chapter 3 Scientific Measurement You need your Calculator! 3.1 SIGNIFICANT FIGURES IN MEASUREMENTS The significant figures in a measurement include all the digits that are known precisely plus one last digit that is esmated. Esmated digit is rounded. Rules for determining significant digits: Every nonzero digit is significant. 24.7 34,000 500,000 Zeros appearing in front of all nonzero digits are NOT significant. The decimal point only maers if it follows a nonzero digit. 0.00247 0.0000050 1
Zeros at the end of a number and to the right of a decimal point are always significant. Ex. 247.00m Zeros appearing between significant digits are significant. 24.07 m 2300.0 m Zeros at the end of a number and to the le of the decimal point may or may not be significant. 2470 m (3) 2470.0 m (5) 30.0 m (3) When a number is in scienfic notaon, all numbers in the coefficient are considered significant. 2.470 x 10 3 m (4) 3.003 x 10 8 (4) 2.3 x 10 6 (2) There are two instances in which measurements have an unlimited number of significant figures. 1. Counng 2. Conversion factors Rules for Significant Digits 1. All numbers 1 9 are always significant. 2. Zeroes between significant digits are always significant. 3. Zeroes after significant digits are only significant if they follow a decimal point. 4. Zeroes in front of significant digits are never significant, even after a decimal point. 5. When using scientific notation, all numbers (including zeroes) in the coefficient are significant. 2
Determine the number of Significant Digits Round to 3 Significant Digits 200 200.0 0.0002 0.0000020 301 301,000 500,000 3.460 x 10 4 0.208 405,450 0.0000601 4030 0.310 3.09 x 10 7 1.0000 413,670 0.034678 6,908,595 0.00000567908 Significant Figures in Calculaons An answer cannot be more precise than the least precise measurement from which it was calculated. To round off a number, you must first decide how many significant figures the answer should have. Rounding does NOT change the value of the number!!!! Use the rule for rounding to arrive at your proper number of significant digits. A calculator does not keep track of significant figures or round off answers correctly. Addion and Subtracon The answer to an addion or subtracon problem can have no more digits to the right of the decimal than the number in the problem with the least numbers past the decimal. 3
Addion: Places aer decimal 4.1 + 5.68 + 6.4370192 2.3333 + 1.12 Subtracon: Places aer decimal 18 4.25891 154.1 8.555555 Mulplicaon and Division Mulplicaon: total significant digits The answer to a mulplicaon and division problem can have no more significant digits than the number in the problem with the least number of significant digits. 3.555555 x 12.3 4.6 x 8.375 x 3300 Division: total significant digits 55.6 2.68754 55500 231.078 3.1 SCIENTIFIC NOTATION Scienfic notaon a number is wrien as the product of two numbers: a coefficient and a power of 10. Ex. 36,000 is wrien as 3.6 x 10 4 4
The coefficient is a number equal to or greater than one and less than ten. 1.0000 9.99999 When wring numbers greater than ten in scienfic notaon the exponent is posive and equal to the number of places that the decimal point has been moved to the le. 4.3 x 10 8 Numbers less than one have a negave exponent when you write them in scienfic notaon. 3.17 x 10 4 Rules for Scienfic Notaon The coefficient must be between 1.0 and 9.99. Your coefficient must contain all significant digits. Move the decimal point as many places as necessary until you create a coefficient between 1.0 and 9.99. The exponent will be the number of places you move your decimal point. Moving the decimal to the left makes the number smaller = POSITIVE EXPONENT Numbers greater than 10 always have exponents that are positive. Moving the decimal to the right makes the number larger = NEGATIVE EXPONENT Numbers less than 1.0 always have exponents that are negative. Put into Scientific Notation 634,000 0.0000346 4,908,000 0.005680 122 Put into Expanded Form 3.062 x 10 11 2.1 x 10-8 9.00 x 10 7 5.098 x 10-4 306200000000.000000021 90000000.0005098 5
