Chemistry 11 Measuring and Recording Scientific Data Measurement tools Graphing Scientific notation Unit conversions Density Significant figures Name: Block:
Measuring and Recording Significant Data SI Units The SI system ( ) is the modern system of measurement and the dominant system of international commerce and trade. The abbreviation SI comes from the French, Système international d unités. The SI system was developed in 1960 and is maintained by the International Bureau of Weights and Measures (BIPM) in France. Table 1.3.1 The Fundamental SI Units Name Unit Symbol Quantity metre m length kilogram kg mass second s time ampere A electric current kelvin K temperature candela cd luminous intensity mole mol amount of substance SI units are given in Table 1.3.1. Unit symbols are always letters unless the unit is named after a person. The one exception to this rule is L for litres. The full names of units are always written in lower case with the exception of degrees Celsius. Unit symbols should never be pluralized. Symbols should only be followed by a period at the end of a sentence. In general, the term replaces the term weight. The symbol cc should not be used in place of ml or cm 3. For values less than 1, in front of the decimal point (e.g., 0.54 g not.54 g). Use fractions rather than common fractions (0.25 rather than ¼). Abbreviations such as sec, cc, or mps are avoided and only standard unit symbols such as for second, for cubic centimeter, and for metre per second should be used. There is a between the numerical value and unit symbol (4.6 g). in preference to words for units attached to numbers (e.g., 5.0 g/mol rather than 5.0 grams/mole). Note that 5.0 grams per mole is incorrect but five grams per mole is correct. A specific temperature and a temperature change both have units of degrees Celsius ( C). Quick Check Locate the SI error(s) in each of the following statements and correct them. 1. Ralph bought 6 kilos of potato salad. 2. The thickness of the oxide coating on the metal was 1/2 c.m. 3. The weight of 1 Ml of water is exactly 1 gms at 4 c. 4. My teacher bought 9.0 litres of gasoline for her 883 c.c. motorcycle. 5. Rama s temperature increased by.9 C.
Accuracy and Precision The Quality of Measurements Measured values like those listed in the Warm Up at the beginning of this section are determined using a variety of different measuring devices. There are devices designed to measure all sorts of different quantities. The collection pictured in Figure 1.3.1 measures,, and. In addition, there are a variety of (exactnesses) associated with different devices. The micrometer (also called a caliper) is more precise than the while the and pipette are more precise than the. Despite the fact that some measuring devices are more precise than others, it is impossible to design a measuring device that gives perfectly exact measurements. Figure 1.3.1 A selection of measuring devices with differing levels of precision All measuring devices have some degree of associated with them. Figure 1.3.2 This kilogram mass was made in the 1880s and accepted as the international prototype of the kilogram in 1889. ( BIPM Reproduced with permission) The 1-kg mass kept in a helium-filled bell jar at the BIPM in Sèvres, France, is the (Figure 1.3.2). All other masses are measured to this and therefore have some degree of associated. Accuracy Accurate measurements depend on careful design and calibration to ensure a measuring device is in proper working order. The term precision can actually have two different meanings. Precision refers to the of a measurement (or the among several measurements of the same quantity). - or - Precision refers to the of a measurement. This relates to uncertainty: the the uncertainty of a measurement, the the precision.
Quick Check Volumetric devices measure liquids with a wide variety of precisions. 1. Which of these is likely the most precise? 2. Which is likely the least precise? 3. Is the most precise device necessarily the most accurate? 4. Discuss your answers. Types of Errors No measurement can be completely precise. In fact, all measurements must have associated with them, so it must be true that every measurement involves some degree of. A group of measurements may tend to consistently show error in the same direction. Such a situation is called error. When a group of errors occurs equally in high and low directions, it is called error. Quick Check Four groups of Earth Science students use their global positioning systems (GPS) to do some geocaching. The diagrams below show the students results relative to where the actual caches were located. 1. Comment on the precision of the students in each of the groups. (In this case, we are using the reproducibility definition of precision.) 2. What about the accuracy of each group? 3. Which groups were making systematic errors? 4. Which groups made errors that were more random?
