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Chapter 2

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.

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Chapter 2

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  1. Chapter 2 Measurements and Calculations 2.2 Units of Measurement

  2. Nature of Measurement Measurement - quantitative observation consisting of 2 parts • Part 1 – number Part 2 - scale (unit) • Examples: • 20grams • 6.63 x 10-34Joule*seconds

  3. The Fundamental SI Units(le Système International, SI)

  4. 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

  5. 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.

  6. Metric Prefixes

  7. Conversions: Factor-Label Method (Dimensional Analysis) Conversion Factor Number, Desired Unit Starting Number, Unit X Number, Starting Unit

  8. Practice D = 3.65 g/cm3 • What is the density of a sample of ore that has a mass of 74.0 g and occupies 20.3 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) Express a time period of exactly 1.00 day in terms of seconds. • 4) How many centigrams are there in 6.25 kg? V = 111 cm3 t = 86400 s m = 625000 cg

  9. Chapter 2 Measurements and Calculations 2.3 Using Scientific Measurement

  10. Uncertainty in Measurement • A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

  11. 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?

  12. Precision and Accuracy • Accuracyrefers to the agreement of a particular value with the truevalue. • Precisionrefers to the degree of agreement among several measurements made in the same manner. Precise but not accurate Precise AND accurate Neither accurate nor precise

  13. 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 directioneach time (high or low), often resulting from poor technique or incorrect calibration.

  14. Percent Error • A way to compare the accuracy of an experimental value with an accepted value. • Percent = Value accepted – Value experimental x 100 • Error 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 %

  15. Rules for Counting Significant Figures • Nonzero integersalways count as significant figures. • 3456has • 4sig figs.

  16. Rules for Counting Significant Figures • Zeros • Leading zeros do not count as significant figures. • 0.0486 has • 3 sig figs.

  17. Rules for Counting Significant Figures • Zeros • Captive zeros always count as significant figures. • 16.07 has • 4 sig figs. Also known as the Hugging Rule!

  18. Rules for Counting Significant Figures • Zeros • Trailing zerosare significant only if the number contains a decimal point. • 9.300 has • 4 sig figs.

  19. Rules for Counting Significant Figures • Exact numbershave an infinite number of significant figures. • 1 inch = 2.54cm, exactly

  20. 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 5 sig figs 100,890 L  3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

  21. Rules for Rounding

  22. 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)

  23. Sig Fig Practice #2 Calculation Calculator says: Answer 22.68 m2 3.24 m x 7.0 m 23 m2 100.0 g ÷ 23.7 cm3 4.22 g/cm3 4.219409283 g/cm3 0.02 cm x 2.371 cm 0.05 cm2 0.04742 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 5870 lb·ft 1818.2 lb x 3.23 ft 5872.786 lb·ft 0.3588 g/mL 0.359 g/mL 1.030 g ÷ 2.87 mL

  24. 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)

  25. Sig Fig Practice #3 Calculation Calculator says: Answer 10.24 m 3.24 m + 7.0 m 10.2 m 100.0 g - 23.73 g 76.3 g 76.27 g 0.02 cm + 2.371 cm 2.39 cm 2.391 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1821.6 lb 1818.2 lb + 3.37 lb 1821.57 lb 0.160 mL 0.16 mL 2.030 mL - 1.870 mL

  26. 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

  27. 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)

  28. 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

  29. 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

  30. 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

  31. Practice D = 1.41 g/mL • Calculate the density of a liquid given that 41.4 mL of it has a mass of 58.24 g. • 2) How many kilometers are there in 6.2 x 107 cm? • 3) How m\any hours are there in exactly 3 weeks? 6.2 x 102 km 504 hours

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