Addion: Places aer decimal (3.04 x 10 22 ) + (2.457 x 10 15 ) Subtracon: Places aer decimal (7.146 x 10 7 ) (5.47 x 10 10 ) Mulplicaon: total sig figs (2.5 x 10 3 ) x (3.620 x 10 5 ) Division: total sig figs (8.040 x 10 11 ) (4.5 x 10 8 ) You need your Calculator! Review Problem A (9.99 x 10 12 ) (3.33 x 10 2 ) 6
Review Problem B (2.00 x 10 16 ) x (2.50 x 10 21 ) Review Problem C (8.00 x 10 8 ) + (2.0 x 10 12 ) ACCURACY AND PRECISION Accuracy is how close a single measurement comes to the actual dimension or true value of whatever is measured. Accuracy 4.55555 vs. 4.56 Which is more accurate? The more decimal places usually makes a number more accurate. Precision is how close several measurements are to the same value. Which archer do you want to shoot the apple off of your head? Rookie Archer Pro Archer 7
1 shot: archer shooting at the round target is more accurate Rookie Archer Pro Archer After multiple shots, which archer is more precise? TQ Precision Which is the most precise group of measurements for the same item with a mass of 1.222 grams? A. 1.2, 1.4, 1.6 B. 1.222, 2.000, 3.666 C. 0.8222, 1.9876, 2.4321 A measurement can be no more reliable than the measuring instrument. Accuracy depends on the measuring device. Precise measurements are reproducible, but not necessarily accurate. If you use a poor quality measuring device, you will always get the same measurement if you are precise. The precision of a measurement depends on more than one measurement. The accuracy of real measurements depends on the quality of the measuring device. 8
Precision depends more heavily on the skill of the person making the measurement. If you have skill at measuring, then you will always get the same measurement even if the measuring device is of poor quality. Determining Error In order to evaluate the accuracy of a measurement, you must be able to compare it to the true or accepted value. The accepted value is the true or correct value based on reliable references. The experimental value is the measured value determined in the experiment in the laboratory. The difference between the accepted value and the experimental value is the error. Error= accepted value experimental value The percent error is the error, divided by the accepted value, expressed as a percentage of the accepted value. % error = Accepted Experimental Accepted value x 100 Temperature Scales Fahrenheit, Celsius, Kelvin Kelvin = Celsius + 273 Celsius = ( F 32) x 0.56 Fahrenheit = (1.8 x C) + 32 Celsius = Kelvin 273 9
An error can be posive or negave; however, in calculang percent error the absolute value of the error is used. This means that the percent error will always be a posive value. Posive Error : greater than accepted value Negave Error : less than accepted value Example Problems Problem 1 Problem 2 Accepted Value = 137.7 ml Accepted Value = 7.14 g/ml Experimental Value = 131.2 ml Experimental Value = 9.27 g/ml Error = Error = % Error = % Error = Temperature Conversions Fahrenheit Celsius Kelvin 100 68 150 K 0 K 3.2 THE INTERNATIONAL SYSTEM OF UNITS, SI The standard of measurement used in science are those of the metric system. All the units are based on 10 or mulples of 10. The Internaonal System of Units (SI) is a revised version of the metric system. There are seven SI base units. Derived units are not measured directly. Examples: volume, density, pressure Which is why L is not an SI Unit. 10
Length = Meter Mass = Kilogram Volume = m 3 or 1 kl Temperature = Kelvin Time = second Energy = joule Amount of Substance = Mole Metric Prefixes Mega M kilo k Hecto h Deca da deci d cen c milli m micro µ nano n pico p Greater than 1 Less than 1 Units of Length The basic SI unit of length or linear measure, is the meter (m) Units of Volume The space occupied by any sample of maer is called its volume. The SI unit of volume is the amount of space occupied by a cube that is 1 meter along each edge. A more convenient unit of volume for everyday use is the liter (L) The units milliliter and cubic cenmeter are used interchangeably. The volume of any solid, liquid, or gas will change with temperature. Units of Mass Mass is the quanty of maer an object contains. Mass is not weight. Weight is a force. (Gravity required) 11