Uncertainty Every measurement has some degree of uncertainty associated with it. The of a measuring device depends on its. The most precise measuring devices have the uncertainties. The most common way to report the uncertainty of a measuring device is as a range uncertainty. The range uncertainty is an range of values within which the true value of a measurement falls. This is commonly presented using the notation (shown as +/ ). 60 50 Figure 1.3.2 A meniscus in a graduated cylinder An acceptable range uncertainty is usually considered plus or minus to the smallest division marked on a measuring scale. For the graduated cylinder in Figure 1.3.2, the smallest division marked on its scale is One-half of this division is ml.. herefore, reading the volume from the bottom of the meniscus would give a value of 56.3 ml +/ 0.5 ml. Because it is possible to go down to one-tenth of the smallest division on the scale as a range, the volume could also be reported as 56.3 ml +/ 0.1 ml. Notice that the unit is given twice, before and after the +/ symbol. This is the correct SI convention. When two uncertain values with range uncertainties are added or subtracted, their. Quick Check 1. Sum the following values: 2. Subtract the following values: 27.6 ml +/ 0.2 ml 19.8 ml +/ 0.2 ml + 14.8 ml +/ 0.2 ml 7.2 ml +/ 0.2 ml uncertainty refers to exactly how much higher or lower a measured value is than an accepted value. In such a case, an accepted value is the value considered to be the Constants such as the speed of light or the boiling point of water at sea level are accepted values. For absolute uncertainties, a sign should be applied to indicate whether the measured value is above or below the accepted value. Another common way to indicate an error of measurement is as a percentage of what the value should be. percentage error =
Quick Check A student weighs a Canadian penny and finds the mass is 2.57 g. Data from the Canadian Mint indicates a penny from that year should weigh 2.46 g. 1. What is the absolute uncertainty of the penny s mass? 2. What is the percentage error of the penny s mass? 3. Suggest a reasonable source of the error. Review Questions 1. Determine the errors and correct them according to the SI system: (a) The package contained 750 Gm of linguini. (b) The car accelerated to 90 km per hour in 10 sec. (c) The recipe called for 1 ML of vanilla and 250 cc of milk. (d) Jordan put 250 gms of mushrooms on his 12 inch pizza. 2. A zinc slug comes from a science supply company with a stated mass of 5.000 g. A student weighs the slug three times, collecting the following values: 4.891 g, 4.901 g, and 4.890 g. Are the student s values accurate? Are they precise (consider both meanings)? 3. A student doing experimental work finds the density of a liquid to be 0.1679 g/cm 3. The known density of the liquid is 0.1733 g/cm 3. What is the absolute error of the student s work? What is the percent error? 5. In an experiment to determine the density of a liquid, a maximum error of 5.00% is permitted. If the true density is 1.44 g/cm 3, what are the maximum and minimum values within which a student s answer may fall into the acceptable range? 6. What is the mass, including uncertainty, arrived at as the result of summing 45.04 g +/ 0.03 g, and 39.04 g +/ 0.02 g? 7. What is the smallest number that could result from subtracting 22 m +/ 2 m from 38 m +/ 3 m? 8. The dimensions of a rectangle are measured to be 19.9 cm +/ 0.1 cm and 2.4 +/ 0.1 cm. What is the area of the rectangle, including the range uncertainty? 9. Read each of the following devices, including a reasonable range uncertainty: 4. Two students weigh the same object with a known mass of 0.68 g. One student obtains a mass of 0.72 g, while the other gets a mass of 0.64 g. How do their percent errors compare? How do their absolute errors compare? (a) (b)
Reading & Recording Measurements
Meniscus:
GRADUATED CYLINDERS Graduated cylinders are used to measure the volume of liquids. Measuring liquids in gradu-ated cylinders can be tricky because the liquid surface is curved. ml 50 40 close - up view 40 graduates This curved surfaces is called the. A meniscus forms because the liquid molecules are more strongly attracted to the container than to each other. To properly measure the volume of a liquid in a graduated cylinder you must be at eye-level and read the bottom point of the meniscus. 30 20 10 graduated cylinder meniscus 30 subgraduates 1. Measure the amount of liquid in the graduated cylinder 2. Record the measurement below. Remember to include ml in your answer. ml 5 ml ml ml ml 50 100 125 500 4 40 80 100 400 3 30 60 75 300 2 20 40 50 200 1 10 20 25 100 ml ml ml ml ml 25 5 50 5 300 20 4 40 4 15 3 30 3 200 10 2 20 2 100 5 1 10 1
A (also buret) is a laboratory equipment used in analytical chemistry for the dispensing of variable amount of a chemical solution and measuring that amount at the same time. Most often used for in acid base chemistry. It is a long, graduated glass tube having a stopcock at its lower end and a tapered capillary tube at the stopcock's outlet.
