Elements are pure substances that cannot be broken down by chemical means.

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1 CHEM Week (Measurement, Calculation) Page of Chemistry is the study of matter. Matter is made up of atoms. The periodic table currently lists different elements or atoms. Elements are pure substances that cannot be broken down by chemical means. Atoms consist of protons and neutrons held tightly in the nucleus. The nucleus is surrounded by a diffuse cloud of electrons. Your periodic table lists element names, element symbols, atomic numbers, molar masses, and electron configuration. Chemistry is known as the central science. Other branches of chemistry include biochemistry (the chemistry of life), geology (the chemistry of soil and rocks), organic chemistry (the chemistry of carbon containing compounds), inorganic chemistry (the chemistry of compounds that do not contain carbon), analytical chemistry (quantitative chemistry), and physical chemistry (the study of natural laws). Science uses the scientific method to systematically investigate nature. The steps of the scientific method include: Planning an experiment to evaluate nature

2 CHEM Week (Measurement, Calculation) Page of Observing the experiment by recording data Explaining the data by forming a hypothesis. A hypothesis is a tentative proposal that explains the results of one or more experiments. A hypothesis can be tested by new experiments. After much experimental testing, a hypothesis may be confirmed, revised, or rejected depending on whether the hypothesis correctly predicted the findings of the new experiments. A natural law is a confirmed hypothesis that has a mathematical relationship to describe the results. A natural law can never be proven to be correct by the scientific method. Instead, natural laws explain observations. An example of a natural law is the ideal gas law. Real gases do deviate from the ideal gas law predictions. Gas laws that make better predictions exist, but they are more complicated to use. The ideal gas law is helpful because it is easier to use and can make reasonable predictions. A scientific theory is a confirmed hypothesis, but unlike a natural law, it does not have a mathematical relationship that describes the results. A scientific theory can never be proven to be correct by the scientific method. Instead, useful theories explain many observations, and they do not have scientific data that overwhelms the usefulness of the theory. An example of a theory is the theory of evolution. Scientific Measurements A measurement is a number with a unit and an uncertainty. Measurements are made all the time while cooking (a cup of sugar), driving (ten gallons of gasoline, 5 miles per hour), working (filling orders per hour), and waking (at :5 a.m.). Using the example measurement of cup from cooking, the number is. The unit is cups. Without additional information about the measuring device, the uncertainty is presumed, conservatively, to be ± cup, which is ± in the last significant digit. The actual measurement is expected to be between cups ( cup minus cup) and cups ( cup plus one cup). Taking a second example from waking up, the number is :5. The units in this measurement are hours (), minutes (5), and a.m. (morning). Without additional information about the measurement, the uncertainty is presumed to be ± minute, which is ± in the last significant digit. The actual time of waking is expected to be between : a.m. (:5 a.m. minus minute) and : a.m (:5 plus one minute). Significant Digits The digits in a correctly recorded measurement are called significant digits. There are rules for determining the number of significant digits in improperly recorded measurements. Rules for counting significant digits in properly recorded measurements will be discussed later.

3 CHEM Week (Measurement, Calculation) Page of Accuracy and Precision Accuracy is the agreement of measurements with the true value. To determine the accuracy of one or more measurements, we need to know the true or accepted value. Precision is the agreement of measurements with each other. Measurements that are closer together are more precise. A dartboard can be a helpful example to describe accuracy and precision. Darts that are accurate and always hit the true value (the bull s eye) are both accurate and precise. If the average of the measurements hits the bull s eye, the measurements are accurate on the average, yet imprecise because the individual darts are far from each other. Darts that consistently hit the wrong target are inaccurate and precise. Darts that inconsistently hit the wrong target are inaccurate and imprecise. Accurate & Precise Accurate & Imprecise Inaccurate & Precise Inaccurate & Imprecise Common Units and Measurements Below are some common units that are used for scientific measurements. Table. Common units, symbols, meaning, and an example measurement Unit Symbol Meaning Example Measurement mass g grams. g chocolate (measure with a balance) length m meters.5 m tall (measure with a ruler) volume L liters. L of soda (measure with a bottle) time s seconds 5.5 s to race (measure with a watch) Mass is a measure of the amount of matter (atoms) in a substance. Mass refers to a different measurement than weight. Weight is affected by the amount of gravity, while mass is not. Mass Measurements, Uncertainty and Counting Significant Digits Additional information about measurement uncertainty comes from the equipment used to make the measurement. Using Table above, the mass measurement,. g of chocolate is known to 5 significant digits. Count each of the digits (,,,, and ) in the number as one significant digit each for a total of 5 significant digits. The balance measures to the nearest

