Part 01 - Notes: Identifying Significant Figures

Transcription

1 Part 01 - Notes: Identifying Significant Figures Objectives: Identify the number of significant figures in a measurement. Compare relative uncertainties of different measurements. Relate measurement precision to the number of significant figures in a measurement. Text Reference: Chang and Goldsby (Chemistry: The Essential Concepts, McGraw-Hill, 2014) pp Text Vocabulary: Significant figures (p15) the number of meaningful digits in a measured or calculated quantity Any measurement involves an estimate and that means that there is some uncertainty in that measurement. When a measurement is recorded, it includes all the digits that are certain plus the single estimated digit. These certain digits plus one uncertain (estimated) digit are referred to as significant figures. The certainty of a particular measurement is indicated by the number of significant figures recorded in that measurement. ***KEY POINT*** The further the estimated digit lies to the right in a measurement, the less relative uncertainty there is, so the more reliable the measurement. Consider the following measurements: 34.3 ml 34 ml ml ml Which of the measurements has the least amount of relative uncertainty? Which of the measurements is the least certain (least reliable)? Which of the measurements has the most precision? Make a statement that relates uncertainty in a measurement and precision of the same measurement? Rules for Identifying Significant Figures These rules #1-5 are from your textbook - page NONZERO INTEGERS: Any digit that is not zero is significant. Thus, 845 cm has three significant figures, kg has four significant figures, and so on. 2. CAPTIVE ZEROS: Zeros between nonzero digits are significant. Thus, 606 m contains three significant figures, 40,501 kg contains five significant figures, and so on. 3. LEADING ZEROS: Zeros to the left of the first nonzero digit are not significant. Their purpose is to indicate the placement of the decimal point. For example, 0.08 L contains one significant figure, g contains three significant figures, and so on. 4. TRAILING ZEROS: If a number is greater than 1, then all the zeros written to the right of the decimal point count as significant figures. Thus, 2.0 mg has two significant figures, ml has five significant figures, and dm has four significant figures. If a number is less than 1, then only the zeros that are at the end of the number and the zeros that are between nonzero digits are significant. This means that kg has two significant figures, L has four significant figures, min has three significant figures, and so on. page 1 of 2

2 Part 01 - Notes: Identifying Significant Figures 5. For numbers that do not contain decimal points, the trailing zeros (that is, zeros after the last nonzero digit) may or may not be significant. Thus, 400 cm may have one significant figure (the digit 4), two significant figures (40), or three significant figures (400). We cannot know which is correct without more information. By using scientific notation, however, we avoid this ambiguity. In this particular case, we can express the number 400 as 4 x 10 2 for one significant figure, 4.0 x 10 2 for two significant figures, or 4.00 x 10 2 for three significant figures. 6. EXACT NUMBERS: Often calculations involve numbers that were not obtained using measuring devices but were determined by counting: 7 beakers, 3 apples, 8 molecules. Such numbers are called exact numbers. They can be assumed to have an unlimited number of significant figures. Exact number may also arise from definitions. For example, 1 meter is defined as 100 cm; neither 100 nor 1 limits the number of significant figures when it is used in calculations. Question: List other exact number that would contain infinite significant figures. Application: How many significant figures in the following measurements and circle the place of uncertainty? (a) 6072 m sigfig (b) g sigfig (c) 6070 L sigfig (d) L sigfig (e) g sigfig (f) L sigfig (g) m sigfig (h) 3000 m sigfig (i) L sigfig (j) m sigfig (k) L sigfig (l) g sigfig page 2 of 2

