. The question will provide you with information that identifies your starting point and your final destination. Starting point = the number and unit provided by the question Final destination = the units desired after converting 2. Using the information gathered from the question, write your starting point and your final destination. 3. Determine the means in which you will get from your starting point to your final destination (simply find connections or conversion factors between your starting and final unit). 4. Create a fraction by placing your starting point over one. 5. Multiply between fractions. 6. Write in the bottom unit of the new fraction. This should be the same as the top unit of the previous fraction. 7. Write one set of connections or conversion factors into the fraction. Your bottom unit will guide you. 8. Ask yourself, Do I have the desired unit (final destination) on the top of the new fraction? (Go back to step 5) NO (Proceed to step 9) YES 9. Cancel any units that are diagonal. (This should leave you with only the units that represent your final destination) 10. Multiply the top of the fractions multiply the bottom of the fractions divide the top by the bottom. Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m + 7.0 m 10.24 m 10.2 m 100.0 g - 23.73 g 76.27 g 76.3 g 0.02 cm + 2.371 cm 2.391 cm 2.39 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb 2.030 ml - 1.870 ml 0.16 ml 0.160 ml Practice Conversions 1. How many seconds are in 6 minutes? 360 seconds 2. How many centimeters are in 27 inches? 68.58 centimeters 3. If a truck weighs 15,356 pounds, how many tons is it? 7.678 tons 4. If you had 10.5 gallons of milk, how many pints would you have? 84 pints 5. Students go to school for 180 days. How many minutes is this equal to? 259,200 minutes 5.6 Problem solving and Dimensional Analysis Dimensional Analysis, also called, Factor-Label problems use multiplication of the units to convert from one unit to another. How many seconds are in 6 minutes? 6 minutes seconds Step 1 Read the question and determine what information it provides you with (starting point & final destination) Step 3 Determine how you will get 1 minute = 60 seconds from your (6 minutes) ( 60 seconds ) = your final 1 ( 1 minute ) starting point to destination (list any connections or Final Destination Starting Point Step 2 Write down your starting point and your final destination (6)(60 seconds) (1)(1) Step 9 Cancel all diagonal units. Step 5 Multiply Once this is done, your final conversion Step 10 Multiply the top of the between fractions = 360 seconds factors) Step 6 Write in Step the 7 bottom Write Step the appropriate 8 destination Determine should if this be top the only fractions; multiply the bottom of Step 4 Create unit of a fraction the new by fraction conversion (this is factor unit into is unit the the desired left in unit this (your case seconds the fractions; divide the product placing your the starting same point as the over top fraction. unit of your Your final bottom destination). of unit the will top by In the this product case of the one previous fraction) guide the you. answer is YES, so bottom we move on to step 9
How many centimeters are in 27 inches? 27 inches centimeters Step 1 Read the question and determine what information it provides you with (starting point & final destination) Step 3 Determine how you will get from your starting point Final Destination (27 inches) ( 2.54 cm ) = 1 ( 1 inch ) Starting Point Step 2 Write down your starting point and your final destination 1 inch = 2.54 centimeters (27)(2.54 cm) (1)(1) to your final destination (list any connections Step 9 Cancel all diagonal units. or conversion Step 5 Multiply Once this is done, your final Step 10 Multiply the top of the factors) between fractions = 68.58 centimeters Step 6 Write in Step the 7 bottom Write Step the appropriate 8 destination Determine should if this be top the only fractions; multiply the bottom of Step 4 Create unit of a fraction the new by fraction conversion (this is factor unit into is the unit the desired left unit in (your this case the fractions; divide the product placing your the starting same point as the over top fraction. unit of your Your final bottom destination). centimeters of unit the will top by In the this product case of the one previous fraction) guide the you. answer is YES, so bottom we move on to step 9 Students go to school for 180 days. How many minutes is this equal to? (180 days) your final 1 180 days minutes Step 1 Read the question and determine what information it provides you with (starting point & final destination) Step 3 Determine how you will get from your starting point to Final Destination 1 day = 24 hours 1 hour = 60 minutes Starting Point Step 2 Write down your starting point and your final destination ( 24 hours ) ( 60 minutes) ( 1 day ) ( 1 hour ) destination (list any connections or Step 9 Cancel all diagonal units. conversion (180)(24)(60 Step 5 Multiply minutes) Step 5 Multiply factors) = between Step 6 fractions Write Step in the 7 bottom Write the appropriate Once this Step is 10 done, Multiply your final the top between fractions Step Step 7 8 Write Determine the appropriate if this top Step 4 Create unit a of fraction the new (1)(1)(1) by fraction conversion Step (this 6 factor is Write destination into = the unit the bottom should be the only unit the desired unit (your placing your starting the same point as the over top fraction. unit unit of of Your your the new bottom fraction Step conversion left final unit bottom will (this 8 Determine factor in this case of the into minutes fractions; if this the top fraction. divide destination). the product In of this the case top one previous fraction) the same guide as the you. top unit unit is of the Your your desired bottom unit unit (your will previous fraction) final the answer destination). by guide the is product NO, you. In so this of we the move case the answer back is to YES, bottom step so 5we move on to step 9 