Grossmont College Chemistry 141 Laboratory Manual 7th Edition Chemistry 141 Laboratory Manual Compiled by J. Lehman, T. Olmstead, and D. ​ Vance​1 Grossmont College 8800 Grossmont College Dr. El Cajon, CA 92020 Prepared for printing on December 11, 2018 1. Original material by, B. Bornhorst, B. Givens, J. Lehman, J. Maley, J. Oakes, T. Olmstead, C. Park, C. Willard, D. Vance and J. George 19-0003-190 Content s EXPERIMENT 1 ​Calibration of Glassware, Density, and Error Analysis 1 ​ Background ​1 ​Procedure Part A: Glassware Calibration ​9 Calculations and Results Part A: Glassware Calibration ​12 Procedure Part B: Density of Coke and Diet Coke using Calibrated Glassware ​13 ​Calculations and Results Part B: Density of Coke and Diet Coke using Calibrated Glassware ​14 ​Post Lab Questions 15 EXPERIMENT 2 ​Propagation of Error: An Error Analysis ​Activity 17 Uncertainty in Measurements ​17 ​Procedure ​25 ​Results and Calculations Part C: Density of Metal Slug by Volume Measurements ​27 ​References ​29 ​Post Lab Questions ​31 EXPERIMENT 3 ​Density: A Study in Precision and Accuracy (Part ​A) ​33 Introduction ​33 ​Theory ​34 Experimental Procedure ​36 Chemistry 141 Grossmont College ​i–1 Contents Post Lab Questions ​39 EXPERIMENT 4 ​Density: A Study in Precision and Accuracy (Part ​B) ​41 Validity ​41 ​The “Experiment” 4​ 5 Least Squares Analysis Of Data ​45 EXPERIMENT 5 ​Conductivity and Net Ionic Equations 4 ​ 7 Background ​47 Procedure ​51 ​Procedure Questions ​53 ​Post Lab Questions ​59 EXPERIMENT 6 ​Writing Redox Reactions 6 ​ 3 Background ​63 ​Post Lab Questions ​69 EXPERIMENT 7 ​Redox Reactions – The Activity Series 7 ​ 3 Background ​73 ​Procedure ​74 Results and Calculations ​77 EXPERIMENT 8 ​Copper Reactions 7 ​ 9 Background ​79 ​Procedure ​80 Calculations and Results ​83 Post Lab Questions ​85 EXPERIMENT 9 ​Analysis of a Two-Component Alloy 8 ​ 7 Background ​87 ​Procedure ​91 Results and Calculations ​92 Post Lab Questions ​93 EXPERIMENT 10 ​Calorimetry–Measuring Heat of Formation ​95 Part A: Introduction to Calorimetry ​95 i–2 ​Chemistry 141 Grossmont College Procedure: Part A ​96 ​Part B: Introduction to Measuring the Heat of Formation of Magnesium Oxide ​97 ​Procedure: Part B ​100 ​Calculations and Data Treatment ​101 EXPERIMENT 11 ​Atomic Spectra 1 ​ 03 Background ​103 ​Procedure 107 ​Results and Calculations 108 ​Post Lab Questions ​109 EXPERIMENT 12 ​Periodicity of Chemical Properties ​113 Background ​113 Procedure ​113 Discussion ​114 ​Sample Report ​114 ​Post Lab Questions ​117 EXPERIMENT 13 ​Molecular Structure 1 ​ 23 Background ​123 Procedure ​123 Prelaboratory Sheet ​129 Report Sheet ​130 ​Post Lab Questions ​142 EXPERIMENT 14 ​Identification of an Unknown Acid 1 ​ 43 Background ​143 ​Procedure ​147 ​Determining the Identity of Your Unknown Acid ​149 ​Sample Data Sheet ​151 ​Sample Calculation Sheet ​152 ​Post Lab Questions ​153 EXPERIMENT 15 ​Determining the Effectiveness of an Antacid ​155 Background ​155 ​Procedure 158 ​Results and Calculations 159 Chemistry 141 Grossmont College i​ –3 Contents Report ​160 ​Post Lab Questions ​163 EXPERIMENT 16 ​Determination of Molar Mass by Freezing Point Depression 1 ​ 64 Background ​164 Procedure ​169 ​Data Treatment ​170 ​Post Lab Questions ​171 EXPERIMENT 17 ​Chemical Equilibrium and Le Châtelier's Principle ​173 Background ​173 ​Sample Write-Up ​174 Equilibrium Experiment Demonstration ​175 Procedure ​176 i–4 ​Chemistry 141 Grossmont College Calibration of EXPERIMENT 1 ​ Glassware, Density, and Error Analysis Backgroun d In Part A of this experiment, you will measure the volume that is delivered from three pieces of lab- oratory glassware: a beaker, a graduated cylinder, and a volumetric pipet. In Part B, you will use the appropriate glassware to measure the density of Coke and Diet Coke. The goal is to determine the accuracy and precision for each type of glassware. The accuracy and precision for a device is deter- mined by a method of calibration. Calibration is a process in which users learn about the inherent limitations of a particular piece of equipment and determine a good estimate of the actual values that one can rely upon when using that equipment for measuring. Chemistry is an experimental sci- ence and therefore chemistry will always involve the taking of measurements. These measurements may be multiple measurements of the same object, measurements of a single object by multiple observers, or one observer’s measurements of many different objects. What is the interpretation of each set of measurements? Are the measurements giving the correct answer (i.e are they accurate)? Are they consistent (i.e. precise)? One thing scientist must keep in mind is that no matter how much care is taken when measuring, uncertainty in measurement is always present. The rules that apply to significant figures are basically an elementary form of error analysis. For the most part, following the rules that govern significant figures are sufficient when there is a single measurement, or at most a duplicate trial. However, there may be times when we have the opportu- nity to carry out more exacting experiments where uncertainties in measurements can be estimated quite accurately. Errors in Observational Data There exists a degree of uncertainty with nearly every type of measurement. A balance that measures to the nearest gram will obviously introduce some uncertainty in the mass of objects that weigh approximately 1.5 grams. If you try to measure the length of a board and the end of the board falls between two of the marks on your measuring tape; you would have to estimate the last frac- tional length. The only measurement that can be determined with certainty is counting a small set of objects, for example, the number of students in your class. However, counting large sets of objects is not an exact measurement. To determine the population of the US for example, actually Chemistry 141 Grossmont College 1 ​ –1 Calibration