REVIEW:
Pictorial Representation of Data Graphing Often in scientific investigations, we are interested in measuring how the value of some property as we vary something that affects it. We call the value that responds to the variation the variable, while the other value is the i variable. For example, we might want to measure the extension of a spring as we attach different masses to it. In this case, the would be the dependent variable, and the would be the independent variable. Notice that the amount of extension depends on the mass loaded and not the other way around. The variable is nearly always The series of paired measurements collected during such an investigation is data. It is usually arranged in a Tables of data should indicate the of measurement at the top of each column. The information in such a table becomes even more useful if it is presented in the form of a. The independent data is plotted on the x-axis. A graph reveals many data points not listed in a data table. Once a graph is drawn, it can be used to find a mathematical (equation) that indicates how the variable quantities depend on each other. The first step to determining the relationship is to calculate the m for the line of best fit. First, the constant is been determined by finding the change in y over the change in x ( y/ x or the rise over the run ). Then substitution of the y and x variable names and the calculated value for m, including its units, into the general equation The result will be an equation that describes the relationship represented by our data. is the general form for the equation of a straight line relationship where: m represents the slope m = In scientific relationships, the slope and represents the that relates two variables. For this reason, it is sometimes represented by a. The three most common types of graphic relationships are shown below in Figure 1.2.2. Direct: y = Kx Inverse: y = K/x Exponential: y = Kx n (y and x increase in direct proportion) (as x increases, y decreases) (as x increases, y increases more quickly) Figure 1.2.2 Three common types of graphic relationships
Sample Problem Determination of a Relationship from Data Find the relationship for the graphed data below: What to Think about 1. Determine the constant of proportionality (the slope) for the straight line. To do this, select two points on the line of best fit. These should be points whose values are easy to determine on both axes. Do not use data points to determine the constant. Determine the change in y ( y) and the change in x ( x) including the units. The constant is y/ x. How to Do It 2. The relationship is determined by subbing in the variable names and the constant into the general equation, y = Kx + b. Often, a straight line graph passes through the origin, in which case, y = Kx. Practice Problem Determination of a Relationship from Data Examine the following graphs. What type of relationship does each represent? (a) (b) (c)
Activity: Graphing Relationships Question Can you produce a graph given a set of experimental data? Background A beaker full of water is placed on a hotplate and heated over a period of time. The temperature is recorded at regular intervals. The following data was collected. Temperature ( o C) Time (min) 22 0 30 2 38 4 46 6 54 8 62 10 70 12 Procedure 1. Use the grid above to plot a graph of temperature against time. (Time goes on the x-axis.) Results and Discussion 1. What type of relationship was studied during this investigation? 2. What is the constant (be sure to include the units)? 3. What temperature was reached at 5 minutes? 4. Use the graph to determine the relationship between temperature and time. 5. How long would it take the temperature to reach 80 C? 6. What does the y-intercept represent? 7. Give a source of error that might cause your graph to vary from that expected.