The weight of an object can change with its locaon, although its mass remains constant regardless of its locaon. On the moon vs. Earth. The mass of an object is measured by comparing it to a standard mass of 1 kilogram (kg), the basic SI unit of mass. A gram is 1/1000 of a kilogram and is a more commonly used unit of mass because a kilogram is relavely large. MEASURING TEMPERATURE Heat moves from the object at the higher temperature to the object at the lower temperature. Temperature is the degree of hotness or coldness of an object. Almost all substances expand with an increase in temperature. Most substances also contract as the temperature decreases. Several temperature scales have been devised. Two readily determined temperatures, the freezing point and the boiling point of water, are used as reference temperature values. On the Celsius temperature scale, the freezing point of water is taken as 0 C and the boiling point of water as 100 C. 12
Another temperature scale used in the physical sciences is the Kelvin scale, or absolute scale. On the Kelvin temperature scale, the freezing point of water is 273 K and the boiling point is 373 K. Noce that with the Kelvin scale, the degree sign is not used. The zero point on the Kelvin scale, K, or absolute Zero, is 273º C. The relaonship between a temperature on the Celsius scale and one on the Kelvin scale is given by: K = ºC + 273 or ºC = K 273 Which is warmer, 0 Celsius or 0 Fahrenheit? To determine the answer, you must convert one of them so both are the same temperature scale and then compare!!! Which is colder, 223 Kelvin or 50 Fahrenheit? To determine the answer, you must convert one of them so both are the same temperature scale and then compare!!! 223 K 273 = 50 C ( 50 F 32) x 0.56 = 46 C Units of Energy Energy is the capacity to do w ork or to produce heat. SI unit is the joule. 1 Joule =0.2390 calorie 1 calorie = 4.184 Joule 13
3.3 CONVERSION PROBLEMS A conversion factor is a ratio of equivalent measurements Dimensional Analysis Different units Converting between units (measurement does NOT Change) Multi step problems Converting complex units Answer should be in scientific notation with the number of significant digits in the measurement. When converting a measurement... The answer should be in scientific notation with the number of significant digits in the actual measurement. One Step Conversions 6.76 L ml 24.5 cm x = inches 7.41 x 10 8 feet miles 3.17 x 10 19 ml x = qts Two Step Conversions 3.29 km cm More Two Step Conversions 6.28 x 10 9 mm km 3.2 x 10 11 lbs grams 4.587 x 10 17 µl cm 3 14
Multi Step Conversions 2.63 hm inches Converting in Multiple Dimensions 4.13 g/cm 3 cg/dm 3 6.23 x 10 13 minutes decades 5.12 x 10 1 cm/s km/h 3.4 DENSITY There is an important relaonship between an object s mass and its volume. This relaonship is called density. Answers in scienfic notaon. Density is the rao of the mass of an object to its volume Densiy = mass volume Density = mass volume Answer expressed in g/ml or g/cm 3. Mass = density x volume Answer expressed in grams, kg, etc. Volume = mass density Answer expressed in ml, cm 3, L, etc. Cover what you are looking for and perform the operaon. Density Mass Volume 15
D = M = 312.0 g V = 1.5 ml D = 3.21 g/ml M = V = 2.81 L D = M = V = D = 1.89 g/ml M = 321.2 g V = When mass is measured in grams and volume in cubic cenmeters, density has units of grams per cubic cenmeter. g/cm 3 or g/ml Mass stays the same: V increase, D decrease V decrease, D increase Volume stays the same: M increases, D increases M decreases. D decreases When do substances sink in water? Liquids Water = 1.0 g/ml Sub A = 1.7 g/ml Sub B = 0.5 g/ml Sub C = 4.6 g/ml SeaWater = 1.025 g/ml http://www.youtube.com/watch?v=_ww6biy3nc0 What happens to the density of a substance as its temperature increases? http://www.bing.com/videos/search?q=youtube +density&form=vire3#view=detail&mid=227066b76c55bc2c7183227066b76c55b C2C7183 The volume of a substance usually increases as the temperature is increased. Meanwhile, its mass remains the same. Thus, the density of a substance usually decreases as its temperature increases. 16
Water is an important excepon. Below 4º C the volume of water increases as its temperature decreases. Ice floats because it is less dense than water. 17