of Glassware, Density, and Error Analysis counting every person is impractical and difficult to the point that estimates must be made to complete the task. When we say that a measurement is uncertain, we mean that the measurement includes error. An error ​is the difference between a measured value and the true value. Errors are typically expressed as the uncertainty in the measurement using statistical quantities. There are three types of errors: systematic, random, and gross errors. Random errors ​are the uncertainties associated with a measuring device; for example, an object measured for length may fall between the smallest divisions on a ruler and its length would have to be estimated. The reality is, no matter how carefully measurements are taken there is always a cer- tain amount of “scatter” in the data. This scatter is due to the inability of an instrument or an observer of that instrument to discriminate between readings differing by less than some small amount. The size of the random error in an individual measurement cannot be anticipated. Since random errors occur in an unpredictable manner, it is impossible to eliminate them. Fortunately, random errors can be dealt with statistically. Often mistakes are made that cannot be accounted for such as imprecise measurements or faulty technique. When an average of a data set is determined, this random error is represented by the deviation of each individual measurements from the average. Errors are an unavoidable part of the scientific process; multiple trials are done in order to increase the precision. There are also techniques that can be used to minimize errors. Systematic errors ​(sometimes referred to as determinate errors) are errors of a definite size and sign that can often be traced to specific sources introduced during the lab. Generally, such errors can be avoided or corrected. For example, this type of error may be caused by improper calibration of an instrument, uncompensated instrumental drift, leakage of material (e.g. gas in a pressure system), incomplete fulfillment of assumed conditions for a measurement (e.g. incomplete reaction in a calorimeter or incomplete drying of a weighed precipitate), personal errors in reading an instrument or a measuring device (e.g. parallax error) or, biased methods implemented during the procedure (e.g. uncompensated human reaction times). Systematic errors often announce their presence in some sort of pattern. Systematic errors must be eliminated (corrected) since there is no statistical method to handle these errors, while random errors are distributed in a way that can be described and understood statistically. When systematic errors occur, accuracy rarely matches precision. Therefore, it is important to calibrate, read instruments, etc. correctly. Gross errors ​are results in a value which is far different than either the true or the mean. They may be caused by sample inconsistencies or technical mistakes (i.e. reading measuring instruments wrong on one trial). Accuracy and Precision When experimental values are discussed it is important to know the information about the quality of the data. What exactly is meant by quality? How can the size of the random and systematic errors be determined? To answer these questions, the terms accuracy and precision need to be defined. Accuracy ​is a measure of the difference between the true value and measured value. However, the true value is not always known. Absolute error is the approximate error of a single measurement: absolute error = ​∆ ​=|true value - observed value| ​(EQ 1.1) Accuracy is frequently described as a percent difference or percent error between the measured (observed) value and the accepted value: 1–2 ​Chemistry 141 Grossmont College Background = % error ​ observed ------------------------------------------------------------------------------ ​ accepted ​ value ​× ​100 % ​ value ​ value – accepted ​ (EQ 1.2) The difficulty with determining the error for a measured (or calculated) value is that it is often dif- ficult or impossible to determine the accepted value. For measurements taken in the undergraduate laboratory, we frequency compare our results with generally accepted results published in the liter- ature as our known or accepted value. In such cases the percent error is calculated. Precision ​is a measure of the variability of individual quantities within a data set. It measures the amount of random error. If we were to take many measurements, how close would they be to each other? The deviation answers this question. deviation = d = |average value - observed value| ​(EQ 1.3) Precision is frequently described by the percent deviation: = % deviation ​ observed ---------------------------------------------------------------------------- ​ average value ​× ​100 % ​ value ​ value – average ​ (EQ 1.4) Unfortunately, precision cannot give much information regarding the accuracy of a measurement. A common illustration for these terms is a “bull's-eye” target. Good accuracy means several arrows close to or in the middle of the target. Good precision means all the arrows are clustered in the same region of the target. Good precision does not guarantee good accuracy; all of the arrows can be grouped close together yet far from the center of the target. FIGURE 1.1 ​Accuracy and Precision Although, both sets of arrows represent a set of precisely thrown arrows, only picture A would be considered accurate and precise, while picture B would be considered precise, but not accurate. How can the best possible results be obtained during an experiment? • ​Perform experiments as carefully as possible to minimize random error. • A ​ nalyze each of the measurements to identify possible sources of systematic error and mini- mize them. • D ​ etermine the result using several entirely independent methods of measurements and compare. If these independent methods give the same final results it is a good indication of