259,200 of the fractions; minutes multiply the If a truck weighs 15,356 pounds, how many tons is it? 15,356 pounds tons Step 1 Read the question and determine what information it provides you with (starting point & final destination) Step 3 Determine how you will get from your Final Destination 2000 pounds = 1 ton starting (15,356 point lbs.) to ( 1 ton ) your final = destination 1(list ( 2000 lbs. ) Starting Point Step 2 Write down your starting point and your final destination (15,356)(1 ton) (1)(2000) any connections or Step 9 Cancel all diagonal units. conversion Step 5 Multiply Once this is done, your final Step 10 Multiply the top of the factors) between fractions = 7.678 tons Step 6 Write in Step the 7 bottom Write Step the appropriate 8 destination Determine should if this be top the only fractions; multiply the bottom of Step 4 Create unit of a fraction the new by fraction conversion (this is factor unit into is the unit the desired left unit in this (your case tons the fractions; divide the product placing your the starting same point as the over top fraction. unit of your Your final bottom destination). of unit the will top by In the this product case of the one previous fraction) guide the you. answer is YES, so bottom we move on to step 9 Online Tutorials http://www2.wwnorton.com/college/chemistry/gilbe rt/tutorials/ch1.htm (click on view tutorial for dimensional analysis) http://www.wfu.edu/~ylwong/chem/dimensionanalys is/practice/index.html (click on examples under dimensional analysis on the left side of the page) http://chemistry.alanearhart.org/tutorials/dimana l/ Interactive Quiz http://chem.lapeer.org/exams/dimanalquiz.html If you had 10.5 gallons of milk, how many pints would you have? 10.5 gallons pints Step 1 Read the question and determine what information it provides you with (starting point & final destination) Step 3 Determine how you will get from your Final Destination 1 gallon = 4 quarts 1 quart = 2 pints Starting Point Step 2 Write down your starting point and your final destination (10.5 starting gallons) point to ( 4 quarts ) ( 2 pints ) your final destination 1 (list ( 1 gallon ) ( 1 quart ) any connections or conversion (10.5)(4)(2 Step 56 Multiply Write pints) Step the Step 7 bottom Write 5 Multiply the Step factors) = Step Step appropriate Step 7 Step Write 8 9 Determine 810 between unit of the fractions new fraction (this is 84 Cancel Determine the Multiply appropriate the conversion factor into pints all if diagonal this if top this top units. top of between fractionsconversion (1)(1)(1) unit = is the fractions; multiply the Step 6 Write Once in unit the this bottom desired the factor is desired into done, unit (your unit the final (your Step 4 Create the same a fraction as the by top fraction. unit of Your your bottom fraction. bottom final destination). unit will of the fractions; divide unit of the new fraction final destination). Your bottom unit (this should is In this be In the case this will only case placing your starting previous point over fraction) guide you. the product of the top by the the same as the answer top the unit unit answer guide you. left product is of NO, your in is of this YES, so the we case so bottom move we pints move one previous fraction) back to on step to step 5 9 5.7 Temperature Scales There are three temperature scales in use that you need to be familiar with.
Temperature: Celsius and Kelvin A measure of the average kinetic energy of the particles in a sample. K= o C + 273 Temperature A measure of how hot or how cold an object is. SI Unit: Kelvin ( K ) Note: not a degree Absolute Zero= 0 K Kelvin Scale English guy, William Kelvin Measures molecular movement Theoretical point of ABSOLUTE ZERO is when all molecular motion stops (no negative numbers) Divisions (degrees) are the same as in Celsius Temperature Scales Absolute Zero Theoretical point where there is absolutely no movement of molecules in matter and a measure of ZERO ENERGY This is not something that we ever witness, scientists have only theorized this point
Farenheit and Celsius Solution 2) Placing the mass and volume of the osmium metal into the density setup, we obtain o F= (1.8 o C ) +32 D = mass = 50.00 g = volume 2.22 cm 3 = 22.522522 g/cm 3 = 22.5 g/cm 3 5.8 Density Density is the amount of matter present in a given volume of subtance. Volume Displacement A solid displaces a matching volume of water when the solid is placed in water. Density compares the mass of an object to its volume D = mass = g or g volume ml cm 3 25 ml 33 ml Note: 1 ml = 1 cm 3 Learning Check Osmium is a very dense metal. What is its density in g/cm 3 if 50.00 g of the metal occupies a volume of 2.22cm 3? 1) 2.25 g/cm 3 2) 22.5 g/cm 3 3) 111 g/cm 3 Learning Check What is the density (g/cm 3 ) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 ml to 33 ml? 1) 0.2 g/ cm 3 2) 6 g/m 3 3) 252 g/cm 3 25 ml 33 ml
Solution 2) 6 g/cm 3 Volume (ml) of water displaced = 33 ml - 25 ml = 8 ml Volume of metal (cm 3 ) = 8 ml x 1 cm 3 = 8 cm 3 1 ml Density of metal = mass = 48 g = 6 g/cm 3 volume 8 cm 3 Density as Conversion Factors A substance has a density of 3.8 g/ml. Density Equality = 3.8 g/ml Conversion factors. 3.8 g = 1 ml 3.8 g and 1 ml 1 ml 3.8 g Learning Check Which diagram represents the liquid layers in the cylinder? (K) Karo syrup (1.4 g/ml), (V) vegetable oil (0.91 g/ml,) (W) water (1.0 g/ml) 1) 2) 3) V W K W K V K V W Learning Check You have 3 metal samples. Which one will displace the greatest volume of water? 1 2 3 25 g Al 2.70 g/ml 45 g of gold 19.3 g/ml 75 g of Lead 11.3 g/ml Discuss your choice with another student. Solution (K) Karo syrup (1.4 g/ml), (V) vegetable oil (0.91 g/ml,) (W) water (1.0 g/ml) Solution 1) 25 g Al x 1 ml = 9.2 ml 2.70 g 1) V W K 25 g Al 2.70 g/ml
Density Practice Problems 1. A student measures the mass of a piece of metal to be 4.0g and it has a volume of 1.5mL what is the density of this metal? 2. The Density of CO 2 gas is 1.8 grams per liter. What is the mass of 0.2L of CO 2 gas?