Chapter 2 Section 3 Continued Objectives • Be able to define: accuracy, precision, percent error, significant figures, scientific notation, conversion factor, hyperbola. • Be able to distinguish between accuracy and precision. • Be able to determine the number of significant figures in a measurement. • Be able to perform mathematical operations and express the result in the proper number of significant figures. • Be able to convert measurements into scientific notation. • Be able to distinguish between inversely and directly proportional. • Be able to perform calculations involving measurements (addition, subtraction, multiplication, division) and express the result in the proper number of significant figures and the proper units. Significant Figures ***** This is IMPERATIVE: You MUST use and recognize significant figures when you work with measured quantities and report your results, and when you evaluate measurements reported by others. So, LEARN THE RULES FOR DETERMINING SIGNIFICANT FIGURES.***** A. Determining the number of Significant Figures: • All nonzero digits are significant. • All zeros between two nonzero digits are significant • Zeros at the end of a number and to the right of a decimal point are significant. • Zeros at the end of a number but to the left of a decimal point may or may not be significant. If a zero has not been measured or estimated but is a placeholder, it is not significant. A decimal point placed after zeros indicates that they are significant. B. Rounding Significant Figures: • The answers given on a calculator can be derived results with more digits than are justified. • Supposed you use a calculator to divide 154g by 327g. • Each of these numbers has three significant figures but the calculator will show an answer of 0.470948012 – • Such an answer has to be rounded off. Rounding Rules Rounding Significant Figures When performing calculations involving measurements certain rules apply whether you are adding, subtracting, multiplying, or dividing. FOLLOW THOSE RULES. Addition and Subtraction with Significant Figures: When adding or subtracting decimals, the answer must have the same number of the digits to the RIGHT of the decimal point as there are in the measurement having the FEWEST digits to the right of the decimal point. Addition and Subtraction Steps when adding or subtracting: 1. Place all measurements in a column. 2. Find the rightmost place where there is a digit for each number. 3. Round all measurements to that place, then add or subtract. Addition and Subtraction + 24.6 cm 120.003 cm 4.68 cm 0.006 cm 149.3 cm Addition and Subtraction 25.1 + 2.03 cm cm 27.1 cm Addition and Subtraction •When working with WHOLE numbers, the answer should be rounded so that the FINAL sig fig digit is in the SAME PLACE as the leftmost uncertain digit. 5400 + 365 = 5800 Multiplication and Division For multiplication and division, the answer can have NO MORE significant figures than are in the measurement with the FEWEST number of significant figures Multiplication and Division 1. Count the number of significant figures in each measurement. 2. Round your answer to the number of significant figures in the measurement with the LEAST number of significant figures. 3.05 g = 0.360094451 g/mL = 0.360 g/mL D= 8.47 mL Scientific Notation • The numbers we deal with in science can be extremely small or extremely large. • Converting to scientific notation or exponential notation makes handling these numbers much easier. Scientific Notation • In scientific notation, numbers are written in the form of M x 10n, where the factor M is a number greater than or equal to 1 but less than 10 and n is a whole number. Example: 6.02 x 1023 atoms/mole Incorrect: 60.2 x 1022 or .602 x 1024 Steps of converting to scientific notation: 1. Determine M by moving the decimal point in the original number to the left or right so that only one nonzero digit remains to the LEFT of the decimal place. 2. Determine n by counting the number of places that you move the decimal place. a. If you move the decimal to the LEFT, n is POSITIVE b. If you move the decimal to the RIGHT, n is NEGATIVE Mathematical Operations using Scientific Notation: Addition and Subtraction: • These operations can be performed ONLY if the values have the same exponent (n factor). • If they do not, adjustments must be made to the values so that the exponents are EQUAL. • Once the exponents are equal, the M factors can be added or subtracted – the exponents can remain the SAME or it may then require adjustment if the M factor of the answer has MORE than ONE digit to the left of the decimal. Addition and Subtraction • Convert all numbers to the same power of 10. • Add or Subtract all the coefficients (M’s). • Convert the sum to proper notation form. 4.2 x 104 kg 0.79 x 104 kg 4.99 x 104 kg rounded to 5.0 x 104 kg Multiplication • Multiply coefficients (M’s) • Add exponents • Convert to proper notation form (5.23 x 106 um) (7.1 x 10-2 um) = (5.23 x 7.1) (106 x 10-2) = 37.133 x 104 um2 (adjust to two sig figs and one nonzero number to left of decimal) = 3.7 x 105 um2 Division • Divide coefficients (M’s) • Subtract exponents • Convert to proper notation form 5.44 x 107 g 8.1 x 104 mol = 5.44 x 107-4 g/mol 8.1 mol = 0.6716049383 x 103 = 6.7 x 102 g/mol Problem Solving Analyzing and solving problems is an integral part of Chemistry. You must follow a logical approach to solving problems in Chemsitry. 1. ANALYZE Analyze the Problem - read the problem at least twice to analyze the information in the problem. If you don’t understand it read it again. Problem Solving 2. PLAN Develop a plan for solving the problem. 3. COMPUTE Substitute the data and necessary conversion factors into the plan you have developed. 4. EVALUATE Examine your answer to determine whether it is reasonable. (Sample Prob. F, pg. 54) Direct Proportions • Two quantities are directly proportional to each other if dividing one by the other gives a constant value. Inverse Proportions • Two quantities are inversely proportional to each other if their product is constant. Inverse Proportion • A graph of variables that are inversely proportional produces a curve called a hyperbola.