Standards for Measurement Chapter 2 Hein and Arena Eugene Passer Chemistry Department Bronx Community College © John Wiley and Sons, Inc Version 2.0 12th Edition

Chapter Outline 2.1 Scientific Notation 2.5 The Metric System 2.2 Measurement and Uncertainty 2.6 Problem Solving 2.7 Measuring Mass and Volume 2.3 Significant Figures 2.8 Measurement of Temperature 2.4 Significant Figures in Calculations 2.9 Density

2.1 Scientific Notation

Very large and very small numbers are often encountered in science. 602200000000000000000000 0.00000000000000000000625 Very large and very small numbers like these are awkward and difficult to work with.

A method for representing these numbers in a simpler form is called scientific notation. 6.022 x 1023 602200000000000000000000 6.25 x 10-21 0.00000000000000000000625

Scientific Notation Move the decimal point in the original number so that it is located after the first nonzero digit. Follow the new number by a multiplication sign and 10 with an exponent (power). The exponent is equal to the number of places that the decimal point was shifted.

Write 6419 in scientific notation. decimal after first nonzero digit power of 10 64.19x102 6.419 x 103 641.9x101 6419. 6419

Write 0.000654 in scientific notation. decimal after first nonzero digit power of 10 0.000654 0.00654 x 10-1 0.0654 x 10-2 0.654 x 10-3 6.54 x 10-4

2.2 Measurement and Uncertainty

Measurements Experiments are performed. Measurements are made.

Form of a Measurement numerical value 70.0 kilograms = 154 pounds unit

Significant Figures The number of digits that are known plus one estimated digit are considered significant in a measured quantity known 5.16143 estimated

Significant Figures The number of digits that are known plus one estimated digit are considered significant in a measured quantity certain 6.06320 uncertain

Reading a Thermometer

The temperature 21.2oC is expressed to 3 significant figures. Temperature is estimated to be 21.2oC. The last 2 is uncertain.

The temperature 22.0oC is expressed to 3 significant figures. Temperature is estimated to be 22.0oC. The last 0 is uncertain.

The temperature 22.11oC is expressed to 4 significant figures. Temperature is estimated to be 22.11oC. The last 1 is uncertain.

Exact Numbers Exact numbers have an infinite number of significant figures. Exact numbers occur in simple counting operations 2 1 3 5 4 Defined numbers are exact. 100 centimeters = 1 meter 12 inches = 1 foot

2.3 Significant Figures

Significant Figures All nonzero numbers are significant. 461

Significant Figures All nonzero numbers are significant. 461

Significant Figures All nonzero numbers are significant. 461

461 Significant Figures All nonzero numbers are significant.

Significant Figures A zero is significant when it is between nonzero digits. 3 Significant Figures 401

Significant Figures A zero is significant when it is between nonzero digits. 5 Significant Figures 9 3 . 6

Significant Figures A zero is significant when it is between nonzero digits. 3 Significant Figures 9 . 3

Significant Figures A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 5 5 .

Significant Figures A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 2 . 1 9 3

Significant Figures A zero is not significant when it is before the first nonzero digit. 1 Significant Figure . 6

Significant Figures A zero is not significant when it is before the first nonzero digit. 3 Significant Figures . 7 9

Significant Figures A zero is not significant when it is at the end of a number without a decimal point. 1 Significant Figure 5

Significant Figures A zero is not significant when it is at the end of a number without a decimal point. 4 Significant Figures 6 8 7 1

Rounding Off Numbers

Often when calculations are performed on a calculator extra digits are present in the results. It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures. When digits are dropped, the value of the last digit retained is determined by a process known as rounding off numbers.

Rules for Rounding Off Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed. 4 or less 80.873

Rounding Off Numbers Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed. 4 or less 1.875377

Rounding Off Numbers Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1. 5 or greater drop these figures increase by 1 5.459672 6

2.4 Significant Figures in Calculations

The results of a calculation based on measurements cannot be more precise than the least precise measurement.