accuracy. Treatment of Errors Since a series of measurements will produce random errors, some positive and some negative val- ues, the true value is best estimated by finding the ​mean ​value ( ). The mean is often called the Chemistry 141 Grossmont College 1 ​ –3 x Calibration of Glassware, Density, and Error Analysis average value and is found by summing all the values and dividing by the number of measure- ments: x= ​ ∑​------------ ​n x​ ​ (​ EQ 1.5) i where ​x i​ s the individual value and ​i ​is the total number of measurements Closely related is the ​median ​value which is the value that has an equal number of measurements above and below the mean. Consider this set of percent chlorine derived from measurements of many sodium chloride samples. The balance is good to four significant figures and five observa- tions were taken: TABLE 1.1 Observed Value for Trial # ​ Percent Chlorine 1 60.50% 2 60.41% 3 60.53% 4 60.54% 5 60.52% Mean 60.500% Median 60.52% Some questions may come to mind: Why wasn’t the same value obtained for each trial? What value should have been obtained for the percent chlorine? One way to analyze the distribution of values is to report the spread or range which is defined as the difference between the highest and lowest val- ues. range = 60.54% – 60.41% = 0.13% (​ EQ 1.6) The range does not indicate anything about the distribution of data points about the mean value. The mean or average value is: = 60.50 -------------------------------------------------------------------------------------------- + 60.41 + 60.53 ​ + 60.54 + x​ 5​ 60.52 = 302.50 ---------------- ​ = 5​ 60.500 ​ (EQ 1.7) The median is found to be 60.52%. The true value for the percent chlorine in sodium chloride can be calculated from the well-known atomic masses of chlorine and sodium: %​Cl (EQ 1.8) Therefore, the accepted or calculated value is 60.66%. The difference between the median and the mean gives us an idea of how skewed our data is (that is, to what extent the data is unevenly distrib- uted about the mean). Then the absolute error for each measurement can be determined for each trial as shown for trial 1: Absolute error = |60.66% - 60.500%| = 0.16% for trial 1 ​(EQ 1.9) 1–4 ​Chemistry 141 Grossmont College 35.45 = --------- ​ ------------------------------------------------------ 35.45 ​ mol ​ g+ ​ mol ​ g​--------- g​ mol ​ × ​100 % = 60.66%Cl 22.99 --------- ​ Background Next the percent error can be calculated: = % error ​ 60.500%Cl ------------------------------------------------------------- ​ – 60.66%Cl ​ 60.66%Cl ​× ​100 % = -0.26% ​ (EQ 1.10) Notice that for this data item the percent error, a measure of accuracy, is a small value implying good accuracy. The negative sign implies the average is lower than the accepted value. The devia- tion for each measurement from the average can also be determined: Deviation = |60.500% - 60.50%| = 0.00% for trial 1 ​(EQ 1.11) Tabulating these results: TABLE 1.2 Observed Value for Trial # ​ Absolute Error Deviation Percent Chlorine ​ 1 60.50% 0.16 0.00 2 60.41% 0.25 0.09 3 60.53% 0.13 0.03 4 60.54% 0.12 0.04 5 60.52% 0.14 0.02 Average 60.50% 0.16 0.04 Using the same data set, the deviation and absolute deviation can be tabulated for each value as shown in Table 1.1 on page 4. Notice that except for rounding errors, the mean deviation is nearly zero as one would expect if the errors in measurement were randomly distributed above and below the mean. The only significant information in this example is the average of the absolute deviation or simply the ​average deviation​, d​ . = d​ ​x​ – ​x ​= 0.00 ----------------------------------------------------------------------------- + 0.09 + 0.03 ∑ --------------------​ n​ ​ i 5 as seen in Table 1.2 . ​(EQ 1.12) Notice that in this data the average deviation, a measure of the precision, is considerably less than the average absolute error. That means that the actual, or true, mass percentage of chlorine in sodium chloride is outside the likely range of the data. Therefore, there must be some systematic error in the data and in the experiment that led to inaccurate data. This example demonstrates how a student might be misled into believing their data was accurate based on good precision. Unfortu- nately, systematic error cannot be described using any simple mathematical theories. So it is often not identified although it is frequently found to be orders of magnitude larger than the random errors. In fact, many published papers have later been shown to be incorrect by amounts far greater than the claimed limits of error. Given the fact that it is very difficult to identify and quantify all sources of systematic error, statis- tical methods of analyzing random error are used as an indication of the error found in experimental data. Although random errors cannot be corrected for, they can be treated statistically in an attempt to establish the reliability of the measurement. The analysis is based on the “normal” distribution, illustrated by the curves inFigure 1.2 on page 6. Curves of this kind describe not only how experi- mental measurements are distributed, but also a wide range of other phenonmea. They are exam- ples of the well-known “bell curve”. + 0.04 + 0.02 = 0.04 Chemistry 141 Grossmont College 1 ​ –5 Calibration of Glassware, Density, and Error Analysis FIGURE 1.2 ​Normal Error Curves The curves show the relative frequency of deviation that can be expected to be found if a large number of measurements are made. Some of the main points to keep in mind regarding these curves are: • ​The curves are symmetric about the midpoint, which is the arithmetic mean. Therefore, positive and negative deviations are equally likely. • ​The curves rise to a maximum at the midpoint, indicating that small deviations occur more often than large deviations. In fact, if a large deviation is observed a systematic error is most likely involved. • ​The shape of the curve is dependent on the inherent