Chapter 2: Metric system, conversion and uncertainty of measurementsObjectives: Apply the concepts of accuracy and precisionApply the concept of significant figures Apply appropriate units to describe the results Use the unit factor method to carry out conversions among units

New vocabularyExponential notation (decimal notation)Exact number: S chnh xcSignificant figures (SF) cc s c nghaAccuracy: chnh xcPrecision: lp liAddition, substraction, multiplication and division (+,-, x, :)

Exponential (scientific) notationWhen dealing with very large and very small numbers (give Ex.)E.g. Avagadro number = ?1.6605 * 10-24 g is ? (1 amu)In the exponential notation, place one nonzero digit to the left of the decimal0.000348 = 3.48 X 10-4

Two kinds of numbers Numbers obtained by counting or from definitions are exact numbers (give ex.)Exact number may be thought of as containing an infinite number of significant figuresNumbers obtained from measurements are not exact. There is some uncertainty (doubts) in all measurements. Every measurement involves an estimation (next example)

Example12.5 cmobjectThe last digit, 5, is a best estimate and is therefore doubtful The smallest divisions (calibration lines) on the ruler are 1 cm. An attemptto measure 0.1 cm (1 mm) requires estimationcmDifferent people measure the same length of the object will probably not givethe same result. 1213

ExampleWhich statements include exact numbers(a) Angel Falls in Venezuela is 32 12 ft high. (b) There are nine known planets in the Solar System. (c) There are 453.59 g in 1 lb. (d) There are 1000 mm in 1 m.

Uncertainty in MeasurementAccuracy: How close you are to the true valuePrecision: How close your values are to one another (internal consistency)Ideally, all measurements should be both accurate and preciseMeasurements are frequently repeated to improve accuracy and precision

Description of precision and accuracy

A laboratory instructor gives a sample of amino-acid powder to each of four students, I, II, III, and IV, and they weigh the samples. The true value is 8.72 g. Their results for three trials are

Significant figures (SF)Are digits believed to be correct by the person who makes measurement.In the above example, because the person making the measurement is not certain that the last digit,5, is correct, it would be meaningless to report the length of the object as 12.54 cm

The number of significant figures in a measurement depends on the measuring device.

Simple rules govern the use of significant figuresNonzero digits are always significantZeroes are sometimes significant, and sometimes they are not (ex. a, b, c)Exact numbers can be considered as having unlimited number of significant figures. We do not apply the rules of significant figures to them

Examples1) 38.57 mL2) a)zeroes at the beginning of a number are never significant: 0.052 gb) zeroes between nonzero digits are always significant: 6.08 kmc) zeroes at the end of a number that contains a decimal point are always significant: 38.0 cm

Rules of rounding offWhen the number to be dropped < 5, the preceding number is left unchanged (e.g., 6.54 rounds to 6.5)When it is > 5, the preceding number is increased by 1 (e.g., 8.48 rounds to 8.5)When the number to be dropped is 5, the preceding number is set to nearest even number (e.g., 7.45 rounds to 7.4, and 7.35 rounds to 7.4) (intended to reduce the accumulation of errors)

Significant figures (Addition and Substraction)In addition and substraction, the last digit retained in the sum or difference is determined by the position of the first doubtful digitExample: (a) Add 37.24 mL and 10.3 mL; (b) Substract 21.2342 g from 27.87 gWhich digit is the first doubtful digit ?

Multiplication and divisionIn multiplication and division, an answer contains no more significant figures than the least number of significant figures used in the operationExample: What is the area of a rectangle 1.23 cm wide and 12.34 long?

Exercise 1Express the following exponentials as ordinary numbers: (a) 5.06 x 103, (b) 4.0010 x 10-3, (c)16.10 x 103, (d) 0.206 x 10-4, (e) 9.000 x 103, (f) 9.000 x 10-3.

Exercise 2For each of the following quantities underline the zeros that are significant figures, determine the number significant figures in the quantity, and rewrite the quantity using scientific notation. (a) 423.06 mL; (b) 0.0001073040 g; (c) 1,081.02 pounds

Exercise 3A box is 252.56 cm wide, 19.23 cm deep and 6.5 cm tall. Calculate the volume of the box. (Show your answer with the correct number of significant figures

Exercise 4Express (a) 1.00 cubic foot in units of liters (b) 1.00 liter in units of pints (c) miles per gallon in kilometers per literWhat is the mass of a rectangular piece copper 24.4 x 11.4 x 7.9 cm ? The density of copper is 8.92 g/cm3Vinegar has a density of 1.0056 g/cm3. What is the mass of 3 L of vinegar

Exercise 5

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