Measurements in Chemistry Fundamentals of General, Organic, and Biological Chemistry 5th Edition Chapter Two Measurements in Chemistry James E. Mayhugh Oklahoma City University 2007 Prentice Hall, Inc.

Outline 2.1 Physical Quantities, Metric System 2.2 Measuring Mass 2.3 Measuring Length and Volume 2.4 Measurement and Significant Figures 2.5 Scientific Notation 2.6 Rounding Off Numbers 2.7 Converting a Quantity from One Unit to Another 2.8 Problem Solving: Estimating Answers 2.9 Measuring Temperature 2.10 Energy and Heat 2.11 Density 2.12 Specific Gravity Prentice Hall © 2007 Chapter Two

2.1 Physical Quantities Physical properties such as height, volume, and temperature that can be measured are called physical quantities. Both a number and a unit of defined size is required to describe physical quantity. Prentice Hall © 2007 Chapter Two

A number without a unit is meaningless. To avoid confusion, scientists have agreed on a standard set of units. Scientists use SI or the closely related metric units. Prentice Hall © 2007 Chapter Two

Scientists work with both very large and very small numbers. Prefixes are applied to units to make saying and writing measurements much easier. The prefix pico (p) means “a trillionth of.” The radius of a lithium atom is 0.000000000152 meter (m). Try to say it. The radius of a lithium atom is 152 picometers (pm). Try to say it. Prentice Hall © 2007 Chapter Two

Frequently used prefixes are shown below. Prentice Hall © 2007 Chapter Two

2.2 Measuring Mass Mass is a measure of the amount of matter in an object. Mass does not depend on location. Weight is a measure of the gravitational force acting on an object. Weight depends on location. A scale responds to weight. At the same location, two objects with identical masses have identical weights. The mass of an object can be determined by comparing the weight of the object to the weight of a reference standard of known mass. Prentice Hall © 2007 Chapter Two

a) The single-pan balance with sliding counterweights a) The single-pan balance with sliding counterweights. (b) A modern electronic balance. Prentice Hall © 2007 Chapter Two

Relationships between metric units of mass and the mass units commonly used in the United States are shown below. Prentice Hall © 2007 Chapter Two

Problem Express 3.27 mg in units of grams. Express 19.3 kg in units of mg. Prentice Hall © 2007 Chapter Two

2.3 Measuring Length and Volume The meter (m) is the standard measure of length or distance in both the SI and the metric system. Volume is the amount of space occupied by an object. A volume can be described as a length3. The SI unit for volume is the cubic meter (m3). Prentice Hall © 2007 Chapter Two

Relationships between metric units of length and volume and the length and volume units commonly used in the United States are shown below and on the next slide. Prentice Hall © 2007 Chapter Two

A m3 is the volume of a cube 1 m or 10 dm on edge A m3 is the volume of a cube 1 m or 10 dm on edge. Each m3 contains (10 dm)3 = 1000 dm3 or liters. Each liter or dm3 = (10cm)3 =1000 cm3 or milliliters. Thus, there are 1000 mL in a liter and 1000 L in a m3. Prentice Hall © 2007 Chapter Two

The metric system is based on factors of 10 and is much easier to use than common U.S. units. Does anyone know how many teaspoons are in a gallon? Prentice Hall © 2007 Chapter Two

2.4 Measurement and Significant Figures Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place. Prentice Hall © 2007 Chapter Two

Figure: 02-03UN03 Title: Properly reporting measurements Caption: What is the temperature reading on the following Celsius thermometer? How many significant figures do you have in your answer? Notes: Recall that measurements should be reported to the first uncertain digit. Therefore this thermometer should be read to the nearest tenth of a degree Celsius. The measurement should also be reported with a unit (the Celsius degree). There would be 3 significant digits in the reported temperature. Keywords: significant figures, digits, measurement

No further insignificant digits should be recorded. To indicate the precision of a measurement, the value recorded should use all the digits known with certainty, plus one additional estimated digit that usually is considered uncertain by plus or minus 1. No further insignificant digits should be recorded. The total number of digits used to express such a measurement is called the number of significant figures. All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. Prentice Hall © 2007 Chapter Two

Below are two measurements of the mass of the same object Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty. Prentice Hall © 2007 Chapter Two

When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four. Prentice Hall © 2007 Chapter Two

RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. Prentice Hall © 2007 Chapter Two

2.5 Scientific Notation Scientific notation is a convenient way to write a very small or a very large number. Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power. 215 is written in scientific notation as: 215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102 Prentice Hall © 2007 Chapter Two

Two examples of converting scientific notation back to standard notation are shown below. Prentice Hall © 2007 Chapter Two