precision of the measurements. Sloppy or crude instrumentation give a high frequency of large deviations, as in curve (b). Refined mea- surements with improved precision show large deviations to be improbable, as in curve (a). For reasons that will not be discussed here, the standard deviation is preferred over average devia- tion because of its statistical meaning. ​Standard deviation ​is the measure of precision, i.e. of the size of the random error in a set of data. The standard deviation gives information about the width or broadness of the error curve that is associated with a set of experimental measurements. In order to determine the value of the standard deviation of a quantity several, preferably many measure- ment must be taken. The equation of standard deviation that we use in this experiment is techni- cally only an approximation of the true standard deviation of a particular distribution of numbers. This estimation of the standard deviation, ​σ​, is given by: (EQ 1.13) Returning to our example the standard deviation of the percent chlorine data can be determined. First the deviation squared is calculated as shown in Table 1.3 on page 6. TABLE 1.3 Observed Value for Trial # ​ Deviation, d d​2 Percent Chlorine ​ 1 60.50% 0.00 0.00 2 60.41% 0.09 0.0081 3 60.53% 0.03 0.0009 4 60.54% 0.04 0.0016 5 60.52% 0.02 0.0004 Sum 0.011 1–6 ​Chemistry 141 Grossmont College x​ x –​ ​( )​ ∑​n 1​ – ------------​ ​ ∑​n ​1 – ---------------------------== ​ ​ d​2 ​ σ​ i 2​ Background The standard deviation of the percent chlorine is calculated to be: = 0.011 σ​ ------------- ​ = 0.052 ​ 5–1​ (EQ 1.14) The standard deviation tells a lot about the distribution of the data. A small value for sigma corre- sponds to a sharp, steeply rising curve, where deviations are close to zero. On the flip side, a broad, squat curve indicates that large deviations are highly probable. The size of the standard deviation may be used to rank the precision of a set of measurements; the larger the standard deviation, the poorer the precision; the smaller the standard deviation, the better the precision. FIGURE 1.3 ​Normalized Distribution The shaded area bound by -​σ ​and ​σ ​is proportional to the probability of an observation with a devi- ation within one unit of ​σ ​of the arithmetic mean (located at the midpoint of the curve). This shaded area represents about two-thirds of the total area, or more exactly 68.2% of the total area under the curve. This mean that if a large number of measurements are made about two-thirds should fall within x - ​σ ​to x + ​σ ​or x ± ​σ ​(i.e. the average plus or minus the standard deviation). So, about one- third of the trials should fall outside of these boundaries and hence, would show a larger deviation. Actually, ± 2​σ ​covers most of the area under the curve or about 95%. This means that 95% of the trials fall within ± 2​σ ​from the arithmetic mean, x - 2​σ ​to x + 2​σ ​or x ± 2​σ. ​This leave about 5% of the measurements out of this range. In other words, about one-twentieth of the measurements will have a deviation of greater than ± 2​σ​. Keep in mind the probability of where a measurement falls is closest to those percentage cut-offs when dealing with a large number of observations. Summary Percent Chlorine Data Analysis Looking at the percent chlorine data, the mean value and its standard deviation may be expressed by 60.500 ± 0.052%. Looking at ranges of plus or minus one standard deviation, two standard devi- ations, and three standard deviations from the mean: Chemistry 141 Grossmont College 1 ​ –7 Calibration of Glassware, Density, and Error Analysis TABLE 1.4 ​Percentage Composition Ranges from the Average Percent Chlorine low end of range 1–8 ​Chemistry 141 Grossmont College high end of range number of trials within range 68% of values fall within 60.448% and 60.552% 4 95% of values fall within 60.396% and 60.604% 5 >99% of values fall within 60.344% and 60.656% 5 So, four out of five (80%) measurements are within one standard deviation 60.448% to 60.552% and all of the measurements are within two standard deviations 60.396% to 60.604%. For this data set it is reasonable to assume that three or four of the measurements of this data set fall within one standard deviation from the mean. As the number of measurements in the data set grows larger, we would expect the proportion of measurements falling within one standard deviation of the mean to approach 68.2%. TABLE 1.5 ​Percentage Composition Ranges from the Median Percent Chlorine low end of range In other words, the standard deviation indicates the distribution of values about the mean. To say this another way, the standard deviation is an indication of the precision involved with a certain set of measurements. The experiment focuses on data analysis where some values represent major data and other values minor, but still necessary, data. Below is a summary of the major data: Standard deviation may be used as a relative measure of ​precision​, so another chlorine data set that with a smaller standard deviation than 0.052 would be judged more precise than this data set. Per- cent error may be used as a relative measure of ​accuracy,​ so another chlorine data set that with a smaller ​magnitude ​of percent error than -0.26% would be judged more accurate than this data set. To sum it up: larger standard deviation corresponds to worse precision; smaller standard deviation corresponds to better precision. Larger percent error corresponds to worse accuracy; smaller per- cent error corresponds to better accuracy. high end of range number of trials within range 68% of values fall within 60.47% and 60.57% 4 95% of values fall within 60.42% and 60.62% 4 >99% of values fall within 60.36% and 60.68% 5 TABLE 1.6 ​Major Data Deviation Standard Median True %error Value ​ Percentage Average 60.500% 0.052 60.52% 60.66% -0.26% Procedure Part A: Glassware Calibration Procedure Part A: Glassware Calibration​ For each piece