Science is Quantitative IMeasurements, units of measurement and uncertaintyProf. Miroslav D. FilipovićScientific Literacy 2021What’s covered in this Lecture?• Science and the scientific method — RESEARCH PROCESS• Measurements – what they are and what do the numbers really mean?• Units – metric system and imperial system• Numbers – exact and inexact• Significant figures and uncertainty• Scientific notation• Dimensional analysis (conversion factors)Scientific Literacy 2021Research ProcessChoose a topic and search the literatureReview the literature and define the researchquestion(s)Design the research and write the proposalCollect the research dataAnalyse and interpret the research dataWrite the dissertation or thesisThe scientific methodIn order to be able to develop explanations for phenomena.After defining a problemExperiments must be designed and conductedMeasurements must be madeInformation must be collectedGuidelines are then formulated based on a pool ofobservationsHypotheses (predictions) are made, using this data, and then tested,repeatedly.Hypotheses eventually evolve to become laws and these aremodified as new data become availableAn objective point of view is crucial in this process. Personal biasesmust not surface.METHODScientific Literacy 2021The scientific methodAt some level, everything is based on a model of behavior.Even scientific laws change because there are no absolutes.Scientific Literacy 2021MeasurementsAn important part of most experiments involves thedetermination (often, the estimation) of quantity, volume,dimensions, capacity, or extent of something – thesedeterminations are measurementsIn many cases, some sort of scale is used to determine a valuesuch as this. In these cases, estimations rather than exactdeterminations need to be made.Scientific Literacy 2021SI UnitsSystème International d’UnitésScientific Literacy 2021Prefix-Base Unit SystemPrefixes convert the base units into units that areappropriate for the item being measured.Know these prefixes and conversions3.5 Gm = 3.5 x 109 m = 3500000000 mSo, and 0.002 A = 2 mAScientific Literacy 2021Temperature:A measure of the averagekinetic energy of the particles in asample.Kinetic energy is the energy anobject possesses by virtue of itsmotionAs an object heats up, itsmolecules/atoms begin to vibrate inplace. Thus, the temperature of anobject indicates how much kineticenergy it possesses.Farenheit: oF = (9/5)(oC) + 32 oFScientific Literacy 2021In scientific measurements,the Celsius and Kelvin scalesare most often used.The Celsius scale is based onthe properties of water.0°C is the freezing point ofwater.100°C is the boiling point ofwater.Scientific Literacy 2021Temperature:The Kelvin is the SI unit oftemperature.It is based on theproperties of gases.There are no negativeKelvin temperatures.K = °C + 2730 (zero) K = absolute zero = -273 oCScientific Literacy 2021Temperature:The most commonly usedmetric units for volume arethe liter (L) and the milliliter(mL).A liter is a cube 1dm long on each side.A milliliter is acube 1 cm long oneach side.1 m = 10 dm = 100 cm1 m = 10 dm(1 m)3 = (10 dm)3 1 m3= 1000 dm3or 0.001 m3 = 1 dm31 dm = 10 cm(1 dm)3 = (10 cm)3 1 dm3= 1000 cm3or 0.001 dm3 = 1 cm3Incidentally, 1 m3 = 1×106 cm3These are conversion factorsScientific Literacy 2021Volume:Another physical property of a substance –the amount of mass per unit volumer =mVmassvolumee.g. The density of water at room temperature (25oC) is ~1.00 g/mL; at 100oC = 0.96 g/mLDensity does not have anassigned SI unit – it’s acombination of mass andlength SI components.Scientific Literacy 2021Density:Density is temperature-sensitive, because the volume that a sampleoccupies can change with temperature.Densities are often given with the temperature at which they weremeasured. If not, assume a temperature of about 25oC.Scientific Literacy 2021Density:Accuracy versus PrecisionAccuracy refers to the proximity of ameasurement to the true value of aquantity.Precision refers to the proximity of severalmeasurements to each other (Precisionrelates to the