Multiplication or Division

In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures.

The correct answer is 440 or 4.4 x 102 2.3 has two significant figures. (190.6)(2.3) = 438.38 190.6 has four significant figures. Answer given by calculator. The answer should have two significant figures because 2.3 is the number with the fewest significant figures. Drop these three digits. Round off this digit to four. 438.38 The correct answer is 440 or 4.4 x 102

Addition or Subtraction

The results of an addition or a subtraction must be expressed to the same precision as the least precise measurement.

The result must be rounded to the same number of decimal places as the value with the fewest decimal places.

Round off to the nearest unit. Add 125.17, 129 and 52.2 Least precise number. 125.17 129. 52.2 Answer given by calculator. 306.37 Correct answer. Round off to the nearest unit. 306.37

0.018286814 Answer given by calculator. Two significant figures. Drop these 6 digits. Correct answer. The answer should have two significant figures because 0.019 is the number with the fewest significant figures.

2.5 The Metric System

The metric or International System (SI, Systeme International) is a decimal system of units. It is built around standard units. It uses prefixes representing powers of 10 to express quantities that are larger or smaller than the standard units.

International System’s Standard Units of Measurement Quantity Name of Unit Abbreviation Length meter m Mass kilogram kg Temperature Kelvin K Time second s Amount of substance m mole Electric Current ampere A Luminous Intensity candela cd

Common Prefixes and Numerical Values for SI Units Power of 10 Prefix Symbol Numerical Value Equivalent giga G 1,000,000,000 109 mega M 1,000,000 106 kilo k 1,000 103 hecto h 100 102 deca da 10 101 — — 1 100

Prefixes and Numerical Values for SI Units Power of 10 Prefix Symbol Numerical Value Equivalent 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 0.00000000000001 10-15

Measurement of Length

The standard unit of length in the SI system is the meter The standard unit of length in the SI system is the meter. 1 meter is the distance that light travels in a vacuum during of a second.

1 meter = 39.37 inches 1 meter is a little longer than a yard

Metric Units of Length Exponential Unit Abbreviation Metric Equivalent Equivalent kilometer km 1,000 m 103 m meter m 1 m 100 m decimeter dm 0.1 m 10-1 m centimeter cm 0.01 m 10-2 m millimeter mm 0.001 m 10-3 m micrometer m 0.000001 m 10-6 m nanometer nm 0.000000001 m 10-9 m angstrom Å 0.0000000001 m 10-10 m

2.6 Problem Solving

unit1 x conversion factor = unit2 Dimensional Analysis Dimensional analysis converts one unit to another by using conversion factors. unit1 x conversion factor = unit2

Basic Steps Read the problem carefully. Determine what is known and what is to be solved for and write it down. It is important to label all factors and units with the proper labels.

Basic Steps Determine which principles are involved and which unit relationships are needed to solve the problem. You may need to refer to tables for needed data. Set up the problem in a neat, organized and logical fashion. Make sure unwanted units cancel. Use sample problems in the text as guides for setting up the problem.

Basic Steps Proceed with the necessary mathematical operations. Make certain that your answer contains the proper number of significant figures. Check the answer to make sure it is reasonable.

Length Conversion

How many millimeters are there in 2.5 meters? The conversion factor must accomplish two things: unit1 x conversion factor = unit2 m x conversion factor = mm It must cancel meters. It must introduce millimeters

The conversion factor takes a fractional form.

The conversion factor is derived from the equality. 1 m = 1000 mm Divide both sides by 1000 mm conversion factor Divide both sides by 1 m conversion factor

How many millimeters are there in 2.5 meters? Use the conversion factor with millimeters in the numerator and meters in the denominator.

Convert 16.0 inches to centimeters. Use this conversion factor

Convert 16.0 inches to centimeters.