Use the grid provided to plot graphs of mass against volume for a series of metal pieces with the given volumes. Plot all three graphs on the same set of axes with the independent variable (volume in this case) on the x-axis. Use a different colour for each graph. Volume (ml) Copper (g) Aluminum (g) Platinum (g) 2.0 17.4 5.4 42.9 8.0 71.7 21.6 171.6 12.0 107.5 32.4 257.4 15.0 134.4 40.5 321.8 19.0 170.2 51.3 407.6 (a) Determine the constant for each metal. (b) The constant represents each metal s density. Which metal is most dense?
Using Scientific Notation Because it deals with atoms, and they are so incredibly small, the study of chemistry is notorious for using very large and very tiny numbers. For example, if you determine the total number of atoms in a sample of matter, the value will be very large. If, on the other hand, you determine an atom s diameter or the mass of an atom, the value will be extremely small. The method of reporting an ordinary, expanded number in scientific notation is very handy for both of these things. refers to the method of representing numbers in form. Exponential numbers have two parts. Consider the following example: 24 500 becomes 2.45 10 4 in scientific notation Convention states that the first portion of a value in scientific notation should always be expressed as a number. This portion is called the mantissa or the. The second portion is the raised to some power. It is called the ordinate or the portion. mantissa 2.45 10 4 and 2.45 10_ 4! ordinate A exponent in the ordinate indicates a in scientific notation, while a exponent indicates a In fact the exponent indicates the number of 10s that must be multiplied together to arrive at the number represented by the scientific notation. If the exponents are negative, the exponent indicates the number of tenths that must be multiplied together to arrive at the number. In other words, the exponent indicates the number of in the mantissa must be moved to correctly arrive at the notation (also called standard notation) version of the number. A exponent indicates the number of places the decimal must be moved to the, while a xponent indicates the number of places the decimal must be moved to the. Quick Check 1. Change the following numbers from scientific notation to expanded notation. (a) 2.75 10 3 = (b) 5.143 10 2 = 2. Change the following numbers from expanded notation to scientific notation. (a) 69 547 = (b) 0.001 68 =
Multiplication and Division in Scientific Notation To two numbers in scientific notation, we multiply the and state their product multiplied by 10, raised to a power that is the. (A 10 a ) (B 10 b ) = (A B) 10 (a + b) To divide two numbers in scientific notation, we divide one mantissa by the other and state their quotient multiplied by 10, raised to a power that is the difference between the exponents. (A 10 a ) (B 10 b ) = (A B) 10 (a b) Sample Problems Multiplication and Division Using Scientific Notation Solve the following problems, expressing the answer in scientific notation. 1. (2.5 10 3 ) x (3.2 10 6 ) = 2. (9.4 10 4 ) (10 6 ) = What to Think about Question 1 1. Find the product of the mantissas. How to Do It 2. Raise 10 to the sum of the exponents to determine the ordinate. 3. State the answer as the product of the new mantissa and ordinate. Question 2 1. Find the quotient of the mantissas. When no mantissa is shown, it is assumed that the mantissa is 1. 2. Raise 10 to the difference of the exponents to determine the ordinate. 3. State the answer as the product of the mantissa and ordinate. Practice Problems Multiplication and Division Using Scientific Notation Solve the following problems, expressing the answer in scientific notation, without using a calculator. Repeat the questions using a calculator and compare your answers. Compare your method of solving with a calculator with that of another student. 1. (4 10 3 ) (2 10 4 ) = 4. 10 9 (5.0 10 6 ) = 2. (9.9 10 5 ) (3.3 10 3 ) = 5. [(4.5 10 12 ) (1.5 10 4 )] (2.5 10 6 ) = 3. [(3.1 10 4 ) (6.0 10 7 )] (2.0 10 5 ) =
