1. Lecture 2 Significant Figures and Dimensional Analysis Ch 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 12 eggs in a dozen * There are 24 hours 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 335.494 g. But because the scale is limited to two decimal places, it rounds up or down. Hence, we use ± to include all possibilities.

4. 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 Significant Figures

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 10 2  2 sig figs 1.30 x 10 2  3 sig figs If we convert 1000 to scientific notation, it can be written as: 1 x 10 3  1 sig fig 1.0 x 10 3  2 sig figs 1.00 x 10 3  3 sig figs 1.000 x 10 3  4 sig figs * Numbers that must be treated as significant CAN NOT disappear in scientific notation

7. 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 Calculations with Significant Figures

8. 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 L = 30.000 cm W =.50 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 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 cm 3 ?? 13 lbs to kg 1 in = 2.54 cm 1 ft = 12 in. 1 mile = 5280 ft 1 quart = 946.3 mL 1 gallon = 4 quarts 1 min = 60 sec 60 min = 1 hr 24 hr = 1 day 1 lb = 453.59 g Non-SI to SI conversions

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 m 3 3 Cube this conversion factor Must use this equivalence to convert between cubic length to capacity

15. Group Examples Convert 48.3 ft 3 to cm 3 Convert 10 mL to m 3 Convert 100 L to µm 3 ** 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 = cm 3 k = 10 3 c = 10 -2 m = 10 -3 μ = 10 -6 1 quart = 946.3 mL 1 gallon = 4 quarts