1 Scientific Measurement Objectives: Section 2.1 List common SI units of measurement and common prefixes used in the SI system. Distinguish mass, volume, density and specific gravity Section 2.2 Convert measurements to scientific notations Use Dimensional Analysis for unit conversions Section 2.3 Distinguish among accuracy, precision, and error of measurement. Calculate values from measurements using the correct number of significant figures.

2 Scientific Notation Estimated number of stars in a galaxy

3 In scientific notation, a number is written as the product of two numbers: a coefficient (  1and <10) and 10 raised to a power. Examples: 3.6 x 10 4 = 3.6 x 10 x 10 x 10 x 10 = x = 8.1. = x 10 x 10 Scientific Notation Convert to scientific notation: 3600 m cm Kg314 ml

4 Using scientific notation makes calculating more straightforward: To multiply numbers written in scientific notation, multiply the coefficients and add the exponents. (3.0 x 10 4 ) x (2.0 x 10 2 ) = (3.0 x 2.0) x = 6.0 x 10 6 To divide numbers written in scientific notation, divide the coefficients and subtract the exponents. (3.0 x 10 4 ) = 3.0 x = 1.5 x 10 2 (2.0 x 10 2 )2.0 (5.5 x 10 5 ) x (2.0 x 10 3 ) = ?(8.8 x 10 9 ) = ? (2.0 x 10 3 )

5 Addition and Subtraction First you must make the exponents the same, because they determine the locations of the decimal points. The decimal points must be “aligned” (use bigger one) Now you add or subtract the coefficients (3.0 x 10 4 ) + (2.0 x 10 3 ) = (3.0 x 10 4 ) + (0.2 x 10 4 ) = 3.2 x 10 4 (3.2 x 10 5 ) + (9.0 x 10 4 ) = ? (3.0 x 10 4 ) - (2.0 x 10 3 ) = ? (5.6 x 10 2 ) + (1.2 x 10 3 ) = ? (6.0 x 10 4 ) - (4.6 x 10 5 ) = ? (3.2 x 10 5 ) + (0.9 x 10 5 ) = 4.1 x 10 5 (3.0 x 10 4 ) - (0.2 x 10 4 ) = 2.8 x 10 4 (0.56x 10 3 ) + (1.2 x 10 3 ) = 1.76 x 10 3 (0.6 x 10 5 ) - (4.6 x 10 5 ) = x 10 5

6 In a conversion factor, the measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom). 1 meter 100 centimeters Smaller numberLarger unit Larger number Smaller unit Dimensional Analysis

77 Whenever two measurements are equivalent (equal), a ratio of the two measurements will equal 1, or unity. For example, you can divide both sides of the equation 1 m = 100 cm by 1 m or by 100 cm. l m = 100cm= 1 or __l m_ =100cm = 1 l m l m 100cm 100cm Conversion Factor A ratio of equivalent measurements, such as __l m is called a conversion factor. 100cm The conversion factors above are read… “one hundred centimeters per meter” and “one meter per one hundred centimeters.”

88 Commonly Used Prefixes in the Metric System

99 Find the correct conversion factors for the following 1 L__1 m__1 kg1 mg ? mL? mm? g? g 1 day1  m__1g 1m 3 ? hrs.? m? cg?cm mL1000 mm1000g g 24 hrs m100 cg 100x100x100 = 10 6 cm 3

10 A Three Step Problem- Approach… A Three Step Problem-Solving Approach… Step 1. Analyze: List knowns and unknowns Step 2. Calculate: Solve for unknowns Step 3. Evaluate: Does the result make sense? How to solve Conversion Problems…

11 How many seconds are there exactly in one day? 1. Analyze: List the knowns and the unknowns KnownUnknown 2. Calculate: Solve for the Unknown 3. Evaluate Does the results make sense? Do units cancel out to receive a unit of mass? Does the numerical value make sense based on estimate? 1 day? s Conversions 1 day = 24 hr, 1 hr = 60 min, 1 min = 60 s day  hours  minutes  seconds day  hours  minutes  seconds 1 day x 24h x 60 min x 60 s = 8.64 x 10 4 s 1 day x 24h x 60 min x 60 s = 8.64 x 10 4 s 1day 1 hr 1 min 1day 1 hr 1 min Yes, a very large number is expected since a second is much smaller than a day