4 CHEM Week (Measurement, Calculation) Page of thousandth of a gram ±. g, so that would be the expected uncertainty of the balance without being able to see the readout of the balance. The next three figures are balances from a lab. To count significant digits in a properly recorded number, start the counting with the first nonzero digit from the left of the number. With the exception of a zero to the left of the decimal point for numbers between zero up and one, all digits from a digital readout are significant. Lead zeroes are not significant. The balance on the left has five significant digits (,,,, and ), and the balance on the right has three significant digits in the measurement (,, and ). Figure. Electronic balance shows five significant digits in a measurement Figure. Electronic balance shows three significant digits in a measurement The balance in Figure has been used to make a mass measurement of a small sliver of chocolate. The mass,.9 g, has one significant digit (9). Start counting from the left with the first nonzero number ( not significant, not significant, not significant, 9 significant). Figure. Electronic balance shows one significant digit in a measurement Volume Measurements, Equipment, Precision, Uncertainty, Measurement, Significant Digits, and Accuracy Table lists some laboratory glassware along with the precision, uncertainty, a typical measurement, the significant digits in the measurement and the accuracy. In order to determine accuracy, the true or correct value would be needed. The correct value can often be determined by measuring a standard. If a suitable standard is not available, then the accuracy of equipment cannot be determined.

5 CHEM Week (Measurement, Calculation) Page 5 of Rules for Counting Significant Digits There are three rules for counting significant digits. Always apply rule, then apply rule or as needed. ) To count significant digits in a properly recorded number, start the counting with the first nonzero digit from the left of the number. ) If a number has a decimal point, count all the digits while observing rule. ) If a number does not have a decimal point, zeroes at the end of the number are presumed not to be significant. Significant digits are underlined in the examples below Example Significant Digits Rule(s) used and. 5 and.5 and.55 and.55 and.5 and 5. and,. and 5 and, and Exact Numbers Exact numbers are considered to have as many significant digits as are needed, so the rules of significant digits do not apply. Items that can be counted, such as coins, eggs, and cars, are examples of exact numbers. Conversion factors within a system of units are exact, such as

6 CHEM Week (Measurement, Calculation) Page of within the British system or within the metric system. Some conversion factors between unit systems are defined as exact, such as inch =.5cm. Counted quantities and conversion factors that are exact will not reduce the number of significant digits in calculations. Rounding All digits in a correctly recorded measurement are significant digits. Digits that are not significant may occur during calculations, and these digits are kept until the end of the calculation. After calculations, the final answer must be rounded to the correct number of significant digits. ) If the first nonsignificant digit is less than 5, drop all nonsignificant digits. ) If the first nonsignificant digit is 5 or more, increase the last significant digit by one then drop all nonsignificant digits. Example: Round 9.59 given that this number has significant digits. Solution: The first nonsignificant digit,, is underlined, Because is less than 5, drop all nonsignificant digits. The rounded answer to four significant digits is 9.5. Example: Round 9.5 given that this number has significant digits. Solution: The first nonsignificant digit, 5, is underlined, 9.5. Because the digit is 5 or more, increase the last significant digit by one and drop all nonsignificant digits. The rounded answer to four significant digits is 9.. Tracking Significant Digits During Calculations The significant digit is used to estimate the uncertainty of the final answer from calculations when there is only one set of data. Calculations for lecture class usually have only one set of data, so the significant digit is used. Laboratory classes typically collect multiple sets of data, so statistics are used to estimate the uncertainty. Addition and Subtraction For addition and subtraction of measurements, the measurement with the biggest uncertainty (least precision) determines the uncertainty and number of significant digits in the answer. In the following example, 5g has the biggest uncertainty (±g, precise to the ones place). The measurement of 5.g has an uncertainty of ±.g (precise to the tenths place), while 5.g is ±.g (precise to the hundredths place). The vertical red line shows the location of the last significant digit for the calculated answer (the ones place). The correct answer is 5g. The uncertainty of the answer is ±g which came from the measurement with the largest uncertainty (the 5g measurement).

