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Chapter 2 Measurements and Calculations 2.2 Units of Measurement Nature of Measurement Measurement - quantitative observation consisting of 2 parts Part 1 – number Part 2 - scale (unit) Examples: 20 grams 6.63 x 10-34 Joule*seconds The Fundamental SI Units (le Système International, SI) P h y sica l Q u a n tity N am e A b b rev ia tio n k ilo g ram kg m eter m T im e seco n d s T em p eratu re K elv in K E lectric C u rren t A m p ere A m o le m ol can d ela cd M ass L en g th A m o u n t o f S u b stan ce L u m in o u s In ten sity Derived Units Volume: Liter - non SI unit commonly used 1 liter = 1 cubic decimeter 1 liter = 1000 ml = 1000 cm3 Density: kg/m3 is inconveniently large commonly used unit is g/cm3 or g/ml 1 g/cm3 = 1 g/ml gas density is reported in g/liter Specific Gravity • A comparison between a substance’s density and that of water, at a specified temperature and pressure. • Since density is divided by density, specific gravity is a dimensionless quantity, meaning it has no units. • It is common to use the density of water at 4 °C (39 ° F) as reference - at this point the density of water is at the highest - 1000 kg/m3. Metric Prefixes Conversions: Factor-Label Method (Dimensional Analysis) Starting Number, Unit X Number, Desired Unit Number, Starting Unit Conversion Factor Practice 1) What is the density of a sample of ore that has a mass of 74.0 g and occupies 20.3 cm3? D = 3.65 g/cm3 2) Find the volume of a sample of wood that has a mass of 95.1 g and a density of 0.857 g/cm3. 3 V = 111 cm 3) Express a time period of exactly 1.00 day in terms of seconds. t = 86400 s 4) How many centigrams are there in 6.25 kg? m = 625000 cg Chapter 2 Measurements and Calculations 2.3 Using Scientific Measurement Uncertainty in Measurement A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty. 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? 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. Neither accurate nor precise Precise but not accurate Precise AND accurate Types of Error Random Error (Indeterminate Error) measurement has an equal probability of being high or low. Systematic Error (Determinate Error) Occurs in the same direction each time (high or low), often resulting from poor technique or incorrect calibration. Percent Error A way to compare the accuracy of an experimental value with an accepted value. Percent = Error Value accepted – Value experimental x 100 Value accepted Ex: The actual density of a certain material is 7.44 g/cm3. A student measures the density of the same material as 7.30 g/cm3. What is the percentage error of the measurement? % Error = 1.9 % Rules for Counting Significant Figures Nonzero integers always count as significant figures. 3456 has 4 sig figs. Rules for Counting Significant Figures Zeros Leading zeros do not count as significant figures. 0.0486 has 3 sig figs. Rules for Counting Significant Figures Zeros Captive zeros always count as significant figures. Also known as the Hugging Rule! 16.07 has 4 sig figs. Rules for Counting Significant Figures Zeros Trailing zeros are significant only if the number contains a decimal point. 9.300 has 4 sig figs. Rules for Counting Significant Figures Exact numbers have an infinite number of significant figures. 1 inch = 2.54 cm, exactly 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 103 s 3 sig figs 0.0054 cm 2 sig figs 3,200,000 2 sig figs Rules for Rounding Rules for Significant Figures in Mathematical Operations Multiplication and Division: # of 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) Sig Fig Practice #2 Calculation Calculator says: 3.24 m x 7.0 m 22.68 m2 100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 Answer 23 m2 4.22 g/cm3 0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2 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 0.3588 g/mL 0.359 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) 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 Scientific Notation A method of representing very large or very small numbers M x 10n • M is a number between 1 and 9 • n is an integer • all digits in M are significant Reducing to Scientific Notation – Move decimal so that M is between 1 and 9 – Determine n by counting the number of places the decimal point was moved Moved to the left, n is positive Moved to the right, n is negative Mathematical Operations Using Scientific Notation Addition and subtraction – Operations can only be performed if the exponent on each number is the same Multiplication – M factors are multiplied – Exponents are added Division – M factors are divided – Exponents are subtracted (numerator denominator) Operations with Units » Cancellation occurs with the units in the same way that it occurs with numbers common to both the numerator and denominator – Units are handled algebraically, just like numbers – Analysis of units can be a clue as to whether a problem was set up correctly » Calculations involving units must have the correct units shown throughout the working of the problem and attached to the answer Direct Proportions The quotient of two variables is a constant k = y/x As the value of one variable increases, the other must also increase As the value of one variable decreases, the other must also decrease The graph of a direct proportion is a straight line Inverse Proportions The product of two variables is a constant k = xy As the value of one variable increases, the other must decrease As the value of one variable decreases, the other must increase The graph of an inverse proportion is a hyperbola Practice 1) Calculate the density of a liquid given that 41.4 mL of it has a mass of 58.24 g. D = 1.41 g/mL 2) How many kilometers are there in 6.2 x 107 cm? 6.2 x 102 km 3) How m\any hours are there in exactly 3 weeks? 504 hours