of glassware, the volume will be determined by weighing the amount of DI water in each sample and then using the density of DI water at the recorded temperature to calculate the vol- ume. For example, a beaker filled with water to the 50 mL mark contains approximately 50 mL of liquid. By weighing the beaker and its water, then subtracting the empty beaker weight, the actual mass of the water sample is known. The density of water is then used to calculate the true volume of that sample. For example -------------------------------------------------------------------------------------------------------------------- ​ ( ​density SAMPLE ​ ( ​mass of water delivered )​ ​of = water at recorded temperature ​) ​ volume delievered ​ (EQ 1.15) Which piece of glassware do you expect to be the most accurate? The most precise? Part I: Beaker Calibration Use a 50 mL beaker that has a calibration line at the 10 mL point. Check out a clean beaker from the stockroom if you do not have one that is suitable for this experiment. Wash your beaker and rinse with DI water. Do not dry the inside of your beaker. Dry the outside then follow the instruc- tions below. Record your data in your laboratory notebook. 1. ​Fill the beaker to the 10 mL mark as carefully as possible. Adjust the level using an eye dropper if needed. What is the uncertainty of the beaker measurement? 2​ . ​Weigh the beaker and its contents. Use the quad-beam balance, do not use an analytical balance. Zero the balance first and continue to use the same balance for each measurement throughout this part of the experiment. What is the uncertainty of the balance? 3​ . ​Pour the water out of the beaker until it stops draining, then weigh the empty beaker (remember to use a glass stir rod, “glass-to-glass-to-glass” to ensure that as much of the liquid is transferred as possible). This is your tare weight. Do not dry out any remaining drops of water before it is weighed. This procedure will measure the volume of water delivered by the beaker, not the actual volume that it holds. 4​ . ​Repeat steps 1 – 3 for five more trials. 5​ . ​Determine the mass of water delivered by your beaker for each trial. This is simply done by subtracting the tare weight from the weight of the beaker plus the water. 6​ . ​Record the temperature of the water. Use the temperature to find the density of water from the CRC Handbook of Chemistry and Physics. This is the density you will use to calculate the vol- ume delivered. TABLE 1.7 ​Beaker Calibration ​Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 temperature of water CRC density of water mass full beaker mass empty beaker mass water delivered volume water delivered Chemistry 141 Grossmont College 1 ​ –9 Calibration of Glassware, Density, and Error Analysis Part II: Graduated Cylinder Calibration Wash a 10.00 mL graduated cylinder and rinse with DI water. As before, do not dry the inside of the cylinder; dry the outside. You will also need a plastic bottle with cap that can hold approxi- mately 125 mL of liquid. Follow the steps below and record the data in your notebook. 1. ​Fill a clean 250 mL beaker with approximately 200 mL DI water. Record the temperature of the water. Use this water for the six trials that you will do in this part of the experiment. 2​ . ​Weigh the plastic bottle and its cap on an appropriate balance. The bottle should be empty but does not have to be completely dry. This is your tare weight for the plastic bottle. 3​ . ​Fill the graduated cylinder to the 10.0 mL mark. Make sure the meniscus is just even with the 10.0 mL mark. Use an eye dropper to adjust final volume if needed. What is the uncertainty of the graduated cylinder? 4​ . ​Pour the contents of the graduated cylinder into the plastic bottle (“glass-to-glass-to-glass”). Some drops of water will adhere to the inside of the cylinder which is fine. Remember, you want to determine the deliverable volume, not the actual volume. Place the cap on the plastic bottle and record the weight. What is the uncertainty of the balance? 5​ . ​Transfer another 10.0 mL of water from the graduated cylinder to the plastic bottle like you did in steps 3 and 4. Do not empty the water in the plastic bottle from the first transfer. You will add each sample to the bottle in consecutive trials. By the end of this exercise, the bottle will be quite full. 6​ . ​Repeat steps 3, 4 and 5 until you have a total of six measurements. 7​ . ​Add another column to the right of your data table and fill it with the mass of the water delivered in each transfer. You will need this value later when interpreting your results. What should the approximate mass of water delivered be? Why is the mass of water delivered about the same for each addition of water? 8. ​Record the temperature of the water. Use the temperature to find the density of water from the CRC Handbook of Chemistry and Physics. TABLE 1.8 ​Graduated Cylinder Calibration mass water mass water delivered volume of water delivered ​temperature of water CRC density of water mass empty plastic bottle mass plastic bottle + 10 mL water mass plastic bottle + 20 mL water mass plastic bottle + 30mL water mass plastic bottle + 40 mL water mass plastic bottle + 50 mL water mass plastic bottle + 60 mL water SAMPLE Part III: Pipet Calibration Before the pipet can be used, it must be clean so that it drains freely (i.e wash three times with soapy water, rinse three times with tap water, rinse three times with D.I. water, and if necessary rinse three times with your solution.). After cleaning, practice with the pipet before you start the procedure. Use DI water to practice until you have mastered the skill needed to deliver an aliquot from the pipet. This exercise is similar to the procedure for the graduated cylinder except you will 1–10 ​Chemistry 141 Grossmont College Procedure Part A: Glassware Calibration use the 10 mL pipet to deliver your samples into a small plastic bottle that has a 50 mL capacity. Follow the steps below and record your data in your notebook. 