The distance from the Earth to the Sun is 150,000,000 km The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures. Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4. Prentice Hall © 2007 Chapter Two

2.6 Rounding Off Numbers Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. How do you decide how many digits to keep? Simple rules exist to tell you how. Prentice Hall © 2007 Chapter Two

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Prentice Hall © 2007 Chapter Two

Figure: 02-03UN13 Title: Rounding off numbers Caption: Calculators often display more digits than are justified by the precision of the data. Notes: Only use the meaningful digits in a measured or calculated quantity. Keywords: rounding numbers, precision

RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers. Prentice Hall © 2007 Chapter Two

Rounding RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures. RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures. If a calculation has several steps, it is best to round off at the end. Prentice Hall © 2007 Chapter Two

Problems Express 1,066,298 to 5 significant figures using scientific notation. Express 2/3 to 4 significant figures. Prentice Hall © 2007 Chapter Two

______________________________ Problem Determine the answer to: 6.023 x 1023 x 2.32 x 10-7 ______________________________ Calculator gives too many significant figures 1.397336 x 1017 ?? 1.40 x 1017 (3 sig. figures) Prentice Hall © 2007 Chapter Two

2.7 Problem Solving: Converting a Quantity from One Unit to Another Factor-Label Method: A quantity in one unit is converted to an equivalent quantity in a different unit by using a conversion factor that expresses the relationship between units. (Starting quantity) x (Conversion factor) = Equivalent quantity Prentice Hall © 2007 Chapter Two

Writing 1 km = 0.6214 mi as a fraction restates it in the form of a conversion factor. This and all other conversion factors are numerically equal to 1. The numerator is equal to the denominator. Multiplying by a conversion factor is equivalent to multiplying by 1 and so causes no change in value. Prentice Hall © 2007 Chapter Two

When solving a problem, the idea is to set up an equation so that all unwanted units cancel, leaving only the desired units. How many km are there in 26.22 mi (a marathon)? Prentice Hall © 2007 Chapter Two

Figure: 02-03UN20 Title: Converting concentration to mass Caption: A patient requires an injection of 0.012 g of a painkiller available as a 15 mg/mL solution. How many milliliters should be administered? Notes: Knowing the amount of painkiller in 1 mL allows us to use the concentration as a conversion factor to determine the volume of solution that would contain the desired amount. Keywords: calculations, conversions

Problem An infant weighs 9.2 lb at birth. How many kg does the infant weigh? 1kg = 2.205 lb is the conversion factor Prentice Hall © 2007 Chapter Two

Problem The directions on the bottle of cough syrup say to give 4.0 fl oz to the child, but your measuring spoon is in units of mL. How many mL should you administer? 1 L = 0.264 gal 1 gal = 4 qt 1 qt = 32 fl oz 1L = 1000 mL Prentice Hall © 2007 Chapter Two

Problem 2.15 from McMurry: One international nautical mile is defined as exactly 6076.1155 ft, and a speed of 1 knot is defined as one nautical mile per hour. What is the speed in m/sec of a boat traveling at 14.3 knots? What we know: 1 knot = 1 naut. mi/hr = 6076.1155 ft/hour 1 ft = 0.3048 m 1 hr = 60 min 1 min = 60 sec Prentice Hall © 2007 Chapter Two

Problem Solved ! Prentice Hall © 2007 Chapter Two

2.9 Measuring Temperature Temperature is commonly reported either in degrees Fahrenheit (oF) or degrees Celsius (oC). The SI unit of temperature is the Kelvin (K). 1 Kelvin, no degree, is the same size as 1 oC. 0 K is the lowest possible temperature, 0 oC = 273.15 K is the normal freezing point of water. To convert, adjust for the zero offset. Temperature in K = Temperature in oC + 273.15 Temperature in oC = Temperature in K - 273.15 Prentice Hall © 2007 Chapter Two

Freezing point of H2O Boiling point of H2O 32oF 212oF 0oC 100oC 212oF - 32oF = 180oF covers the same range of temperature as 100oC - 0oC = 100oC covers. Therefore, a Celsius degree is exactly 180/100 = 1.8 times as large as a Fahrenheit degree. The zeros on the two scales are separated by 32oF. Prentice Hall © 2007 Chapter Two

Fahrenheit, Celsius, and Kelvin temperature scales. Prentice Hall © 2007 Chapter Two

Converting between Fahrenheit and Celsius scales is similar to converting between different units of length or volume, but is a little more complex. The different size of the degree and the zero offset must both be accounted for. oF = (1.8 x oC) + 32 oC = (oF – 32)/1.8 Prentice Hall © 2007 Chapter Two

