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Measurement and Significant Figures

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Precision and Accuracy What is the difference between precision and accuracy in chemical measurements? Accuracy refers to how close you are to the true value. Precision refers to how close several measurements are to each other.

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Precision and Instruments Your measurement will only be as good as the instrument you use. Precision is limited by the gradations -or markings -on your instrument. We can typically estimate to one-tenth of a gradation mark when using graduated instruments in Chemistry.

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Measurement Practice Accurate Not accurate Accurate (the average), but is precise and precise not precise BEAKER CYLINDER BURET 47 +/- 1 mL 36.5 +/- 0.1 mL 20.38 +/- 0.01 mL

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Uncertainty in Measurement Any measurement will have some degree of uncertainty associated with it. For example, if you are 1.6 meters tall, we know that you are exactly ONE meter tall (not 0 or 2) but the second digit is an estimate and contains some uncertainty (it could be 0.58 rounded up, or 0.62 rounded down). Scientific measurements are rounded off so that the last digit is the only one that is uncertain. Preceding digits are known with certainty.

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Significant figures What is the difference between the measurements 25.00 mL and 25 mL? The first measurement is known to a greater degree of precision and contains more significant figures –it could be 25.01 or 24.99 whereas the second measurement lies between 24 and 26. The number of significant figures tells us how well we know a measurement. The known numbers PLUS the last uncertain number in a measurement are significant.

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Rules for determining significant figures Any non-zero number is significant. example: 762 has 3, and 2500 has 2 Zeros: (a) leading zeros are not significant, they are just place holders. Ex: 006471 has 4, and 0.00284 has 3 (b) “Captive” zeros between nonzeros are significant. Ex: 1.008 has 4 and 12046 has 5 (c) Trailing zeros are significant ONLY if the number contains a decimal point. Ex: 1.0 x10 2 has 2, and 3000. has 4 Exact numbers are numbers that are determined by counting (not measurement) or by definition are assumed to have an infinite number of significant figures. example: 1 minute equals 60 seconds 15 students are in class today

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Practice Problem How many sig figs are in each? 4.5090 0.00607 6.7 x10 3 200. 250 698,000.1 2.0000 x10 6

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Addition and Subtraction Using Significant Figures The answer must have the same number of decimal places as the least precise measurement used in the calculation. For example, consider the sum 12.11 18.0 + 1.013 31.123 The answer is 31.1 since 18.0 only has one decimal place.

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Multiplication and division using sig. figs. The number of significant figures in the answer is the same as the least precise measurement (lowest number of sig. figs.) used in the calculation. For example, consider the calculation 4.56 x 1.4 = 6.38 The correct answer is 6.4 (it should only have two sig figs since 1.4 has only two)

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% Error calculations To determine the percent error of a measurement, use the following formula: % Error = accepted value – experimental value x 100% accepted value Example: 4.50 g – 4.31 g x 100% = 4.2% error 4.50 g Activity: density of Zn metal and % error calculations.. Use sig figs, find % error.

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Scientific Notation The primary reason for converting numbers into scientific notation is to make calculations with unusually large or small numbers less cumbersome. Because zeros are no longer used to set the decimal point, all of the digits in a number in scientific notation are significant, as shown by the following examples: 2.4 x 10 22 has 2 significant figures 9.80 x 10 -4 has 3 significant figures 1.055 x 10 -22 has 4 significant figures

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Converting to Sci. Not. The following rule can be used to convert numbers into scientific notation: The exponent in scientific notation is equal to the number of times the decimal point must be moved to produce a number between 1 and 10. Example: In 1990 the population of Chicago was 6,070,000. To convert this number to scientific notation we move the decimal point to the left six times. 6,070,000 = 6.07 x 10 6 To convert numbers smaller than 1 into scientific notation, we have to move the decimal point to the right. The decimal point in 0.000985, for example, must be moved to the right four times. 0.000985 = 9.85 x 10 -4

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Exponent Review Some of the basics of exponential mathematics are given below: –Any number raised to the zero power is equal to 1. ex: 1 0 = 1 and 10 0 = 1 –Any number raised to the first power is equal to itself. ex: 1 1 = 1 and 10 1 = 10 –Any number raised to the nth power is equal to the product of that number times itself n-1 times. ex: 2 2 = 2 x 2 = 4 and 10 5 = 10 x 10 x 10 x 10 x 10 = 100,000 –Dividing by a number raised to an exponent is the same as multiplying by that number raised to an exponent of the opposite sign. ex: 5 ÷ 10 2 = 5 x 10 -2 = 0.05

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Practice Problems Convert the following numbers into sci. not.: (a) 0.004694 (b) 19.8 (c) 4,679,000 ANSWER: (a) 4.694 x 10 -3 (b) 1.98 x 10 1 (c) 4.679 x 10 6 What is the percent error of a measurement that is 2.51 cm if the accepted value is 2.54 cm? ANSWER: (2.54 – 2.51) x 100% = 0.03 x 100% 2.54 2.54 = 1.18% = 1% error

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Review: metric unit prefixes Prefix Symbol Factor Numerically Name Giga G 10 9 1 000 000 000 billion Mega M 10 6 1 000 000 million kilo k 10 3 1 000 thousand Deca D 10 1 10 ten deci d 10 -1 0.1 tenth centi c 10 -2 0.01 hundredth milli m 10 -3 0.001 thousandth micro μ 10 -6 0.000 001 millionth nano n 10 -9 0.000 000 001 billionth

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More Examples: Convert 50.0 mL to liters: How many seconds are in two years?

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Dimensional Analysis We will often need to convert from one unit to another when solving problems in Chemistry. The best way to do this is by a method called dimensional analysis (a.k.a. factor-label method). For example, consider a pin measuring 2.85 cm in length. Given that one inch is equal to 2.54 cm, what is its length in inches? ? in = 2.85 cm x 1 in = 1.12 in 2.54 cm

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Tips for using the method… In math you use numbers, in chemistry we use quantities. A quantity is described by a number and a unit. 100 is a number: 100 Kg is a quantity (notice that in chemistry we give meaning to the numbers). In science we solve a lot of the "math" by watching the units of the quantities There are two main rules to solving science problems with the factor-label method: 1. Always carry along your units with any measurement you use. Cancel units when appropriate. 2. You need to form the appropriate labeled ratios, (which means conversion factors have equal numerators and denominators). NO NAKED NUMBERS!

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Unit Conversion Practice A pencil is 7.00 inches long. How long is it in cm? ANSWER: 17.8 cm A student has entered a 10.0 km race. How long is this in miles? ANSWER: 6.22 mi The speed limit on many highways in the U.S. is 55 mi/hr. What is this in km/hr? ANSWER: 89 km/hr

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Linking conversion ratios Sometimes you will need to multiply by more than one ratio to get to your desired units, you can do this by using linking units. Your setup will look like this: Example: How many inches are in 1.00 meter given the equality 1 inch = 2.54 cm and 1 meter = 100 cm? ? in = 1.00 m x 100 cm x 1 inch = 39.4 in 1 m 2.54 cm

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Advanced Problem A Japanese car is advertised as having a gas mileage of 15 km/L. Convert this rating to mi/gal. (Given conversion factors 1 mi = 1.609 km, 1L=1.06 qt and 4 qt = 1 gal) ANSWER: 15 km x 1mi x 1 L x 4 qt = 35.18 mi/gal L 1.609 km 1.06 qt 1 gal With correct sig figs this rounds to 35 mi/gal

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