3 Part 02 - Notes: Calculations with Significant Figures - 1 Objectives: Correctly record an answer to a single-step math problem with the appropriate number of significant figures. Determine, utilize, and explain the rule for the number of significant figures in a sum or a difference. Determine, utilize, and explain the rule for the number of significant figures in a product or quotient. Explain how the rules for determining the number of significant figures in single-step math problems are derived. Text Reference: Wilbraham, Staley, Matta, and Waterman (Chemistry, Prentice Hall, 2002) pp Adding and Subtracting Significant Figures When adding or subtracting measurements, the sum or difference can only be as certain as the least certain measurement. Remember, properly recorded measurements are recorded using significant figures, all certain digits plus a single uncertain estimated digit. Let s Practice: Add the values and report the answers to the correct number of significant figures. Indicate/track your uncertainty. (1) m m (2) 12.7 g g (3) cm cm (4) 9.1 m m + 10 m The position of the uncertain digit with relation to the decimal point in the least certain measurement determined the position of the uncertain digit with relation to the decimal point in the answer. For our purposes, the uncertain digit should be rounded up or rounded down, just as in math. Complete the simplified rule for addition and subtraction. The answer for an addition or subtraction problem is rounded to... Multiplying and Dividing Significant Figures When multiplying and dividing measurements, the answer may only contain as many significant figures as does the measurement used with the least amount of significant figures. Let s Practice: Find the following products and report the answers to the correct number of significant figures. Uncertainty? (5) 42.0 m x 4.0 m (6) 1.78 cm x 0.05 cm (7) 1001 mm x 40 mm (8) 1001 mm x 40. mm Complete the simplified rule for multiplication and division. The answer for a multiplication or division problem is rounded to... page 1 of 1

4 Part 03 - Notes: Calculations with Significant Figures - 2 Unit 01 - ChemSkills Objectives: Correctly record an answer to multiple-step math problems, averages, and conversions with the appropriate number of significant figures. Additional objective see objectives from Unit 1 Part 2. Text Reference: Chang and Goldsby (Chemistry: The Essential Concepts, McGraw-Hill, 2014) pp Case 1: Multiplication, Division, and Exponent Operations, only When multiplying, dividing, and using exponents only, perform all operations first and then apply the rule for significant figures to the answer. Be careful with parentheses; it s easy to get tricked and round in the middle. Example 1: (2.48 m) 2 (2.518 m) 5.78 m Example 2: (680 m 2 x 1.250m) ( m x 298 m) Case 2: Addition and/or Subtraction in combination with other operations When addition or subtraction operations are used in conjunction with other operations, follow the algebraic order of operations, applying the rules of significant figures after addition/subtraction is completed. Then perform the multiplication/division as necessary and then apply the rules for significant figures, again. Example 3: (4.238 m m) 2.5 s Example 4: [(4.56 m m) (2.00 m + 9 m)] 27.9 m Rule 3: Numbers with NO uncertainty All conversion fractions, constants, and numbers with no uncertainty are treated as if they have an infinite number of significant figures. More on this part later. Example 5: Convert hours to seconds. Case 4: Averages and uncertainty When finding the average (mean) of measurements, the mean cannot be more certain than the least certain measurement. If you are going to go wrong, here it where it is!!! It s about place value. No more than one place of uncertainty. Example 6: Calculate the mean of the following measurements: g and g Example 7: Calculate the mean of the following measurements: 220 g, 137 g, and g page 1 of 1

5 Part 04 - Notes: Observations and Scientific Notation Objectives: Convert a number in standard notation to correct scientific notation, and vice versa. Identify the part of a number in scientific notation and state appropriate values for each part. Distinguish between qualitative and quantitative measurements. Perform simple math problems with number in scientific notation without a scientific calculator. Text Reference: Chang and Goldsby (Chemistry: The Essential Concepts, McGraw-Hill, 2014) pp Text Vocabulary: Qualitative (p3) consisting of the general observations about the system Quantitative (p3) comprising numbers obtained by various measurements of the system Types of Observations Qualitative observation and Quantitative observation Question: Consider this chemistry class. List several qualitative observations. List several quantitative observations. Clarifying Quantitative Observations Scientists use large and small numbers frequently, but with so many zeros, problems are frequently encountered. Writing large numbers of zeros is time consuming and there is an increased chance of an error being made when writing many zeros. Scientists handle large and small numbers using scientific notation, also called standard exponential form. A number written in scientific notation has the following appearance and parts: a.bc x 10 d Part a.bc 10 d What is it? coefficient base exponent Acceptable Values 1 coefficient < 10 we are using base 10 May be any real number. contains as many digits as there are significant figures There are other bases we encounter later. Ex 1: Write the following numbers in scientific notation. (a) (b) Generally we will encounter only whole numbers. Positive exponents for numbers larger than one. Makes the coefficient larger. Negative exponents for numbers smaller than one. Makes the coefficient smaller. Exponent of 0 = 1 Changing the Form of Exponential Numbers You can increase either the coefficient or the exponent of a number in scientific notation by any factor without changing the overall value of the number as long as we reduce the other portion by the same factor. In other words, as one part increases, the other part decreases. Ex 2: Change the following to correct scientific notation: (a) x10 6 (b) 303x10-4 (c) 0.012x10-7 page 1 of 2

