Conversion and Dimensional Analysis PDF

Topic 1: Measurement and Dimensional Analysis Prepared by: Michelle Joy M. Velasco, LPT, MAEd, CHRA Objectives: define measurement and dimensional analysis describe the properties and units of length, mass, volume, density, temperature, and time perform basic...

Topic 1: Measurement and Dimensional Analysis Prepared by: Michelle Joy M. Velasco, LPT, MAEd, CHRA Objectives: define measurement and dimensional analysis describe the properties and units of length, mass, volume, density, temperature, and time perform basic unit calculations and conversions in the metric and other unit systems  6 3 1 2 7 CHEMISTRY IS EVERYWHERE!! 4 9 10 5 8 Are the lines parallel? What is measurement? ☺A measurement is a quantity that has both a number and a unit. ☻Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct. What is measurement? Chemistry is about observation! Therefore, all lab experiments involve data collection. There are two types of data we can collect in any lab: qualitative and quantitative. What is the difference between them? Qualitative: Descriptive, non-numerical data (ex) color, phase, texture Quantitative: Numbers with units  HOW W E MEASURE ❖ We measure quantities using a suitable unit and measuring instrument. ❖ For e.g. to measure 1 liter of milk we can use a measuring jug.  The International System of Units In the signs shown here, the distances are listed as numbers with no units attached. Without the units, it is impossible to communicate the measurement to others. When you make a measurement, you must assign the correct units to the numerical value. ✓ All measurements depend on units that serve as reference standards. The standards of measurement used in science are those of the metric system. ✓ The International System of Units (abbreviated SI, after the French name, Le Système International d’Unités) is a revised version of the metric system.  UNITS OF MEASUREMENT BASIC UNITS IN METRIC SYSTEM Mass Kilogram Length Meter Volume Liter Time Second Temperature Kelvin Energy joule Amount of Mole substance Which five SI base units do chemists commonly use? The five SI base units commonly used by chemists are the meter, the kilogram, the kelvin, the second, and themole.  What metric units are commonly used to measure length, volume, mass, temperature and energy?  Units of Length In SI, the basic unit of length, or linear measure, is the meter (m). For very large or and very small lengths, it may be more convenient to use a unit of length that has a prefix. Common metric units of length include the centimeter, meter, and kilometer. Units of Volume ❖ The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge. This volume is the cubic meter (m)3. A more convenient unit of volume for everyday use is the liter, a non-SI unit. ❖ A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm  10 cm  10 cm = 1000 cm3 = 1 L).  Common metric units of volume include the liter, milliliter, cubic centimeter, and microliter. The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. A sugar cube has a volume of 1 cm3. 1 mL is the same as 1 cm3. A gallon of milk has about twice the volume of a 2-L bottle of soda. Units of Mass ❖ The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass. ❖ A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g. Common metric units of mass include kilogram, gram, milligram, and microgram. ❖ Weight is a force that measures the pull on a given mass by gravity. ❖ The astronaut shown on the surface of the moon weighs one sixth of what he weighs on Earth. Units of Temperature ❖ Temperature is a measure of how hot or cold an object is. ❖ Thermometers are used to measure temperature. Scientists commonly use two equivalent units of temperature, the degree Celsius and the Kelvin. ▪ On the Celsius scale, the freezing point of water is 0°C and the boiling point is 100°C. ▪ On the Kelvin scale, the freezing point of water is 273.15 kelvins (K), and the boiling point is 373.15 K. ▪ The zero point on the Kelvin scale, 0 K, or absolute zero, is equal to −273.15 °C. Units of Energy ❖ Energy is the capacity to do work or to produce heat. ❖ The joule and the calorie are common units of energy. ❖ The joule (J) is the SI unit of energy. ❖ One calorie (cal) is the quantity of heat that raises the temperature of 1g of pure water by 1°C. This house is equipped with solar panels. The solar panels convert the radiant energy from the sun into electrical energy that can be used to heat water and power appliances. Conversion Factors What happens when a measurement is multiplied by a conversion factor?  Conversion Factors – If you think about any number of everyday situations, you will realize that a quantity can usually be expressed in several different ways. – For example: 1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies – These are all expressions, or measurements, of the same amount of money.  Conversion Factors – The same thing is true of scientific quantities. – For example: 1 meter = 10 decimeters = 100 centimeters = 1000 millimeters – These are different ways to express the same length.  