7 CHEM Week (Measurement, Calculation) Page of Greatest Uncertainty Example: Add g, 55g, and g. Solution: The answer before rounding is 99g. The correctly rounded answer to the hundreds place is g. The uncertainty of the answer is ±g, which is the same as the uncertainty of the measurement with the greatest uncertainty. Greatest Uncertainty Example: Add.5g,.g, and.g. Solution: The measurement.g has the most uncertainty (±.g) and limits the significant digits in the answer. The vertical red line in the figure below indicates the position of the last significant digit. In the figure below, the answer before rounding is.99. The correct, rounded answer is.g. Note that the answer can have a different number of significant digits than the measurements. Biggest Uncertainty Example: Subtract.g from.999g Solution: Follow the same rules as for addition. The answer before rounding is.99g. Because.g has a greater uncertainty (±.g) than.999 (±.g), it limits the number of significant digits in the answer. The correct, rounded answer is.9g. Bigger Uncertainty

8 CHEM Week (Measurement, Calculation) Page of Example: Subtract.999g from.9999g Solution: The answer before rounding is.9g. Because.999g (±.g) has a greater uncertainty than.9999g (±.g). The correct, rounded answer is.g. The last significant digit is in the thousandths place. Multiplication and Division The rule for multiplication and division is to keep as many significant digits in the final answer as in the measurement with the fewest significant digits. Example: cm x.cm =.5cm Because.cm has only one significant digit, the correct, rounded answer is.5cm. Example: 99.cm x.cm x.5cm = 59.55cm Because.5cm has only significant digits, the final answer is 5cm. Example: 9.5g / cm = 5.g/cm Because cm has only significant digits, the final, rounded answer is 5.5g/cm. Scientific Notation Writing numbers in scientific notation helps to specify the correct number of significant digits. The format for a number in scientific notation is: D.D x n The D are the significant digits, where the first D must be a nonzero digit. n is an integer that indicates the power of to represent the original number. Positive values of n indicate big numbers (bigger than ), while negative values of n indicate fractions (smaller than and bigger than ). Example:. x g = g (SD) Explanation: To convert the scientific notation to a number, move the decimal two places to the right because of the positive value for n. Scientific notation makes it clear that the measurement has significant digits. The number g is presumed to have only one significant digit (review rules for counting significant digits above). Example:. x - g =.g

9 CHEM Week (Measurement, Calculation) Page 9 of Explanation: To convert the scientific notation to a number, move the decimal three places to the left because of the negative value for n. Example:,5,g =.5 x g Explanation: To convert a number to scientific notation, put a decimal point after the first nonzero digit and indicate the positive power of for a large number. The decimal was moved places to the left to get.5, so n=. Example:.m =. x -5 m Explanation: To convert a number to scientific notation, put a decimal point after the first nonzero digit and indicate the negative power of for a fractional number. The decimal had to be moved 5 places to the right to get., so n=-5. Be sure that you know how to use your calculator for scientific notation. The EE or EXP button is usually used to enter numbers in scientific notation. Unit Equations Unit equations describe a mathematical equality between two different units, for example, hour = minutes. Many such equations exist including mm=x - m, lb=5g, inch =.5cm, and cm = ml = cc. Any unit equation can be written as two unit factors that convert one set of units into another. Unit equations and unit factors (conversion factors) are usually known (or can be looked up) to enough significant digits so that they do not reduce the number of significant digits in a calculated answer. Recall that conversions within a unit system are exact. Exact numbers do not reduce the number of significant digits in a calculated answer. Unit Factors Any unit equation can be written as two unit factors. For example, hour = minutes (this conversion is exact) can be written as two possible unit factors. The correct unit factor (conversion) to use depends on the problem. hour minutes or minutes hour Example: Convert. hours to minutes. Solution: We have hours. We need minutes. Select the conversion factor with hours in the denominator and minutes in the numerator so that the answer will be given in minutes. The exact conversion factor does not reduce the number of significant digits in the answer. minutes. hours x = minutes =. x minutes hour (SD) (exact) (SD)

10 CHEM Week (Measurement, Calculation) Page of Example: Convert the speed 5 miles/hour into yards/second ( mile= yards). Solution: In the numerator, we have miles that needs to be converted into yards. In the denominator, we have hour that needs to be converted to seconds. 5 miles yards hour. yards x x = hour mile seconds second (SD) (exact) (exact) (SD) = yards/second ( significant digits because 5 miles/hour has significant digits using the multiplication/division rule). Percents A percent gives the number of parts out of total parts. For example, an exam score of 95% means 95 points awarded out of total points. Example: If the exam consisted of 5 points, how many points were awarded? Solution: Start with the 5 total points and calculate points awarded using percent as a unit factor. 95 points awarded 5 total points x =.5 points awarded total points (counted, exact) (counted, exact) (exact answer) Because 5 points are counted (exact), the answer is.5 points awarded (keep all the digits). Similarly, percents can be used to solve many types of scientific problems with measurements. Example: An ore is known to contain 5.5% silver by mass. How many pounds of silver can be found in pounds of ore? Solution: Start with the measurement of pounds of ore and calculate the pounds of silver using the measured percent as a unit factor. The percentage (5.5%) has significant digits that came from measurements, such as mass. All of the uncertainty is placed in either the numerator or the denominator of the fraction while the other part of the fraction is exact. The fraction reads there are approximately 5.5 pounds (±.pound) of silver in exactly pounds of ore. 5.5 pounds silver pounds ore x = 5. pounds silver pounds ore (SD) (SD) (SD) The measurement, pounds, has three significant digits, so the answer must be expressed to significant digits, 5 pounds silver or.5 x pounds silver using the multiply/divide rule.