6 Part 04 - Notes: Observations and Scientific Notation Multiplication and Division with Scientific Notation To multiply or divide numbers in scientific notation, handle the coefficient portions and the exponent portions separately. For multiplication, multiply the coefficients and add algebraically add the exponents. (a x 10 m ) (b x 10 n ) = (a x b) x 10 (m + n) For division, divide the coefficients and algebraically subtract the exponents. (a x 10 m ) (b x 10 n ) = (a b) x 10 (m n) Make sure that the final product or quotient has a coefficient that is greater than or equal to 1 or less than 10. Addition and Subtraction with Scientific Notation When we add or subtract numbers in exponential notation, the exponents must be the same. This rule is the same as making sure the decimals of your numbers you add or subtract are aligned. The answer is the sum or difference of the coefficients with the same exponent as each number in the problem. Remember, the final answer needs to be in correct scientific notation form. Suggestion: Change the exponent on the number that has the SMALLER exponent. Ex 5: Perform the following operations. Show the necessary intermediates. (a) (3.0x10 6 ) (5.0x10 3 ) = (b) 4.25x10 5 ) (7.7x10 2 ) = (c) (5.0x10 3 ) (6.75x10 5 ) = (d) (2.5x10 5 ) (5.6x10 2 ) = (e) (3.45x10 6 ) + (4.56x10 7 ) = (f) (2.45x10 4 ) + (3.75x10 2 ) = (g) (2.98x10 5 ) (3.25x10 4 ) = (h) (9.68x10 3 ) (4.67x10 4 ) = page 2 of 2

7 Part 07 - Notes: Accuracy and Precision Objectives: Define, explain, and distinguish between accuracy and precision and when each is relevant. Calculate the absolute and relative error and the deviation and standard deviation of a set of experimental measurements. Contrast random and systematic errors, their impact on measurements, and how they may be reduced. Text Reference: Chang and Goldsby (Chemistry: The Essential Concepts, McGraw-Hill, 2014) pp Text Vocabulary: Accuracy (p17) the closeness of the measurement to the true value of the quantity that is being measured Precision (p17) the closeness of agreement of two or more measurements of the same quantity Concept Charts: Accuracy & Precision and Random & Systematic Error A measurement is An error in measurement can occur PRECISE ACCURATE CONSISTENTLY INCONSISTENTLY if it is in agreement with in which case it is classified as repeated measurements by the same procedure the true value random and it causes poor systematic in other words, it is PRECISION ACCURACY reproducible correct The extent of agreement can be expressed as the range or standard deviation of repeated measurements absolute error or percent error of the measurement The effect of the error can be reduced by repeating the measurement and averaging the results running a standard and applying a correction factor A Tale of Four Graduated Cylinders Consider the four graduated cylinders in the figures below. Each graduated cylinder contains exactly ml of liquid. Compare the accuracy and precision of these cylinders. Why do the cylinders on the left side in each set ( cylinders A and C) give more precise measurements? Why are the set of graduates on the in the right set (cylinders C and D) inaccurate when compared with the set on the left? Precise Imprecise Precise Imprecise What type of errors are associated with each cylinder? Do you know why? page 1 of 3