Conversion Factors – Whenever two measurements are equivalent, a ratio of the two measurements will equal 1, or unity. – For example, you can divide both sides of the equation 1 m = 100 cm by 1 m or by 100 cm. 1 m = 100 cm = 1 or 1 m = 100 cm = 1 1m 1m 100 cm 100 cm  Conversion Factors – The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors. – A conversion factor is a ratio of equivalent measurements. 1 m = 100 cm = 1 or 1 m = 100 cm = 1 1m 1m 100 cm 100 cm conversion factors  Conversion Factors – The measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom). – The conversion factors shown below are read “one hundred centimeters per meter” and “one meter per hundred centimeters.” 1 m = 100 cm = 1 or 1 m = 100 cm = 1 1m 1m 100 cm 100 cm conversion factors  Conversion Factors 1 meter 100 centimeters Smaller number 1 m Larger unit Larger number 100 cm Smaller unit The figure above illustrates another way to look at the relationships in a conversion factor. – The smaller number is part of the measurement with the larger unit. – The larger number is part of the measurement with the smaller unit.  Conversion Factors When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same. For example, even though the numbers in the measurements 1 g and 10 dg (decigrams) differ, both measurements represent the same mass.  Conversion Factors – In addition, conversion factors within a system of measurement are defined quantities or exact quantities. Therefore, they have an unlimited number of significant figures and do not affect the rounding of a calculated answer. Conversion Factors – Here are some additional examples of pairs of conversion factors written from equivalent measurements. – The relationship between grams and kilograms is 1000 g = 1 kg. The conversion factors are 1000 g and 1 kg 1 kg 1000 g  Conversion Factors The figure at right shows a scale that can be used to measure mass in grams or kilograms. – If you read the scale in terms of grams, you can convert the mass to kilograms by multiplying by the conversion factor 1 kg/1000 g.  Conversion Factors The relationship between nanometers and meters is given by the equation 109 nm = 1 m. – The possible conversion factors are 109 nm 1m and 1m 109 nm  Conversion Factors – Common volumetric units used in chemistry include the liter and the microliter. – The relationship 1 L = 106μL yields the following conversion factors: 1L 106 μL and 106 μL 1L English System The English system of measurement is a system of measure that uses body parts and familiar objects. An example is ground distances that uses human foot for short distance and paces for longer distances. This system allows discrepancies between measurements obtained by different individuals. A standard was eventually set to ensure that all measurements represented the same amount for everyone. Some conversion factors for English System Length Weight Capacity 1inch (in)=2.54 centimeters 1 pound (lb)= 16 ounces (oz) 1 cup (c)=16 tablespoons (cm) 1 foot (ft) = 12 inches (in) 1 ton = 2000 lbs 1 cup = 8 fluid ounces 1 yard (yd) = 3 feet 1 lb=0.4536 kilograms 1 pint(pt)=2 cups 1 mile (mi) = 5280 feet 1 quart (qt)= 2 pints 1 mile = 1760 yards 1 gallon(gal)= 4 quarts Factor Label Method A conversion ratio or unit factor is a ratio equal to one. This conversion ratio is based on the concept of equivalent values. It can be used for conversion within English system and metric system or between the systems. In the example below one foot is substituted for its equivalent measure of 12 inches. (12 inches)/(12 inches)=(1 foot)/(12 inches) conversion ratios always equal to 1 This is the usual way of converting units that most students are comfortable with or just using the conversion factor directly. Factor Label Method In the factor label technique, the conversion factors are obtained from the unit equivalents and the two conversion factors are reciprocals of each other, and both of them are equal to one. Thus, when multiplying a measurement by a conversion factor only the units are changing and not the value. Unit Equivalents Conversion Factors Conversion factors (longer to shorter (shorter to longer units units of measurement) of measurement) 1 inch (in)= 2.54 2.54 𝑐𝑚 1 𝑖𝑛 centimeters (cm) 1 𝑖𝑛 2.54 𝑐𝑚 1 foot(ft) = 12 inches(in) 12 𝑖𝑛 1 𝑓𝑡 1 𝑓𝑡 12 𝑖𝑛 1 yard(yd) =3 feet (ft) 3 𝑓𝑡 1 𝑦𝑑 1 𝑦𝑑 3 𝑓𝑡 What is the relationship between the two components of a conversion factor? What is the relationship between the two components of a conversion factor? The two components of a conversion factor are equivalent measurements with different units. They are two ways of expressing the same quantity. Dimensional Analysis What kinds of problems can you solve using dimensional analysis? Dimensional Analysis – Many problems in chemistry are conveniently solved using dimensional analysis, rather than algebra. – Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements. Sample Problem below explains dimensional analysis by using it to solve an everyday situation. Sample Problem Using Dimensional Analysis How many seconds are in a workday that lasts exactly eight hours?  