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Lecture 2 Significant Figures and Dimensional Analysis Ch 1.7-1.9

Lecture 2 Significant Figures and Dimensional Analysis Ch 1.7-1.9. Dr Harris 8/23/12 HW Problems: Ch 1: 31, 33, 37. Significant Figures. Precision is indicated by the number of significant figures. Significant figures are those digits required to convey the precision of a result .

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Lecture 2 Significant Figures and Dimensional Analysis Ch 1.7-1.9

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  1. Lecture 2Significant Figures and Dimensional AnalysisCh 1.7-1.9 Dr Harris 8/23/12 HW Problems: Ch 1: 31, 33, 37

  2. Significant Figures • Precision is indicated by the number of significant figures.Significant figures are those digits required to convey the precision of a result. • There are two types of numbers: exact and inexact • Exact numbers have defined values: * There are 12eggs in a dozen * There are 24hours in a day * There are 1000 grams in a kilogram • Inexact number are obtained from measurement. Any number that is measured has some error because: • Limitations in equipment • Human error

  3. Significant Figures • Example: Laboratory balances are precise to the nearest cg (.01g). Lets say you measure the mass of a particular sample and you find the sample to have a mass of 335.49 g. • If you choose to report error, you would give the mass of the sample as 335.49±.01g because there is uncertainty in the last digit (9). • The actual mass may be 335.485 g, or 335.494g. But because the scale is limited to two decimal places, it rounds up or down. Hence, we use ± to include all possibilities.

  4. Significant Figures • The value 335.49 has 5 significant figures, with the hundredths place (9) being the uncertain digit. • Exact numbers have infinite sig. figs because there is no limit of confidence. • Other examples of inexact numbers? • Speedometer • Thermometer • Scale

  5. How to Determine if a Digit is Significant • All non-zeros and zeros between non-zeros are significant • 457 (3) ; 2.5(2) ; 101(3) ; 1005(4) • Zeros at the beginning of a number aren’t significant. They only serve to position the decimal. • .02 (1) ; .00003 (1) ; 0.00001004 (4) • For any number with a decimal, zeros to the right of the decimal are significant • 2.200(4) ; 3.0(2)

  6. Ambiguity • Zeros at the end of a number with no decimal may or may not be significant • 130(2 or 3), 1000(1, 2, 3, or 4) • This is based on scientific notation • 130 can be written as: 1.3 x 102  2 sig figs 1.30x 102  3 sig figs • If we convert 1000 to scientific notation, it can be written as: 1x 103 1 sig fig 1.0x 103  2 sig figs 1.00x 103  3 sig figs 1.000x 103  4 sig figs * Numbers that must be treated as significant CAN NOT disappear in scientific notation

  7. Calculations with Significant Figures • You can not get exact results using inexact numbers • Multiplication and division • Result can only have as many sig figs as the least precise number

  8. Calculations with Significant Figures • Addition and Subtraction • Result must have as many digits to the right of the decimal as the least precise number 20.4 1.322 83 + 104.722 211.942 212

  9. Group Problems • Solve the following. Use proper scientific notation for all answers. Also, include correct units. • Using scientific notation, convert 0.000976392 to 3 significant figures • Using scientific notation, convert 198207.6 to 1 significant figure H=10.000 cm W = .50 cm L = 30.000 cm • Volume of rectangle (volume = LWH) ? • Surface area (SA = 2WH + 2LH + 2LW) ? note: the constants in an equation are exact numbers

  10. Dimensional Analysis • Dimensional analysis is an algebraic method used to convert between different units • Conversion factors are required • Conversion factors are exact numbers (infinite sig figs), that are equalities between one unit and another. • For example, we can convert between inches and feet. The conversion factor can be written as: • In other words, there are 12 inches per foot, or 1 foot per 12 inches.

  11. Dimensional Analysis conversion factor (s) • Example. How many feet are there in 56 inches? • Our given unit of length is inches • Our desired unit of length is feet • We will use a conversion factor that equates inches and feet to obtain units of feet. The conversion factor must be arranged such that the desired units are ‘on top’ 4.7 ft

  12. Group Examples Non-SI to SI conversions 1 in = 2.54 cm 1 ft = 12 in. 1 mile = 5280 ft • Answer the following using dimensional analysis. Consider significant figures • 35 minutes to hours • Convert 40 weeks to seconds • Convert 4 gallons to Liters • 4 gallons to cm3 ?? • 13 lbs to kg 1 quart = 946.3mL 1 gallon = 4 quarts 1 min = 60 sec 60 min = 1 hr 24 hr = 1 day 1 lb = 453.59 g

  13. Solutions

  14. Converting Cubic Units • As we previously learned, the units of volume can be expressed as cubic lengths, or as capacities. When converting between the two, it may be necessary to cube the conversion factor • Ex. How many mL of water can be contained in a cubic container that is 1 m3 Must use this equivalence to convert between cubic length to capacity 3 Cube this conversion factor

  15. Group Examples • Convert 48.3 ft3 to cm3 • Convert 10 mL to m3 • Convert 100 L to µm3 • ** A certain gasoline tank can hold 12.50 gallons of fuel. Assuming a gasoline density of 0.797 g/cm3, calculate the mass of gasoline in a full tank. 1 in = 2.54 cm 1 ft = 12 in. mL = cm3 k = 10 3 c = 10 -2 m = 10 -3 μ = 10 -6 1 quart = 946.3mL 1 gallon = 4 quarts

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