uncertainty of ameasurement).For a measured quantity, we can generally improve its accuracy by making moremeasurementsScientific Literacy 2021Measured Quantities and UncertaintyWhenever possible, you should estimate ameasured quantity to one decimal placesmaller than the smallest graduation on a scale.The measured quantity, 3.7, is an estimation;however, we have different degrees of confidencein the 3 and the 7 (we are sure of the 3, but notso sure of the 7).Scientific Literacy 2021Uncertainty in Measured QuantitiesWhen measuring, for example, how much an apple weighs, the mass canbe measured on a balance. The balance might be able to report quantitiesin grams, milligrams, etc.Let’s say the apple has a true mass of 55.51 g. The balance we are usingreports mass to the nearest gram and has an uncertainty of +/- 0.5 g.The balance indicates a mass of 56 gThe measured quantity (56 g) is true to some extent and misleading tosome extent.The quantity indicated (56 g) means that the apple has a true mass whichshould lie within the range 56 +/- 0.5 g (or between 55.5 g and 56.5 g).Scientific Literacy 2021Significant FiguresThe term significant figures refers to the meaningfuldigits of a measurement.The significant digit farthest to the right in themeasured quantity is the uncertain one (e.g. for the56 g apple)When rounding calculated numbers, we pay attentionto significant figures, so we do not overstate theaccuracy of our answers.In any measured quantity, there will be some uncertainty associatedwith the measured value. This uncertainty is related to limitations of thetechnique used to make the measurement.Scientific Literacy 2021Exact quantitiesIn certain cases, some situations will utilize relationshipsthat are exact, defined quantities.For example, a dozen is defined as exactly 12 objects(eggs, cars, donuts, whatever…)1 km is defined as exactly 1000 m.1 minute is defined as exactly 60 seconds.Each of these relationships involves an infinite numberof significant figures following the decimal place whenbeing used in a calculation.Relationships between metric units are exact (e.g. 1 m = 1000 mm, exactly)Relationships between imperial units are exact (e.g. 1 yd = 3 ft, exactly)Relationships between metric and imperial units are not exact (e.g. 1.00 in = 2.54 cm)Scientific Literacy 2021Significant Figures1. All nonzero digits are significant. (1.644 has foursignificant figures)2. Zeroes between two non-zero figures are themselvessignificant. (1.6044 has five sig figs)3. Zeroes at the beginning (far left) of a number are neversignificant. (0.0054 has two sig figs)4. Zeroes at the end of a number (far right) are significant ifa decimal point is written in the number. (1500. has foursig figs, 1500.0 has five significant figures)(For the number 1500, assume there are two significantfigures, since this number could be written as 1.5 x 103.)When a measurement is presented to you in a problem, you need to know how manyof the digits in the measurement are actually significant.Scientific Literacy 2021RoundingReporting the correct number of significant figures forsome calculation you carry out often requires that youround the answer to the correct number of significantfigures.Rules: round the following numbers to 3 sig figs5.4835.486(this would round to 5.48, since 5.483 is closer to5.48 than it is to 5.49)(this would round to 5.49)If calculating an answer through more than one step,only round at the final step of the calculation.Scientific Literacy 2021Significant FiguresWhen addition or subtraction is performed, answers are rounded to the leastsignificant decimal place.When multiplication or division is performed, answers are rounded to the numberof digits that corresponds to the least number of significant figures in any of thenumbers used in the calculation.Example: 6.2/5.90 = 1.0508… = 1.1Example: 20.4 + 1.332 + 83 = 104.732 = 105“rounded”Scientific Literacy 2021Significant FiguresIf both addition/subtraction and multiplication/division are usedin a problem, you need to follow the order of operations, keepingtrack