1. ​Obtain the tare weight of your plastic bottle on an appropriate balance. As before, the bottle does not have to be completely dry on the inside. ​2. ​Fill the pipet to the 10 mL mark and deliver the contents to the plastic bottle. What is the uncertainty of the pipet? ​3. ​Weigh the bottle plus its contents. What is the uncertainty of the balance? ​4. Repeat steps 2 and 3 until you have data for six trials. Remember - don’t empty the bottle between trials. ​5. ​Record the temperature of the water. Use the temperature to find the density of water from the CRC Handbook of Chemistry and Physics. TABLE 1.9 ​Pipet Calibration mass water mass water delivered volume of water delivered temperature of water CRC density of water mass empty plastic bottle mass plastic bottle + 10 mL water mass plastic bottle + 20 mL water mass plastic bottle + 30mL water mass plastic bottle + 40 mL water mass plastic bottle + 50mL water SAMP LE mass plastic bottle + 60 mL water Chemistry 141 Grossmont College 1 ​ –11 Calibration of Glassware, Density, and Error Analysis Calculations and Results P ​ art A: ​Glassware Calibration Remember to convert your masses to volumes since the beaker, gradu- ated cylinder, and pipet are used to measure volumes and not masses. Part I: Beaker Calibration 1. ​Calculate the average and standard deviation for the mass of water contained in the beaker. does the standard deviation tell you about the precision of the beaker? Remember that you will use your data to support your choice in your discussion. TABLE 1.10 ​Beaker Calibration Calculations a. ​What mass water delivered volume water delivered average volume delivered deviation from average volume deviation squared sum of deviation squared standard deviation SAMPLE Trial 1 ​ Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 TABLE 1.11 ​Beaker low end of Calibration Ranges high end of range range 2. ​Assuming that the density of water is 1.0 g/mL, what is the most likely volume contained in your beaker when filled to the 50 mL mark (i.e the “true value” for the volume)? a. ​What is the percent error with respect to the volume? What does the percent error tell you about the accuracy of your measurements? Remember that you will use your data to support your choices in your discussion Part II: Graduated Cylinder Calibration the average and standard deviation for the mass of water contained in the cylinder. 2​ . ​Find the density of water at the temperature you recorded in the ​Handbook of Chemistry and Physics​. Use this density to convert the mean value of the mass of water and its standard devia- tion to the corresponding volume. a. ​What does the standard deviation tell you about the precision of the beaker? You will use your data to support your choices in your discussion. 1. ​Calculate 1–12 ​Chemistry 141 Grossmont College number of trials within range ​68% of values fall within 95% of values fall within >99% of values fall within and and and SAMPLE Procedure Part B: Density of Coke and Diet Coke using Calibrated Glassware 3. ​What is the percent error? What does the percent error tell you about the accuracy of your measurements? Classify the accuracy as good, fair, or poor as you did before. Part III: Volumetric Pipet Calibration the same calculations as listed above for the graduated cylinder. 2​ . ​What do standard deviation and percent error tell you about the precision and accuracy of your measurements? Remember that you will use your data to support your choices in your discus- sion. NOTE: ​Your results and calculations section should include a table like the one below to summarize your major data. TABLE 1.12 ​Analysis of Volume Instruments 1. ​Do Average ​ Instrument ​beaker graduated cylinder pipet​Uncertainty ​ Volume SAMPLE ​ Deviation Standard Value True %error 3. ​Which of your glassware was the most accurate? Which of your glassware was the most precise? In your discussion be sure to discuss and explain any differences. Procedure Part B: Density of Coke and Diet Coke using Calibrated Glassware in your calculations (not the volume Remember to use the best glassware you calibrated previously. Use that volume ​ it says it delivers). Density is a physical property of matter. It is defined as the ratio of the mass of an object divided by its volume: (EQ 1.16) For solids and liquids, density is usually expressed in units of g/cm​3​. It is evident that the mass and volume of an object must be known to determine its density. Determine the density of each soft drink using 10 mL samples. From your results in Part A, you need to decide which piece of glassware is the best choice for measuring the density of each. Chemistry 141 Grossmont College 1 ​ –13 m​ ---- = d ​ V​ Calibration of Glassware, Density, and Error Analysis Procedure 1. ​The student should develop and follow a procedure similar to that used in Part A. You should do a minimum of three trials on each soda. You may want to set-up a table like this to record your data: TABLE 1.13 temperature volume mass density SAMPLE ​ Trial 1 ​ Trial 2 Trial 3 Calculations and Results Part B: Density of Coke and Diet Coke using Calibrated Glassware the average density and standard deviation for each soft drink. Show your calculations. ​2. ​Create a table summarizing your major data. TABLE 1.14 ​Density of Coke and Diet Coke 1. ​Report average density standard deviation ​1–14 C ​ hemistry 141 Grossmont College C ​ oke SAMPLE Diet Coke Post Lab Questions Post Lab Questions 1. ​What factors did you consider in choosing the particular piece of glassware for Part B of the experiment? Include comments on the accuracy and precision of your glassware choice. 2. ​Is the density of the two soft drinks the same, greater or less than that of water? Suggest reasons for your answer (hint- examine the contents label on the aluminum cans). 