12 What is m in micrometer? 1. Analyze: List the knowns and the unknowns Known Unknown length = m = 7.3 x m length = ?  m Conversions Conversions 1m = m = 10 6  m 2. Calculate: Solve for the Unknown x 10 6 = 7.3 x x m x 10 6  m = 7.3 x 10 4  m 1 m 1 m 3. Evaluate Does the results make sense? Do units cancel out to receive a unit of micrometer? yes Does the numerical value make sense based on estimate? yes, micrometer is a smaller unit than m, thus value must increase correct numbers of significant digits? yes

13 ! Remember ! The unit you want to get rid of has to appear on the other side of the division bar to cancel out x 10 6 = 7.3 x x m x 10 6  m = 7.3 x 10 4  m 1 m 1 m End with the unknown Start with the known

14 The accuracy of a measurement describes how close a measurement comes to the true value. The precision of a measurement depends on its reproducibility. a) good accuracy b) poor accuracy c) poor accuracy good precision good precision poor precision Section 2.3 Uncertainty in Data

15 To evaluate the accuracy of a measurement, it must be compared with the correct value. The difference between the accepted value and the experimental value is called error. This scale has not been properly zeroed. The reading for the person’s weight is inaccurate. Error = accepted value — experimental value (Error can be positive or negative)

16 Often, it is useful to calculate the relative error, or percent error. The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%. Using the absolute value of the error means that the percent error will always be a positive value. Percent Error |error | Percent Error = X 100% = accepted value | accepted value — experimental value | Percent Error = X 100% accepted value Calculate the percent error, if the accepted value is 103°C and the experimental value is 99.3 °C

17 Laboratory Glassware used to measure volume Erlenmeyer Buret Graduatedbeaker Volumetric flask cylinder Flask Which graduated cylinder will give a more precise measurement?

18 Measurements and calculations must always be reported to the correct number of significant figures Significant figures include all the digits that can be known accurately plus a last digit that must be estimated Significant Figures 0.6 m 0.61 m m

19 Rules to Determine The Number of Significant Figures 1. Digits other than zero are always significant 61.4 g 3 significant digits 2. Zeros between two significant digits are always significant 306 m3 significant digits 3. Zeros in front of nonzeros are not significant, they are used solely for spacing the decimal point (placeholder) g3 significant digits 4. Zeros at the end of a number after the decimal points are always significant km5 significant digits 5. Zeros at the right end of a number to the left of an understood decimal point are not significant (placeholders) 7000 g1 significant digit

20 6. Unlimited numbers of significant figures: Numbers that are counted instead of measured, e.g. 23 students is understood to be exact (cannot be 23.8 students) = Exactly defined quantities, like mathematical constants (pi) or relations e.g. 60 min = 1 hr unlimited How many significant digits are there in each: a. 20 kgd sg kg b ge °Ch L c. 11. cmf cmI °C j. 20 cars If 7000 g was accurately measured to the nearest gram, all four digits are significant. In that case write: x 10 3 (or 7000.)

21 Rounding Significant Figures 7.7 meters 5.5 meters The calculated area cannot be more precise than the measured values used to obtain it. The calculated area must be rounded. 5.5 m X 7.7 m = m 2 # sig.figs.: ! too much ! Rounding: 1)If the digit immediately to the right of the last significant digit is less than 5, it is simply dropped. 2)If it is 5 or greater than 5, round up (by one). round to 2 significant digits: m m m m m 2 42 m 2 72 m m m x x 10 3 m 2

22 A round to 4 significant digits a) L b) L c) L d) L e) 1100 L a) Lor x 10 1 L b) L or x 10 4 L c) Lor x L d) L or x 10 2 L e) L or x 10 3 L

23 Significant Figures in Calculation Multiplication and Division The answer must have the same number of significant digits as the measurement with the least number of significant digits m x m = = m 2 (4)(5)= (8) wrong = (4) is correct (4)(5)= (8) wrong = (4) is correct Addition and Subtraction The answer must have the same number of decimal places (not significant digits) as the measurement with the least number of decimal places m (3) m (2) m (2) m (2) m (2)

24 Calculate and round the answer correctly a) m m = b) 9.86 m m = c) 1.2 L L L = d) 7.89 m x 8.222m = e) 89.3 cm ÷ 6.2 cm = f) 2.0 x = a) m  8.7 m b) m  6.64 m c) L  61.0 L d) m 2  64.9 m 2 e)  14 f) 5.85 = 5.8 (because 8 is even)