Addition and Subtraction in Scientific Notation Remember that a number in proper scientific notation will always have a mantissa between and. Sometimes it becomes necessary to a decimal in order to express a number in proper scientific notation. The number of places shifted by the decimal is indicated by an equivalent change in the value of the exponent. If the decimal is shifted, the exponent becomes ; shifting the decimal to the causes the exponent to become. Another way to remember this is if the mantissa becomes smaller following a shift, the exponent becomes larger. Consequently, if the exponent becomes larger, the mantissa becomes smaller. Consider AB.C 10 x : if the decimal is shifted to change the value of the mantissa by 10 n times, the value of x changes n times. For example, A number such as 18 235.0 10 2 (1 823 500 in standard notation) requires the decimal to be places to the to give a mantissa between 1 and 10, that is 1.823 50. A shift places, means the exponent in the ordinate becomes (from 10 2 to 10 6 ). The correct way to express 18 235.0 10 2 in scientific notation is 1.823 50 10 6. Notice the new mantissa is 10 4 smaller, so the exponent becomes 4 numbers larger. Quick Check Express each of the given values in proper scientific notation in the second column. Now write each of the given values from the first column in expanded form in the third column. Then write each of your answers from the second column in expanded form. How do the expanded answers compare? Given Value Proper Notation Expanded Form Expanded Answer 1. 6 014.51 10 2 2. 0.001 6 10 7 3. 38 325.3 10 6 4. 0.4196 10 2 When adding or subtracting numbers in scientific notation, it is important to realize that we add or subtract only the mantissa. Do not add or subtract the exponents! Shift the decimal to obtain the same value for the exponent in the ordinate of both numbers to be added or subtracted. Then simply sum or take the difference of the mantissas. Convert back to proper scientific notation when finished.
Sample Problems Addition and Subtraction in Scientific Notation Solve the following problems, expressing the answer in proper scientific notation. 1. (5.19 10 3 ) (3.14 10 2 ) = 2. (2.17 10 3 ) + (6.40 10 5 ) = What to Think about Question 1 1. Begin by shifting the decimal of one of the numbers and changing the exponent so that both numbers share the same exponent. For consistency, adjust one of the numbers so that both numbers have the larger of the two ordinates. The goal is for both mantissas to be multiplied by 10 3. This means the exponent in the second number should be increased by one. Increasing the exponent requires the decimal to shift to the left (so the mantissa becomes smaller). 2. Once both ordinates are the same, the mantissas are simply subtracted. Question 1 Another Approach 1. It is interesting to note that we could have altered the first number instead. In that case, 5.19 10 3 would have become 51.9 10 2. 2. In this case, the difference results in a number that is not in proper scientific notation as the mantissa is greater than 10. 3. Consequently, a further step is needed to convert the answer back to proper scientific notation. Shifting the decimal one place to the left (mantissa becomes smaller) requires an increase of 1 to the exponent. How to Do It Question 2 1. As with differences, begin by shifting the decimal of one of the numbers and changing the exponent so both numbers share the same ordinate. The larger ordinate in this case is 10 3. 2. Increasing the exponent in the second number from 5 to 3 requires the decimal to be shifted two to the left (make the mantissa smaller). 3. Once the exponents agree, the mantissas are simply summed. 4. The alternative approach involves one extra step, but gives the same answer.