Convert 3.7 x 103 cm to micrometers. Centimeters can be converted to micrometers by writing down conversion factors in succession. cm  m  meters

Convert 3.7 x 103 cm to micrometers. Centimeters can be converted to micrometers by a series of two conversion factors. cm  m  meters

2.7 Measuring Mass and Volume

Mass

The standard unit of mass in the SI system is the kilogram The standard unit of mass in the SI system is the kilogram. 1 kilogram is equal to the mass of a platinum-iridium cylinder kept in a vault at Sevres, France. 1 kg = 2.205 pounds

Metric Units of mass Exponential Unit Abbreviation Gram Equivalent Equivalent kilogram kg 1,000 g 103 g gram g 1 g 100 g decigram dg 0.1 g 10-1 g centigram cg 0.01 g 10-2 g milligram mg 0.001 g 10-3 g microgram g 0.000001 g 10-6 g

Convert 45 decigrams to grams. 1 g = 10 dg

An atom of hydrogen weighs 1. 674 x 10-24 g An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh? Grams can be converted to ounces using a series of two conversion factors. 1 lb = 454 g 16 oz = 1 lb

An atom of hydrogen weighs 1. 674 x 10-24 g An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh? Grams can be converted to ounces using a single linear expression by writing down conversion factors in succession.

Volume

Volume is the amount of space occupied by matter. In the SI system the standard unit of volume is the cubic meter (m3). The liter (L) and milliliter (mL) are the standard units of volume used in most chemical laboratories.

Convert 4.61 x 102 microliters to milliliters. Microliters can be converted to milliliters using a series of two conversion factors. L  L  mL

Convert 4.61 x 102 microliters to milliliters. Microliters can be converted to milliliters using a linear expression by writing down conversion factors in succession. L  L  mL

2.8 Measurement of Temperature

Heat A form of energy that is associated with the motion of small particles of matter. Heat refers to the quantity of this energy associated with the system. The system is the entity that is being heated or cooled.

Temperature A measure of the intensity of heat. It does not depend on the size of the system. Heat always flows from a region of higher temperature to a region of lower temperature.

Temperature Measurement The SI unit of temperature is the Kelvin. There are three temperature scales: Kelvin, Celsius and Fahrenheit. In the laboratory, temperature is commonly measured with a thermometer.

degrees Fahrenheit = oF Degree Symbols degrees Celsius = oC Kelvin (absolute) = K degrees Fahrenheit = oF

To convert between the scales, use the following relationships:

It is not uncommon for temperatures in the Canadian plains to reach –60oF and below during the winter. What is this temperature in oC and K?

It is not uncommon for temperatures in the Canadian planes to reach –60oF and below during the winter. What is this temperature in oC and K?

2.9 Density

Density is the ratio of the mass of a substance to the volume occupied by that substance.

The density of gases is expressed in grams per liter. Mass is usually expressed in grams and volume in mL or cm3.

Density varies with temperature

Examples

A 13. 5 mL sample of an unknown liquid has a mass of 12. 4 g A 13.5 mL sample of an unknown liquid has a mass of 12.4 g. What is the density of the liquid?

A graduated cylinder is filled to the 35. 0 mL mark with water A graduated cylinder is filled to the 35.0 mL mark with water. A copper nugget weighing 98.1 grams is immersed into the cylinder and the water level rises to the 46.0 mL. What is the volume of the copper nugget? What is the density of copper? 35.0 mL 46.0 mL 98.1 g

The density of ether is 0. 714 g/mL. What is the mass of 25 The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether? Method 1 (a) Solve the density equation for mass. (b) Substitute the data and calculate.

The density of ether is 0. 714 g/mL. What is the mass of 25 The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether? Method 2 Dimensional Analysis. Use density as a conversion factor. Convert: mL → g The conversion of units is

(b) Substitute the data and calculate. The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature? Method 1 (a) Solve the density equation for volume. (b) Substitute the data and calculate.

The conversion of units is The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature? Method 2 Dimensional Analysis. Use density as a conversion factor. Convert: g → L The conversion of units is

The End