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1 Chemistry 121 Lecture 3: Physical Quantities Measuring Mass, Length, and Volume; Measurement and Significant Figures; Scientific Notation; Rounding Review Sections in McMurry, Ballantine, et. al. 7 th edition HW #1: 1.42, 1.52, 1.54, 1.64, 1.66, 1.70, 1.76, 1.78, 1.80, 1.82, 1.84, 1.86, 1.92, 1.94, 1.98, 1.106, 1.110, Learning Objectives: 1. Define the base units in the metric system 2. Review metric prefixes 3. Describe the difference between accuracy and precision 4. Define uncertainty in measurements 5. Represent precision in measurements by the use of significant figures a. Define significant figures b. Determine the number of significant figures in a measured value 6. Recognize exact vs. inexact (measured) numbers a. Explore why exact numbers have no bearing on the number of significant figures when measured quantities are treated mathematically 7. Define and utilize scientific notation 8. Carry out basic mathematical manipulations on measured values, retaining the correct number of significant figures, by using the rules of rounding 1

2 1.7: Physical Quantities A number without a unit does not have a physical meaning make certain you include units when you solve problems Measurement and the Système International d Unites (SI) The fundamental physical parameters are mass, length, and time (note the SI system is generally broken into the MKS or CGS systems). Important derived parameters include volume, temperature, and energy. Of particular interest to chemists is the concept of the mole. 2

3 The metric system is a base 10 system. Different prefixes are used to describe different orders of magnitude (or scale size). These are very important to know. 1.8: Measuring Mass, Length, and Volume 3

4 Questions: What is mass? How does it differ from weight? Mass: Weight: Question: If we weigh objects to determine mass, how do we know what a mass of, say, exactly 1 kg is? A few common conversions between British and metric 1 kg = lb 1 lb = 454 g 1 oz = g Question: For the purposes of one size fits all dosing, the standard human is taken to be 70 kg. What is this in lbs? The Relationship between Length and Volume in the Metric System 4

5 1.9: Measurement, Uncertainty, and Significant Figures Accuracy vs. Precision The Bull s-eye Model: High Precision, High Accuracy High Precision, Low Accuracy Low Precision, High Accuracy Low Precision, Low Accuracy Notice determination of precision requires repeated measurements Imagine taking the high precision, high accuracy case and magnifying by Would it still look as accurate and precise, how would this look when represented numerically? Accuracy vs. Precision On the Electronic Balance 5

6 Establishing Significant Figures Consider measuring an 8 ml volume with the following 3 devices a 500 ml graduated cylinder, a 100 ml graduated cylinder, and a 50 ml buret Clearly, the greatest precision in measuring volumes is obtained using a buret 1, which is reflected by the number of significant figures reported. The significant figures in a measurement are all digits that are known with certainty plus the first uncertain digit that is guesstimated Question: What is the number of significant figures in each of the above measurements? 1 The buret would also be the most accurate volume measuring device, which stands to reason since one would not take the trouble to build a highly precise measuring device without comparing it to an accurate standard. HOWEVER, one should avoid this assumption generally, since the capacity of instruments - particularly electronic instruments - to measure accurately over time varies. Complex instruments need to be calibrated to insure accuracy 6

7 Question: What is the correct number of significant figures for the following measured values? 19.3 g.0193 kg g g 1900 g Exact vs. Inexact Numbers Exact numbers are just that there is no uncertainty associated with them. While they may be represented by a minimal number of digits for convenience, it is understood there is an infinitely large number of digits associated with them, so they do not impact the number of significant figures in a calculation result. The question then becomes, Which types of numbers are exact? The general rules are: Defines conversions between units within the English System are exact o 12 in = 1 ft or 12 in/1 ft (12 and 1 are both exact) Defined conversions between units within the Metric System are exact o 1 m = 100 cm or 1 m/100 cm (1 and 100 are both exact). Conversions between English and Metric system are typically NOT exact. Exceptions exist and will be noted o 454 g = 1 lb or 454 g/1 lb (454 has 3 sig. fig., 1 is exact) o g = 1 lb or g/1 lb ( has 5 sig. fig., 1 is exact) o 1 in = 2.54 cm exactly (1 and 2.54 are both exact) 7

8 Per means out of exactly one o 45 miles per hour means 45 mi = 1 hr or 45 mi/1 hr. (45 has 2 sig. fig. but 1 is exactly one.) Percent means out of exactly one hundred o 25.9% means 25.9 out of exactly 100 or 25.9/100 (25.9 has 3 sig. fig., but 100 is exact.) Counting numbers are exact o If you were trying to determine the total volume in 6 12 oz drinks, the 6 would be exact, the 12 oz would be an inexact measured quantity 2.5: Scientific Notation Do you need to clearly represent the number of significant figures? Have a very large or small number to represent? Try scientific (exponential) notation! When expressing numbers in scientific notation the convention is to place the decimal point after the first digit Bear in mind that you show all significant figures in the lead term when expressing a value in scientific notation Express the following values using scientific notation 19.3 g.0193 kg g g 1900 g g s 1,000,000 s 8

9 2.6: Calculations, Significant Figures, and Rounding Conventions Multiplication and Division: The number of significant figures does not change; of course you can have no greater precision than your least precise measurement, so round to the number of significant figures in that measurement Example: An object of mass g. has a volume of 523 ml. How would you report the density? Answer: g/523 ml = 1.53 g/ml. A bit counterintuitive, but bear in mind the new value has units of g/ml not grams, not ml Compare the density measurement above to addition and subtraction, where significant figures end when one of the measurements has no digit as you move to the right of the decimal. That is, you cannot add a known value to an unknown value and have a known value Rounding Numbers 2.23 g reported to 2 significant figures: 2.26 g reported to 2 significant figures: g reported to 3 significant figures: g reported to 3 significant figures: The convention I show here is at odds with the book. I will accept either, but the one shown above is statistically more accurate 9

10 The Use of Scientific Notation in Calculations The use of scientific notation is straightforward for multiplication & division (use the number of significant figures in the least precise measurement). For addition and subtraction consider adding the following: 4.63 x x 10 2 Clearly, while the convention for scientific notation is to place the decimal after the first value, we can change this and the corresponding exponent for ease of addition and subtraction 10

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