8 Part 07 - Notes: Accuracy and Precision Let s look at the quantitative aspects of reporting accuracy and precision with error and deviation calculations. For our calculations, let s consider an experiment where we determined the boiling point of pure isopropyl alcohol (rubbing alcohol) at sea level in three different trials. We need to compare our trials to each other to check our precision. We need to compare our trials to the accepted value for the boiling point of isopropyl alcohol to check our accuracy. Our experimental boiling point data: Trial 1: 79.3 o C Trial 2: 79.1 o C Trial 3: 79.7 o C Accepted value for the boiling point: 82.5 o C Mean For our purposes, unless otherwise specified, mean refers to arithmetic average. The mean represents the gravitational center of a distribution of data. This is where the distribution would balance (like a seesaw). Calculating the Mean x = i = n x i i = 1 n Standard Deviation and Variance Standard deviation is the most common measure of the spread of data, in other words, how far apart are our measurements from one another. This is a measure of the precision of our data. Deviation The difference between a data value and the mean ( x i x) Find the deviation for EACH data measurement. Square each deviation value. This makes the negative or positive sign associated with the deviation unimportant. Sum of Squares The sum of all the squares of the deviations SS = x i x ( ) 2 Variance The average of the sum of squares Standard Deviation The square root of the variance s = s 2 May also express the relative standard deviation (RDS) RDS = s x x 100 page 2 of 3

9 Part 07 - Notes: Accuracy and Precision Absolute and Relative Error Relative error, also known as percent error, is the most common measure of how accurate your measurement set is, in other words, how close did you come to the accepted or actual value. Calculating the percent error is a shorter process than finding RDS but important in helping find any potential systematic error. Calculate the mean of your data Absolute Error This is the difference between mean and accepted Also called Experimental Error E absolute = x x actual Percent Error Also called Relative Error E relative = E absolute x actual x 100 = x x actual x actual x 100 The process above was completed using the mean of your lab measurements. It may also be done with each individual measurement and then all of the individual relative errors may be averaged. Unless there is a significant outlier, the values are the same. Recall the Tale of Four Graduated Cylinders from above. Look at the data from 5 different students each recording the volume shown in the graduates. Table 1. Data from Graduated Cylinders Illustrated in Figure 1 Measured Precision Accuracy Cylinder Volume / ml ** Mean / ml Range / ml Error / ml Standard Deviation / ml Percent Error A B C D ** Each cylinder contains exactly ml page 3 of 3

10 Part 08 - Notes: Matter Web and Changes Unit 01 - The Atom Text Reference: Chang and Goldsby (Chemistry: The Essential Concepts, McGraw-Hill, 2014) pp Text Vocabulary: Matter (p4) anything that occupies space and possesses mass Element (p5) a substance that cannot be separated into simpler substances by chemical means Heterogeneous mixture (p5) the individual components of such a mixture remain physically separate and can be seen as separate component Homogeneous mixture (p5) the composition of the mixture is the same throughout the solution Mixture (p5) combination of two or more substances in which the substances retain their individual identities Substance (p5) a form of matter that has a definite or constant composition (the number and type of basic units present) and distinct properties Compound (p6) a substance composed of two or more elements chemically united in fixed proportions Chemical property (p7) a property of a substance that cannot be studied without converting the substance into some other substance Physical property (p7) property that can be observed without transforming into some other substance Extensive property (p8) a property that depends on how much matter is being considered Intensive property (p8) a property that does not depend on how much matter is being considered Macroscopic property (p8) properties that can be measured directly Microscopic property (p8) properties that must be measured indirectly with the aid of special instrumentation Task: Fill in the word or terms needed to complete the matter concept web. NO NO MATTER Is it uniform throughout? YES NO Does it have variable composition? Can it be separated into simpler substances? YES YES page 1 of 2

11 " Part 08 - Notes: Matter Web and Changes Unit 01 - The Atom The following figures show the representative particles of a given substance in various states of matter. E F D G A B C Solid Liquid Gas 1. In which phase of matter is there the least spacing between particles? 2. In which phase of matter is there the most potential for movement? 3. Which phase of matter does not have a definite shape yet the particles will not fill the container? Information: The change between one state of matter and another is called a phase change or a change of state. ALL changes between different states of matter are physical changes. The changes between different states of matter occur because energy is put in (endothermic) or energy is released (exothermic). 4. Which arrow(s) in the Model indicate a change of state due to an increase in energy? D E F G 5. Are the above changes of state endothermic or exothermic? 6. Which arrow(s) in the Model indicate a change of state due to an decrease in energy? D E F G 7. Are the above changes of state endothermic or exothermic? 8. The diagram below shows all three major states of matter. A change of state may occur between any two of the states. Label the arrows with the name of the various phase changes. Pay attention to the direction of the arrows when labeling the change of state. SOLID LIQUID GAS page 2 of 2