Practice Problems Addition and Subtraction in Scientific Notation Solve the following problems, expressing the answer in scientific notation, without using a calculator. Repeat the questions using a calculator and compare your answers. Compare your use of the exponential function on the calculator with that of a partner. 1. 8.068 10 8 2. 6.228 10 4 3. 49.001 10 1 4.14 10 7 +4.602 10 3 + 10 1 Scientific Notation and Exponents Occasionally a number in scientific notation will be raised to some power. When such a case arises, it s important to remember when one exponent is raised to the power of another, the by one another. Consider a problem like (10 3 ) 2. This is really just (10 10 10) 2 or (10 10 10 10 10 10). So we see this is the same as 10 (3 x 2) or 10 6. (A 10 a ) b = A b 10 (a x b) Topic Review: Solve the following problems, expressing the answer in scientific notation, without the use of a calculator. Repeat the problems with a calculator and compare your answers. 1. (10 3 ) 5 2. (2 10 2 ) 3 3. (5 10 4 ) 2 4. (3 10 5 ) 2 (2 10 4 ) 2 5. Convert the following numbers from scientific notation to expanded notation and vice versa (be sure the scientific notation is expressed correctly). Scientific Notation Expanded Notation 3.08 10 4 960 4.75 10 3 0.000 484 0.0062 10 5 6. Give the product or quotient of each of the following problems (express all answers in proper form scientific notation). Do not use a calculator. (a) (8.0 10 3 ) x (1.5 10 6 ) = (b) (1.5 10 4 ) (2.0 10 2 ) = (c) (3.5 10 2 ) (6.0 10 5 ) = (d) (2.6 10 7 ) (6.5 10 4 ) =
7. Give the product or quotient of each of the following problems (express all answers in proper form scientific notation). Do not use a calculator. (a) (3.5 10 4 ) (3.0 10 5 ) = (b) (7.0 10 6 ) (1.75 10 2 ) = (c) (2.5 10 3 ) (8.5 10 5 ) = (d) (2.6 10 5 ) (6.5 10 2 ) = 8. Solve the following problems, expressing the answer in scientific notation, without using a calculator. Repeat the questions using a calculator and compare your answers. (a) 4.034 10 5 (b) 3.114 10 6 (c) 26.022 10 2 2.12 10 4 +2.301 10 5 + 7.04 10 1 9. Solve the following problems, expressing the answer in scientific notation, without using a calculator. Repeat the questions using a calculator and compare your answers. (a) 2.115 10 8 (b) 9.332 10 3 (c) 68.166 10 2 1.11 10 7 +6.903 10 4 + 10 1 10. Solve each of the following problems without a calculator. Express your answer in correct form scientific notation. Repeat the questions using a calculator and compare. (a) (10 4 ) 3 (b) (4 10 5 ) 3 (c) (7 10 9 ) 2 d. (10 2 ) 2 (2 10) 3 11. Solve each of the following problems without a calculator. Express your answer in correct form scientific notation. Repeat the questions using a calculator and compare. (a) (6.4 10 6 + 2.0 10 7 ) (2 10 6 + 3.1 10 7 ) (b) 3.4 10 17 1.5 10 4 1.5 10 4 (c) (2 10 3 ) 3 [(6.84 10 3 ) (3.42 10 3 )] (d) (3 102 ) 3 + (4 10 3 ) 2 1 10 4
Unit Converstions
Significant Figures When determining the correct value indicated by a measuring device, you should always record all of the significant figures (sometimes called sig figs ). The number of significant figures in a measurement includes all of the certain figures plus the first uncertain (estimated) figure. Figure 1.3.3 Scale on a measuring device Say the scale on a measuring device reads as shown in Figure 1.3.3. (Note that scale may refer to the numerical increments on any measuring device.) The measured value has three certain figures (1, 5, and 6) and one uncertain figure (5 or is it 4 or 6?). Hence the measurement has four significant figures. The following rules can be applied to determine how many figures are significant in any measurement. Counting Significant Figures in a Measured Value 1. All non-zero digits are significant. 2. All zeros between non-zero digits are significant. Such zeros may be called sandwiched or captive zeros. 3. Leading zeros (zeros to the left of a non-zero digit) are never significant. 4. Trailing zeros (zeros to the right of a non-zero digit) are only significant if there is a decimal in the number. Another way to determine the number of significant figures in a measured value is to simply express the number in scientific notation and count the digits. This method nicely eliminates the non-significant leading zeros. However, it is only successful if you recognize when to include the trailing (right side) zeros. Remember that trailing zeros are only significant if there is a decimal in the number. If the trailing zeros are significant, they need to be included when the number is written in scientific notation. Edvantage Interactive 2011 Chapter 1 Skills and Processes of Chemistry 37