3. ​Compare the density of Coke with the density of Diet Coke. Are they the same or different? Explain your answer. Chemistry 141 Grossmont College 1 ​ –15 Calibration of Glassware, Density, and Error Analysis 1–16 ​Chemistry 141 Grossmont College Propagation of Error: An EXPERIMENT 2 ​ Error Analysis Activity Uncertainty in Measurements In chemistry significant figures are used to convey the uncertainty associated with all measurements made in the lab. There are rules dictating how those quantities should then be added and sub- tracted, and/or multiplied and divided. These rules are a substitute for statistical propagation of error. The “true” error or uncertainty in a given measurement is actually given by the range of pos- sible error for a given instrument use to make the measurement. This uncertainty is then approxi- mated using significant figures. You will determine the density of a metal slug. The relative and percent errors will be determined for each method. More often than not one measurement is used to determine derived quantities. For instance, mass and volume measurements are used to make den- sity determinations. This lab will introduce a method of statistical analysis to represent error that is propagated through calculations. The range of possible error for an instrument (“true” error) is known as the ​absolute error ​(​∆​) . The absolute error is the uncertainty from the instrument used for the measurement, the ± value. This may be written on the instrument or interpolated from the instrument. It is important to know how large this error is relative to the measurement being made. EXAMPLE 2.1 A 10 mL pipet has ± 0.02 mL written on the side. This is the “true uncertainty in any measurement using this pipet. If we measure 10 mL using this pipet, the error is given by 10.00 ± 0.02 mL. Meaning the range of the measurement is 9.98 mL to 10.02 mL. This error is approximated, using significant figures, to the highest digit of uncertainty written on the instrument. In this case, if 10 mL is measured with the volumetric pipet, a value of 10.00 mL ​would be recorded. In ​ this way, significant figures can be seen as a shorthand for the “true” error in the measurement. Propagation of Error As discussed previously, all measurements made in the lab contain an uncertainty associated with them. This uncertainty helps to determine the number of significant figures in a particular measure- Chemistry 141 Grossmont College 2 ​ –17 Propagation of Error: An Error Analysis Activity ment. There are rules dictating how those quantities should then be added and subtracted, and/or multiplied and divided. These rules are a substitute for statistical propagation of error. The “true” error or uncertainty in a given measurement is actually given by the range of possible error for a given instrument used to make the measurement. This uncertainty is then approximated using sig- nificant figures. Errors of a Calculated Result If a balance with a ± 0.0001 g error and a pipet with a ± 0.01 mL error are used, how should these errors be combined to determine the error in the density? The use of significant figures is an approximation of error, but for a more exact representation the following methods are used. Absolute error ​(​∆​) is the approximate error of a single measurement. The absolute error is best estimated as the standard deviation for a measure- ment. Absent this data an estimation should be made based upon the confi- dence the experimenter has with their ability to read the measuring device. Relative error ​is the ratio of the size of the absolute error to the size of the measurement being made. = --------------------------------------------- ​ absolute error Relative Error ​ experimental ​ = --------------------------------- ​ value ​ ∆ measurement ​ = ∆​R-​ ------ ​ R​(EQ 2.1) %Relative Error = Relative Error ​× ​100 % ​(EQ 2.2) EXAMPLE 2.2 ​Calculating Relative Error ​When reading the volume of liquid in the graduated cylinder to the right, you would estimate 64 mL, but it might be 63 or 65 mL. So, the value would be reported as 64 ± 1 mL and the absolute error in the volume measurement, ​∆​V = 1 mL. The relative error for the example would be: = --------------- ​ 1 mL Relative Error ​ 64 ​ = 0.016 ​ mL ​ or reported as 1.6% relative error. 2–18 ​Chemistry 141 Grossmont College Uncertainty in Measurements Relative Error ​A 250 mL beaker has an error of ± 5% written on the side. This means that the beaker has an absolute error of ± 5% of 250 mL or ± 12.5 mL. It is difficult to gauge how large this error EXAMPLE 2.3 ​Calculating will be in a measurement without comparing the error to the measurement. If we were to measure 50 mL using this beaker the measuring would read 50. mL to the correct number of significant figures. The absolute error would be represented as 50. ± 12.5 mL, with a range of possible values for the measurement of 37.5 - 62.5 mL. The error relative to the measurement will be = 12.5 --------------------- ​ mL Relative Error ​ 50. mL ​ = 0.25 ​ This means that the error is 25% of the measurement! Using these values we can approximate the error in calculated values. For addition and subtraction the absolute error of the sum or difference can be roughly approximated as the sum of the absolute errors. FIGURE 2.1 This method of error propagation overestimates the combined error because of the possibility that errors can cancel when more than one measurement is made. In fact, if this process is looked at statistically, a better approximation of error in a sum or a differ- ence is given by the formula: (EQ 2.3) Chemistry 141 Grossmont College 2 ​ –19 ∆​R ∆ ​ ​a2​ ​∆​b​2 ​... ​+ + = Propagation of Error: An Error Analysis Activity Absolute Error in Addition ​If the volume of a rock is measured by the displacement of water two volume mea- surements will be found: Volume water = 2.5 ± 0.3 mL Volume water + rock = 6.8 ± 0.3 mL Volume rock = (Volume water + rock) - (Volume water) = 6.8 mL - 2.5 mL = 4.3 mL EXAMPLE 2.4 ​Calculating 2​ 2​ 