11 CHEM Week (Measurement, Calculation) Page of Example: How would the answer be different in the above example if the ore contained the same percentage of silver, the mass of ore was grams, and the answer is to be expressed in grams of silver? Solution: The answer is numerically identical. The units change. 5.5 grams silver grams ore x = 5. grams silver grams ore (SD) (SD) (SD) The measurement, grams, has three significant digits, so the answer must be expressed to significant digits, 5 grams silver or.5 x grams silver. Percents can be calculated using experimental data. See the examples below. Example: If a test contains points of multiple-choice questions and points of worked problems, what percent of the test is worked problems? Solution: The definition of percent is the quantity of interest over the total quantity available times %. The numbers are counted, so the answer is exact. points worked problems x % = % points worked problems + points multiple choice Because points are counted (exact numbers), the answer (%) is exact and has as many significant digits as are needed. Example: If a sample ore that contains both copper and oxygen contains 9. g of copper and. g of oxygen, what is the percent of copper in the ore? Solution: Use the definition of percent to express the grams of copper ore over the total grams of ore (sample). Unlike points, masses are measurements with uncertainty and significant digits. 9. g copper x % = 5.95 % copper 9. g copper +. g oxygen The answer has four significant digits because the measurements have four significant digits and the steps instruct to keep four significant digits. The result from addition in the denominator has four significant digits. The correct answer to the correct number of significant digits is 5.% copper or.5 x % copper.

12 CHEM Week (Measurement, Calculation) Page of Density The density of a substance can be calculated by dividing the mass of a substance by its volume. Often, the unit for mass is the gram (g), and the unit of volume is the milliliter (ml). Density is an intensive property, one that does not depend on the amount of material. Temperature is also intensive. Mass and volume are extensive properties that depend on the amount of material. density = mass / volume = m/v Example: Problem: A bar of copper has a mass of 5.g. It is 9.5cm long.5cm wide and.9mm thick. Calculate the density of the bar. Solution: Get length measurements in the same units. The solution uses the conversion factors that are given to you on exams.: thickness =.9mm (m / x mm) (x cm/m) =.9cm (SD) (exact) (exact) (SD) Calculate the volume of the bar, a rectangular solid, from the length measurements. V=l x w x h Volume = 9.5cm x.5cm x.9cm = 5.95cm ( SD, keep all the digits) (SD) (SD) (SD) Calculate density of the bar D=m/V Density = 5.g (5SD) / 5.95cm (SD) = 9.g/cm = 9.g/cm (SD). Because density is extensive, we can compare the the density of copper from this experiment 9.g/cm ±g/cm to a published value of.9g/cm 9 9. The rounded results agree to significant digits (9.g/cm ), so the calculated density is consistent with a sample of copper. Volume of an object can be determined by measuring the water it displaces. Subtract the volume of water with the object 5submerged 5 (final) 55 5from 5 the 5volume 5 of water before adding the object (initial). Vobject = Vfinal Vinitial. The convention when measuring on a scale with marks is to estimate the last digit depending on whether the scale has wide spacings (estimate to / of the spacing) or fine spacings between scale marks (estimate to / of spacing). Example: 5 Determine 5 5 the volume of the marble from the information below

13 CHEM Week (Measurement, Calculation) Page of Solution: The scales have fine spacing because the distance between adjacent marks is small. For this graduated cylinder, the scale spacing is.ml between marks. Using the convention given above, estimate the measurements as half of the scale spacing or ±.5mL. Vinitial =.ml Vfinal =.5mL Vobject =.5mL.mL =.5mL The answer has SD because both measurements are made to the hundredths place, so the answer is known to the hundredths place ±.ml. Example: Mercury (Hg) has a density of.g/cm. What is the mass of 5.mL of mercury in grams? in pounds? Answer: Use dimensional (unit) analysis. Recall from above that ml = cm = cc. Find that lb=5g on the page of information for the exam. Given measurement Conversions (ml) Units of answer (g) cm.g 5.mL x x =.g = g ml cm (SD) (exact) (SD) (SD or ±g) Convert the answer above to pounds lb g x =.559lb =.lb. 5g (SD) (exact) (SD or ±.lb)