of sig figs at each step, before reporting the final answer.1) Calculate (68.2 + 14). Do not round the answer, but keep in mind how many sig figsthe answer possesses.2) Calculate [104.6 x (answer from 1st step)]. Again, do not round the answer yet, butkeep in mind how many sig figs are involved in the calculation at this point.3) , and then round the answer to the correct sig figs.Scientific Literacy 2021Significant FiguresIf both addition/subtraction and multiplication/division are usedin a problem, you need to follow the order of operations, keepingtrack of sig figs at each step, before reporting the final answer.Despite what our calculatortells us, we know that thisnumber only has 2 sig figs.Despite what our calculatortells us, we know that thisnumber only has 2 sig figs.Our final answer shouldbe reported with 2 sig figs.Scientific Literacy 2021An example using significant figuresIn the lab, you are required to measure the heightand diameter of a metal cylinder, in order to get itsvolume Sample data:V = pr2hheight (h) = 1.58 cmdiameter = 0.92 cm; radius (r) = 0.46 cmVolume = pr2h =p(0.46 cm)2(1.58 cm)2 sig figs 3 sig figs = 1.050322389 cm3If you are asked toreport the volume,you should round yourOnly operation here Answer = 1.1 cm3 answer to 2 sig figsis multiplicationScientific Literacy 2021Calculation of DensityIf your goal is to report the density of the cylinder(knowing that its mass is 1.7 g), you would carry outthis calculation as follows: 1.050322389 31.7cmg= 1.61855066… 3cmg=Use the non-rounded volume figure for the calculation of the density. If a rounded volumeof 1.1 cm3 were used, your answer would come to 1.5 g/cm3Then round the answer to the propernumber of sig figs1.6 3cmg=Please keep in mind that although the “non-rounded”volume figure is used in this calculation, it is still understoodthat for the purposes of rounding in this problem, it containsonly two significant figures (as determined on the last slide)Scientific Literacy 2021𝜌 =𝑚 𝑉Dimensional Analysis(conversion factors)The term, “dimensional analysis,” refers to a procedure that yields the conversion ofunits, and follows the general formula:Desired UnitsGiven UnitsDesired UnitsGiven Units ____ ÷ ÷ =ö øæ ç ç èconversion factorScientific Literacy 2021Some useful conversionsThis chart shows all metric – imperial(and imperial – metric) systemconversions. They each involve acertain number of sig figs.Metric – to – metric and imperial –to – imperial conversions are exactquantities.Examples:16 ounces = 1 pound1 kg = 1000 gexactrelationshipsScientific Literacy 2021Sample ProblemA calculator weighs 180.5 g. What is its mass, inkilograms?Desired UnitsGiven UnitsDesired UnitsGiven Units ____ ÷ ÷ =ö øæ ç ç èkggkggGiven UnitsDesired Unitsg 0.180510001180.5__180.5 ÷ ÷ =ö øç ç è æ ÷ ÷ ø ö = ç ç è æ“given units” are grams, g“desired units” are kilograms. Make a ratio that involves both units.Since 1 kg = 1000gBoth 1 kg and 1000 g are exact numbershere (1 kg is defined as exactly 1000 g);assume an infinite number of decimalplaces for theseThe mass of the calculator has four sig figs.(the other numbers have many more sig figs)The answer should be reported with four sig figsconversion factor is made using this relationshipScientific Literacy 202Dimensional AnalysisAdvantages of learning/using dimensional analysis for problemsolving:• Reinforces the use of units of measurement• You don’t need to have a formula for solving mostproblemsHow many moles of H2O are present in 27.03g H2O?Scientific Literacy 2021 Sample ProblemA car travels at a speed of 50.0 miles per hour (mi/h).What is its speed in units of meters per second (m/s)?Two steps involved here:a measured quantity • Convert miles to meters• Convert hours to secondsDesired UnitsGiven UnitsDesired UnitsGiven Units ____ ÷ ÷ =ö øæ ç ç è0.621 mi = 1.00 km1 km = 1000 m1 h = 60 min1 min = 60 shmi50.0 ÷ö øæçèmikm0.6211ö÷øæçèkmm11000ö÷øæçè60min1hö÷øæçè60s1minms= 22.3653605296…ms= 22.4should be 3 sig figsScientific Literacy 2021

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