CHEM 139: Zumdahl Chapter 2 page 1 of 17 CHAPTER 2: MEASUREMENT AND CALCULATIONS Active Learning Problems: 1-3, 6, 9-13, 16-21 End-of-Chapter Problems: 1-97, 99-101, 103-105, 107, 109-110, 112-119, 121-137, 141-153 measurement: a number with attached units When scientists collect data, they record the measurements as accurately as possible, and they must also report the measurements taken to reflect the accuracy and precision of the instruments they used to collect that data. Consider the following plot of global land-ocean temperatures based on measurements taken from meteorological stations and ship and satellite temperature (SST) measurements: Source: Hansen, J., Mki. Sato, R. Ruedy, K. Lo, D.W. Lea, and M. Medina-Elizade, 2006: Global temperature change. Proc. Natl. Acad. Sci., 103, 14288-14293, doi:10.1073/pnas.0606291103. (http://pubs.giss.nasa.gov/abstracts/2006/Hansen_etal_1.html) The plot above shows annual mean (average) temperatures in black, 5-year mean temperatures in red, and the uncertainty as green bars. Example: According to the plot, global land-ocean temperatures have changed by how much since the 1960s?CHEM 139: Zumdahl Chapter 2 page 2 of 17 2.3 SIGNIFICANT FIGURES (or SIG FIGS): Writing Numbers to Reflect Precision To measure, one uses instruments = tools such as a ruler, balance, etc. All instruments have one thing in common: UNCERTAINTY!  INSTRUMENTS CAN NEVER GIVE EXACT MEASUREMENTS! When a measurement is recorded, all the given numbers are known with certainty (given the markings on the instrument), except the last number is estimated.  The digits are significant because removing them changes the measurement's uncertainty. – Thus, when measurements are recorded, – they are recorded to one more decimal place than the markings for analog instruments; – they are recorded exactly as displayed on electronic (digital) instruments. LENGTH – generally reported in meters, centimeters, millimeters, kilometers, inches, feet, miles – Know the following English-English conversions: 1 foot  12 inches 1 yard  3 feet Example: Using Rulers A, B, and C below, indicate the measurement to the line indicated for each ruler. Assume these are centimeter rulers, so show the each measurement has units of cm. Circle the estimated digit for each measurement. A B C Measurement Increment of the smallest markings on ruler # of sig figs Thus, a measurement is always recorded with one more digit than the smallest markings on the instrument used, and measurements with more sig figs are usually more accurate. Ruler ARuler B0 1 2 3 4 50 1 2 3 4 5Ruler C4.1 4.2 4.3 4.4CHEM 139: Zumdahl Chapter 2 page 3 of 17 2.4 UNCERTAINTY IN MEASUREMENT Guidelines for Sig Figs (if measurement is given): Count the number of digits in a measurement from left to right: 1. When there is a decimal point: – For measurements greater than 1, count all the digits (even zeros). – 62.4 cm has 3 sig figs, 5.0 m has 2 sig figs, 186.100 g has 6 s.f. – For measurements less than 1, start with the first nonzero digit and count all digits (even zeros) after it. – 0.011 mL and 0.00022 kg each have 2 sig figs 2. When there is no decimal point: – Count all non-zero digits and zeros between non-zero digits – e.g. 125 g has 3 sig figs, 1007 mL has 4 sig figs – Placeholder zeros may or may not be significant – e.g. 1000 may have 1, 2, 3 or 4 sig figs Example: Indicate the number of significant digits for the following: a. 165.3 g _____ c. 90.40 m _____ e. 0.19600 g _____ b. 105 cm _____ d. 100.00 L _____ f. 0.0050 cm _____ 2.3 MEASUREMENTS OF LENGTH, VOLUME, AND MASS VOLUME: Amount of space occupied by a solid, gas, or liquid. – generally in units of liters (L), milliliters (mL), or cubic centimeters (cm3) – Know the following: 1 L  1 dm3 1 mL  1 cm3 (These are both exact!) Note: When the relationship between two units or items is exact, we use the “” to mean “equals exactly” rather than the traditional “=” sign. – also know the following equivalents in the English system 1 gallon  4 quarts 1 quart  2 pints 1 pint  2 cups MASS: a measure of the amount of matter an object possesses – measured with a balance and NOT AFFECTED by gravity – usually reported in grams or kilograms WEIGHT: a measure of the force of gravity – usually reported in pounds (abbreviated lbs) mass ≠ weight = mass  acceleration due to gravityCHEM 139: Zumdahl Chapter 2 page 4 of 17 Mass is not affected by gravity! 2.1 SCIENTIFIC NOTATION Some numbers are very large or very small  difficult to express. Avogadro’s number = 602,000,000,000,000,000,000,000 an electron’s mass = 0.000 000 000 000 000 000 000 000 000 91 kg To handle such numbers, we use a system called scientific notation. Regardless of their magnitude, all numbers can be expressed in the form N10n where N =digit term= a number between 1 and 10, so there can only be one number to the left of the decimal point: #.#### n = an exponent = a positive or a negative integer (whole #). To express a number in scientific notation: – Count the number of places you must move the decimal point to get N between 1 and 10. Moving decimal point to the right (if # < 1)  negative exponent. Moving decimal point to the left (if # > 1)  positive exponent. Example: Express the following numbers in scientific notation (to 3 sig figs): Height of Mt. Rainier: 14,400 ft.  __________________ Diameter of a human hair: 0.0110 cm  __________________ Avogadro’s Number: 602,000,000,000,000,000,000,000  _____________________CHEM 139: Zumdahl Chapter 2 page 5 of 17 Some measurements may be rounded to a number of sig figs requiring scientific notation. For example, Express 100.0 g to 3 sig figs: ___________  ______________ Express 100.0 g to 2 sig figs: ___________  ______________ Express 100.0 g to 1 sig fig: ___________  ______________ UNBIASED ROUNDING (or ROUND-TO-EVEN METHOD) How do we eliminate nonsignificant digits? • If first nonsignificant digit < 5, just drop the nonsignificant digits • If first nonsignificant digit ≥ 6, raise the last sig digit by 1