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Introduction to General Chemistry Lecture 2 Suggested HW: 33, 37, 40, 43, 45, 46 Ch. 1.6 - 1.9 Accuracy and Precision • Accuracy defines how close to the correct answer you are. Precision defines how repeatable your result is. Ideally, data should be both accurate and precise, but it may be one or the other, or neither. Accurate, but not precise. Reached the target, but could not reproduce the result. Precise, but not accurate. Did not reach the target, but result was reproduced. Accurate and precise. Reached the target and the data was reproduced. Measuring Accuracy • Accuracy is calculated by percentage error (%E) %E average value true value x100 true value •We take the absolute value because you can’t have negative error. •GROUP PROBLEM - A certain brand of thermometer is considered to be accurate if the %E is < 0.8%. The thermometer is tested using water (BP = 100oC). You bring a pot of distilled water to a boil and measure the temperature 5 times. The thermometer reads: 100.6o, 100.4o, 99.8o, 101.0o, and 100.4o. Is it accurate? Measuring Precision: Significant Figures • Precision is indicated by the number of significant figures. Significant figures are those digits required to convey a result. – There are two types of numbers: exact and inexact – Exact numbers have defined values and possess an infinite number of significant figures because there is no limit of confidence: * There are exactly 12 eggs in a dozen * There are exactly 24 hours in a day * There are exactly 1000 grams in a kilogram – Inexact number are obtained from measurement. Any number that is measured has error because: • Limitations in equipment • Human error Measuring Precision: Significant Figures • Example: Some laboratory balances are precise to the nearest cg (.01g). This is the limit of confidence. The measured mass shown in the figure is 335.49 g. • The value 335.49 has 5 significant figures, with the hundredths place (9) being the uncertain digit. Thus, the (9) is estimated, while the other numbers are known. • It would properly reported as 335.49±.01g - The actual mass could be anywhere between 335.485… g and 335.494… g. The balance is limited to two decimal places, so it rounds up or down. We use ± to include all possibilities. Determining the Number of Significant Figures In a Result • 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) Determining the Number of Significant Figures In a Result • Zeros at the end of an integer 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 1.30 x 102 3 sig figs – If we convert 1000 to scientific notation, it can be written as: 1 x 103 1 sig fig 1.0 x 103 2 sig figs 1.00 x 103 3 sig figs 1.000 x 103 4 sig figs *Numbers that must be treated as significant CAN NOT disappear in scientific notation Calculations Involving Significant Figures • You can not get exact results using inexact numbers • Multiplication and division – Result can only have as many significant figures as the least precise number 6.2251 𝑐𝑚 𝑥 𝟓. 𝟖𝟐 𝑐𝑚 = 36.230082 𝑐𝑚2 = 36.2 𝑐𝑚2 (3 s.f.) 105.86643 𝑚 𝑚 𝑚 𝑚 = 108.0269694 = 110 𝑜𝑟 1.1 𝑥 102 0. 𝟗𝟖 𝑠 𝑠 𝑠 𝑠 (2 s.f.) 𝑚 𝑘𝑔 𝑚 𝑘𝑔 𝑚 𝑘𝑔 𝑚 5 43270.0 𝑘𝑔 𝑥 𝟒 2 = 173080 = 200000 𝑜𝑟 2 𝑥 10 𝑠 𝑠2 𝑠2 𝑠2 (1 s.f.) Calculations Involving Significant Figures • Addition and Subtraction – Result is only as precise as the least precise number. 20.4 1.322 83 + 104.722 211.942 Limit of certain is the ones place 212 Group Work • Using scientific notation, convert 0.000976392 to 3 sig. figs. • Using scientific notation, convert 198207.6 to 1 sig. fig. H=10.000 cm W = .40 cm L = 31.00 cm • • Volume of rectangle ? Surface area (SA = 2WH + 2LH + 2LW) ? note: constants in an equation are exact numbers = 2 4.0𝑐𝑚2 + 2 310.0𝑐𝑚2 + 2(1𝟐. 4𝑐𝑚2 ) = 8.0𝑐𝑚2 + 620.0𝑐𝑚2 + = 65𝟐. 8𝑐𝑚2 = 653𝑐𝑚2 2𝟒. 8𝑐𝑚2 Limit of certainty is the ones place Dimensional Analysis • Dimensional analysis is an algebraic method used to convert between different units • Conversion factors are required – Conversion factors are exact numbers which are equalities between one unit set and another. – For example, we can convert between inches and feet. The conversion factor can be written as: 12 inches 1 foot or 1 foot 12 inches • In other words, there are 12 inches per 1 foot, or 1 foot per 12 inches. Dimensional Analysis conversion factor (s) desired units given units x given units desired units • 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’ 1 𝑓𝑜𝑜𝑡 𝟓𝟔 𝑖𝑛𝑐ℎ𝑒𝑠 𝑥 = 4.6666 𝑓𝑡 12 𝑖𝑛𝑐ℎ𝑒𝑠 4.7 ft Group Work • Answer the following using dimensional analysis. Consider significant figures. – Convert 35 minutes to hours – Convert 40 weeks to seconds – Convert 4 gallons to cm3 (1 gal = 4 quarts, 1 quart = 946.3 mL) – *35 𝒎𝒊𝒍𝒆𝒔 𝒉𝒓 to 𝒊𝒏𝒄𝒉𝒆𝒔 𝒔𝒆𝒄 (1 mile = 5280 ft and 1 ft = 12 in) High Order Exponent Unit Conversion (e.g. 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 1 𝑚3 𝑥 𝑐𝑚 3 𝒎𝑳 𝑥 10−2 𝑚 𝒄𝒎𝟑 Must use this equivalence to convert between cubic length to capacity Cube this conversion factor =1 𝑚3 𝒄𝒎𝟑 𝑚𝐿 𝑥 𝑥 𝟏𝟎−𝟔 𝒎𝟑 𝑐𝑚3 = 𝟏 𝒙 𝟏𝟎𝟔 𝒎𝑳 Group Work • Convert 10 mL to m3 (c = 10-2) • Convert 100 L to µm3 (µ = 10-6) • Convert 48.3 ft2 to cm2 (1 in. = 2.54 cm)