1 Analyze List the knowns and the unknown. To convert time in hours to time in seconds, you’ll need two conversion factors. First you must convert hours to minutes: h → min. Then you must convert minutes to seconds: min → s. Identify the proper conversion factors based on the relationships 1 h = 60 min and 1 min = 60 s. KNOWNS UNKNOWN time worked = 8 h seconds worked = ? s 1 hour = 60 min 1 minute = 60 s 2 Calculate Solve for the unknown. The first conversion factor is based on 1 h = 60 min. The unit hours must be in the denominator so that the known unit will cancel. 60 min 1h 2 Calculate Solve for the unknown. The second conversion factor is based on 1 min = 60 s. The unit minutes must be in the denominator so that the desired units (seconds) will be in your answer. 60 s 1 min  2 Calculate Solve for the unknown. Multiply the time worked by the conversion factors. 60 min 60 s 8h x x = 28,000 s = 2.8800 x 104 s 1h 1 min 3 Evaluate Does the result make sense? The answer has the desired unit (s). Since the second is a small unit of time, you should expect a large number of seconds in 8 hours. The answer is exact since the given measurement and each of the conversion factors is exact. Sample Problem Using Dimensional Analysis The directions for an experiment ask each student to measure 1.84 g of copper (Cu) wire. The only copper wire available is a spool with a mass of 50.0 g. How many students can do the experiment before the copper runs out? 1 Analyze List the knowns and the unknown. From the known mass of copper, use the appropriate conversion factor to calculate the number of students who can do the experiment. The desired conversion is mass of copper → number of students. KNOWNS mass of copper available = 50.0 g Cu Each student needs 1.84 grams of copper. UNKNOWN number of students = ? 2 Calculate Solve for the unknown. The experiment calls for 1.84 grams of copper per student. Based on this relationship, you can write two conversion factors. 1.84 g Cu 1 student and 1 student 1.84 g Cu  2 Calculate Solve for the unknown. Because the desired unit for the answer is students, use the second conversion factor. Multiply the mass of copper by the conversion factor. 1 student = 27.174 students = 27 students 50.0 g Cu x 1.84 g Cu Note that because students cannot be fractional, the answer is rounded down to a whole number. 3 Evaluate Does the result make sense? The unit of the answer (students) is the one desired. You can make an approximate calculation using the following conversion factor. 1 student 2 g Cu Multiplying the above conversion factor by 50 g Cu gives the approximate answer of 25 students, which is close to the calculated answer. Using Density as a Conversion Factor What is the volume of a pure silver coin that has a mass of 14 g? The density of silver (Ag) is 10.5 g/cm3. 1 Analyze List the knowns and the unknown. You need to convert the mass of the coin into a corresponding volume. The density gives you the following relationship between volume and mass: 1 cm3 Ag = 10.5 g Ag. Multiply the given mass by the proper conversion factor to yield an answer in cm3. KNOWNS mass = 14 g density of silver = 10.5 g/cm3 UNKNOWN volume of a coin = ? cm3 2 Calculate Solve for the unknown. Use the relationship 1 cm3 Ag = 10.5 g Ag to write the correct conversion factor. 1 cm3 Ag 10.5 g Ag  Sample Problem 3.12 2 Calculate Solve for the unknown. Multiply the mass of the coin by the conversion factor. 1 cm 3 Ag 14 g Ag x = 1.3 cm3 Ag 10.5 g Ag 3 Evaluate Does the result make sense? Because a mass of 10.5 g of silver has a volume of 1 cm3, it makes sense that 14.0 g of silver should have a volume slightly larger than 1 cm3. The answer has two significant figures because the given mass has two significant figures. Dimensional Analysis Simple Unit Conversions – In chemistry, as in everyday life, you often need to express a measurement in a unit different from the one given or measured initially. Dimensional analysis is a powerful tool for solving conversion problems in which a measurement with one unit is changed to an equivalent measurement with another unit. Dimensional Analysis Derived Units What kind of problems can you solve using dimensional analysis? What kind of problems can you solve using dimensional analysis? Problems that require the conversion of a measurement from one unit to another can be solved using dimensional analysis. Key Concepts When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same. Dimensional analysis is a powerful tool for solving conversion problems in which a measurement with one unit is changed to an equivalent measurement with another unit. – conversion factor: a ratio of equivalent measurements used to convert a quantity from one unit to another – dimensional analysis: a technique of problem-solving that uses the units that are part of a measurement to help solve the problem Why is measurement important?  References https://www.keyence.com/ss/products/measure/measurement_librar y/basic/measurement/ https://www.brainkart.com/article/Methods-of-Measurements_5808/ https://en.wikipedia.org/wiki/SI_base_unit https://manoa.hawaii.edu/exploringourfluidearth/physical/world- ocean/map-distortion/practices-science-precision-vs-accuracy https://courses.lumenlearning.com/boundless- statistics/chapter/measurement-error/ https://www.isobudgets.com/estimate-uncertainty-in-measurement- for-chemistry-laboratories/ https://courses.lumenlearning.com/trident-boundless- chemistry/chapter/dimensional-analysis/