Working With Numbers Objectives: 1. Define significant digits. 2. Explain how to determine which digits in measurement are significant. 3. Convert measurements in to scientific notation. Key Terms: significant digit, percent error, density

Estimated Digits & Measurements When making measurements in science it is important to write down all of the digits that a device can give you and one estimated digit. Remember that measurements are never completely accurate for the following two reasons: measuring equipment is never completely free of flaws measurements always involve some degree of estimation Note: Electronic devices take care of estimation for you and record the last digit as the estimated digit.

Rules having to do with ZEROS! Zeros that simply hold places are not significant Rule #1: Trailing zeros without a decimal are NEVER significant (13000… only the 1 & 3 are significant) This number is said to have 2 significant digits Rule #2: Leading zeros are NEVER significant (0.0027… only the 2 & 7 are significant) Rule #3: Trailing zeros after a decimal are ALWAYS significant (0.0024500… 24500 are all significant and add to the accuracy of the measurement) Rule #4: Zeros found between numbers are ALWAYS significant (987001… all numbers are significant) This number is said to have 6 significant digits

Significant Digits In Calculations – Multiplication and Division The answer cannot have more precision than the least precise measurement. Example: 0.3287g x 45.2g = ? 1) determine the operand with the smallest amount of significant digits (0.3287 (___ sig digits) & 45.2 (__ sig digits)) 2) perform the operation on your calculator and round to the correct amount of digits 0.3287g x 45.2g = 14.85724g (too many digits) ____g (rounded to ___ sig digits)

Significant Digits In Calculations – Multiplication and Division The measurement with the smallest amount of significant digits determines the significant digits of the answer Example: 0.3287g x 45.2g = ? 1) determine the operand with the smallest amount of significant digits (0.3287 (4 sig digits) & 45.2 (3 sig digits)) 2) perform the operation on your calculator and round to the correct amount of digits 0.3287g x 45.2g = 14.85724g (too many digits) 14.9g (rounded to 3 sig digits)

Significant Digits In Calculations – Addition and Subtraction The total cannot be more accurate than the least accurate measurement. This time the precision of significant digits of each number does not matter. The quantity with the least digits to the right of the decimal point determines the accuracy of the answer Example: 125.5kg + 52.68kg + 2.1kg = ? 1) determine the precision of your least accurate measurement. (125.5kg and 2.1kg are both accurate to the ___ place while 52.68kg is accurate to the ___ place) 2) perform the operation 125.5kg + 52.68kg + 2.1kg = 180.28kg (too precise!) ___kg (precise to the ___place)

Significant Digits In Calculations – Addition and Subtraction The total cannot be more accurate than the least accurate measurement. This time the amount of significant digits of each number does not matter. The quantity with the least digits to the right of the decimal point determines the accuracy of the answer Example: 125.5kg + 52.68kg + 2.1kg = ? 1) determine the precision of your least accurate measurement. (125.5kg and 2.1kg are both accurate to the tenths place while 52.68kg is accurate to the hundredths place) 2) perform the operation 125.5kg + 52.68kg + 2.1kg = 180.28kg (too precise!) 180.3kg (precise to the tenths place)

Rules with Mixed Operations When doing combinations of addition/subtraction and multiplication/division, each step determines its significant digits. The digits in the end answer are determined by the last operation performed. Example: 1250cal – (234.207cal/52.69cal) = ? 1250cal – (4.445cal) = (parenthesis 1st … answer has 4 sig digits) 1250cal – (4.445cal) = 1245.555cal (too precise!) 1250cal (accurate to the tens place)

Scientific Notation Scientific notation is a way that scientists make incredibly large numbers used in science easier to work with. There are 602,000,000,000,000,000,000,000 atoms in a mole of a substance. It is much easier to use the answer as 6.02 x 1023 Rules: 1) The answer must be in the form of a real number followed by a decimal point while retaining the correct amount of significant digits. 2) If the magnitude of the number is to be reduced the exponent will be positive. Example: 398700 = 3.987 x 105 (the exponent is equal to the number of times that the decimal point was moved) 3) If the magnitude of the number is to be increased the exponent will be negative. Example: 0.00501 = 5.01 x 10-3

Percent Error Percent error calculations are used to compare test results to a known accepted quantity. The formula is as follows: Percent error = ((measured value - accepted value) / accepted value ) * 100% Percent error = ((experimental value – theoretical value) / theoretical value ) * 100% Note: The result can be positive or negative but the answer is always represented as the absolute value Example: The accepted mass of an object is 5.00g. When you measure it on your digital scale the reading shows 5.02g. What is the percent error of your measurement? (5.02g – 5.00g) / 5.00g = 0.004 or 0.4%

Metric Prefixes Prefix Prefix Symbol Meaning Scientific Notation exa- 1,000,000,000,000,000,000 1018 peta- P 1,000,000,000,000,000 1015 tera- T 1,000,000,000,000 1012 giga- G 1,000,000,000 109 mega- M 1,000,000 106 kilo- k 1,000 103 hecto- h 100 102 deka- da 10 101 meter 1 deci- d 0.1 10-1 centi- c 0.01 10-2 milli- m 0.001 10-3 micro- 0.000001 10-6 nano- n 0.000000001 10-9 pico- p 0.000000000001 10-12 femto- f 0.000000000000001 10-15 atto- a 0.000000000000000001 10-18

Molarity is simply the moles of a solute per liter of solution. (mol/L) What is the molarity of a solution containing 100.0g of NaCl in 500.L of H2O? Dilutions with molarity are performed using the equation M1V1 = M2V2 How many ml of 3M HCl solution are required to make 100.ml of a 0.1M solution? Molarity

Concentrations % Concentrations (m/m) or (v/v) Are always the (part/whole) How many ml of ethyl alcohol are required to make 50.0ml of a 60% solution? V/50.0ml = .60 What is the %(m/m) of a 3M solution of NaOH? (mass 3mol NaOH)/(mass 3mol NaOH + mass 1L H2O) Dilutions are made using the equation C1V1 = C2V2 How many ml of a 50% EtOH solution are needed to make 100.ml of a 10% solution? Concentrations