Problem Problem A patient has a temperature of 38.8oC. What is her temperature in oF? We know 2 equations: oF = (1.8 x oC) + 32 oC = (oF – 32)/1.8 Prentice Hall © 2007 Chapter Two

2.10 Energy and Heat Energy: The capacity to do work or supply heat. Energy is measured in SI units by the Joule (J); the calorie is another unit often used to measure energy. One calorie (cal) is the amount of heat necessary to raise the temperature of 1 g of water by 1°C. A kilocalorie (kcal) = 1000 cal. A Calorie, with a capital C, used by nutritionists, equals 1000 cal. An important energy conversion factor is: 1 cal = 4.184 J Prentice Hall © 2007 Chapter Two

Figure: 02-05 Title: Energy and heat Caption: The reaction of aluminum with bromine releases energy in the form of heat. When the reaction is complete, the products undergo no further change. Notes: All chemical reactions are accompanied by a change in energy, which is defined in scientific terms as the capacity to do work. Keywords: energy, heat, reaction, aluminum, bromine

Specific heat is measured in units of cal/gC Not all substances have their temperatures raised to the same extent when equal amounts of heat energy are added. One calorie raises the temperature of 1 g of water by 1°C but raises the temperature of 1 g of iron by 10°C. The amount of heat needed to raise the temperature of 1 g of a substance by 1°C is called the specific heat of the substance. Specific heat is measured in units of cal/gC Prentice Hall © 2007 Chapter Two

(Heat Change) = (Mass) x (Specific Heat) x (Temperature Change) Knowing the mass and specific heat of a substance makes it possible to calculate how much heat must be added or removed to accomplish a given temperature change. (Heat Change) = (Mass) x (Specific Heat) x (Temperature Change) H in cal; mass in g; C in cal/goC; Temp in oC Using the symbols D for change, H for heat, m for mass, C for specific heat, and T for temperature, a more compact form is: DH = mCDT Prentice Hall © 2007 Chapter Two

Problem What is the specific heat of Aluminum if it takes 161 cal to raise the temperature of 75 g of Al by 10.0oC? DH = mCDT solve for C (specific heat, cal/g.oC) Prentice Hall © 2007 Chapter Two

2.11 Density Density relates the mass of an object to its volume. Density is usually expressed in units of grams per cubic centimeter (g/cm3) for solids, and grams per milliliter (g/mL) for liquids. Mass (g) Density = Volume (mL or cm3) Prentice Hall © 2007 Chapter Two

Problem What volume in mL should you measure out if you need 16 g of ethanol which has a density of 0.79 g/mL? (You only have a graduated cylinder, no balance.) Prentice Hall © 2007 Chapter Two

Which is heavier, a ton of feathers or a ton of bricks? Which is larger? If two objects have the same mass, the one with the higher density will be smaller. Prentice Hall © 2007 Chapter Two

2. 12 Specific Gravity Specific gravity (sp gr): density of a substance divided by the density of water at the same temperature. Specific gravity is unitless. The density of water is so close to 1 g/mL that the specific gravity of a substance at normal temperature is numerically equal to the density. Density of substance (g/ml) Specific gravity = Density of water at the same temperature (g/ml) Prentice Hall © 2007 Chapter Two

The specific gravity of a liquid can be measured using an instrument called a hydrometer, which consists of a weighted bulb on the end of a calibrated glass tube. The depth to which the hydrometer sinks when placed in a fluid indicates the fluid’s specific gravity. Prentice Hall © 2007 Chapter Two

Chapter Summary Physical quantities require a number and a unit. Preferred units are either SI units or metric units. Mass, the amount of matter an object contains, is measured in kilograms (kg) or grams (g). Length is measured in meters (m). Volume is measured in cubic meters in the SI system and in liters (L) or milliliters (mL) in the metric system. Temperature is measured in Kelvin (K) in the SI system and in degrees Celsius (°C) in the metric system. Prentice Hall © 2007 Chapter Two

Chapter Summary Cont. The exactness of a measurement is indicated by using the correct number of significant figures. Significant figures in a number are all known with certainty except for the final estimated digit. Small and large quantities are usually written in scientific notation as the product of a number between 1 and 10, times a power of 10. A measurement in one unit can be converted to another unit by multiplying by a conversion factor that expresses the exact relationship between the units. Prentice Hall © 2007 Chapter Two

Chapter Summary Cont. Problems are solved by the factor-label method. Units can be multiplied and divided like numbers. Temperature measures how hot or cold an object is. Specific heat is the amount of heat necessary to raise the temperature of 1 g of a substance by 1°C. Density relates mass to volume in units of g/mL for a liquid or g/cm3 for a solid. Specific gravity is density of a substance divided by the density of water at the same temperature. Prentice Hall © 2007 Chapter Two