2​ To find the absolute error statistically: ​∆​V = ​ ​( ∆​V water ​ )​​ 2 ​+ ​( ∆​V water ​ ​+ ​rock ​)​ = ​( ​0.3 ​mL ​)​ + ​( ​0.3 ​mL ​)​ = 0.4 ​mL ​So the range of values for the rock volume is 3.9 - 4.7 mL or 4.3 ± 0.4 mL. The absolute error in this measurement is 0.4 mL, which gives a much smaller range than estimating the errors as in Example 2.3. When performing multiplication and division, the propagation of error must use relative rather than absolute errors. This is illustrated when density is calculated. The error in mass is in grams, while the error in volume is in mL. Grams and milliliters cannot be added together to calculate the total error; the relative error is unitless. The sum of the relative errors is an approximation of the total relative error (although it is an overestimation as before. Again, a better approximation of error can be obtained by applying statistics, as shown in the following: ∆​R-​ ------ ​ ⎛ ​ ∆​a​------ ​ ⎞​ 2​ Relative Error = ​ R= ​ ⎝ ​ a2 ​ –20 ​Chemistry 141 Grossmont College ​ ⎠ 2​ ​+ ​⎛ ​⎝ ∆​ ​ b​------ ​b⎞​ ​⎠ ​ + ​⎛ ​⎝ ∆​c​------ ​c​⎞ ​⎠ 2​ ​+ ​... ​(EQ 2.4) where R is a calculated value and the final absolute error in the result is given by: ∆​R = (relative error)(R) ​(EQ 2.5) Uncertainty in Measurements The table below summarizes the rules for addition, subtraction, multiplication, and division. Often calculations will involve more than one operation, so these rules need to be applied several times EXAMPLE 2.5 ​Calculating Absolute Error in Multiplication ​Boyle’s Law shows that pressure is indirectly related to volume, P ​α ​1/V, when moles of gas and temperature are held constant. So, pressure times volume is equal to a constant, k. (EQ 2.6) where P is the pressure in bar and V is the volume in mL. A gas is put into a con- tainer with a movable piston; the pressure of the gas is found to be 0.15 ± 0.01 bar and the volume is measured to be 120. ± 1 mL. What is the gas constant, k, and its uncertainty? Answer: The gas constant is Since the gas constant is the product of pressure and volume, the relative uncer- tainty in the gas constant, k, is ∆​k = 0.0673 (k) = (0.0673)(18 bar · mL) = 1.2 bar · mL The ​∆​k is rounded to the correct number of significant figures to agree with the precision in k. k = 18, The constant, k, is precise to the ones place, so the abso- lute error 1.2 bar · mL is rounded to the ones place 1 bar · mL. Therefore, the gas constant, k, is reported as 18 ± 1 bar · mL. TABLE 2.6 Operation Example Error Addition R = a + b Subtraction R = a - b Multiplication Division ⎛ ​ ∆​P-​ ------ ​ ⎞ ​ 2 ​ 2 ​ ⎛ ​ 0.01 P​ ⎠ ​ + ​⎛ ​⎝ ∆​ ​ V-​ ------- ​V⎞​ ​⎠ ​ = ​ ⎝ ​ P× ​ ​V ​= ​k k = ​ ​( ​0.15 ​bar )​ ( ​120 ​mL )​ ​= 18 ​bar ·​ ​mL ​∆​k​------ ​k​= ​ ⎝ ​ --------------------- ​ bar 0.15 ​ ⎞​ ⎛ ​ -------------------- ​ bar ​ ⎠ 2​ ​+ ​ ⎝ ​ 120. 1 ​mL ⎞​ mL ​ ⎠ 2​ ​= 0.0672 ​∆​R ​= ​∆​a​2 ​+ ​∆​b​2 ​∆​R ​= ​∆​a​2 ​+ ​∆​b​2 ​R ​= ⎛ ​ ∆​a​------ ​ ⎞ ​ 2 ​ ⎛ ​ ∆​a​------ ​ ⎞ ​ 2 ​ 2 ​ = ​a-​ - ​ ∆​R​------- ​ 2 a ​× ​b ∆ ​ ​R-​ ------ ​R​= ​ ⎝ ​ a​ ⎠ ​ + ​⎛ ​⎝ ∆​ b​ R= ​ ​ ⎝​ a​ ⎠ ​ + ​⎛ ​⎝ ∆​ ​ b-​ ----- ​b⎞​ ​⎠ ​ R ​ ​ b-​ ----- ​b⎞​ ​⎠ ​ Chemistry 141 Grossmont College 2 ​ –21 Propagation of Error: An Error Analysis Activity Absolute Error ​A student finds the density of a liquid by allowing a 10.00 mL volumetric pipet filled with liquid to drain into a previously weighed Erlenmeyer flask. The fol- lowing data was recorded: Volume liquid = 10.04 ± 0.01 mL Mass empty flask = 22.452 ± 0.002 g Mass flask + liquid = 33.629 ± 0.002 g Answer: m = 33.629 - 22.452 g = 11.177 g V = 10.04 mL EXAMPLE 2.7 ​Calculating = ​m-​ --- ​ = --------------------- ​ 11.177​g V​ 10.04​mL ​ d​ g​ mL ​ For the error in mass, use the addition rule: = 1.113 ------- ​ 2​ 2​ 2​ 2​ ∆​m ​= ​( ∆​m flask ​ ​)​ + ​( ∆​m ​flask ​+ ​liquid )​​ = ​( ​0.002​g )​ ​ + ​( ​0.002​g )​ ​ = 0.003 ​For the error in density, use the rule for division: ⎛ ​ ∆​m​-------- ​ ⎞ ​ ∆​d-​ ------ ​d= m​ ⎠ ​2–22 ​Chemistry 141 Grossmont College ​ ​ ⎝​ 0.003​g 2​ 2 ​ ⎛ ​ ------------------- ​ + ​⎛ ​⎝ ∆​ 11.177​g ​ ​ V​-------- ​V​⎞ ​⎠ ​ = ​ ⎝ ​ ⎞ ​ 2 ​ ⎛ ​ --------------------- ​ ⎠​ +​ ⎝​ 10.04​mL 0​ .01​mL ⎞​ ⎠ 2​ ​= 0.035 ​∆​d = 0.035 (d) ∆​d = (0.035)(1.1113 g/mL) = 0.039 g/mL Remember that the error can only be as precise as the least precise measure- ment. The density measurement has two decimal places therefore the error will be reported to two decimal places: ∆​d = 0.04 g/mL Therefore, the student can now report the error in disunity as: d = 1.12 ± 0.04 g/mL Note that the final density and the error in the density end at the same decimal place. See the following web sites for a good discussion of error propagation: http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart2.html http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm http://teacher.nsrl.rochester.edu/phy_labs/AppendixB/AppendixB.html Uncertainty in Measurements Using Vernier calipers: A Vernier allows a for better precision for a measurement. In Figure 2.2 on page 23, the Vernier moves up and down to measure a position on the Scale. The “pointer” is the line on the Vernier labeled “0”, which is used to determine the “certain digit” for a measurement. Thus the measured position is almost exactly 756 in whatever units the scale has been calibrated, for example, this may be mmHg as on a barometer. The last digit reading, i.e., the “doubtful digit” is read from the Ver- nier. In the examples below, the Vernier delineates 10 lines, which are used to estimate how far the measurement is between the gap of the lines printed on the scale. Since a single gap between the lines on the scale represent 1 unit increment (750 to 751, 751 to 752, etc.), the 10 lines drawn on the Vernier represent that increment divided by 10, or 1/10 of an increment Therefore, this Vernier allows preci- sion to 0.1 units. FIGURE 2.2 ​Barometric Pressure Readings Using Vernier Calipers Main scale Vernier scale Main scale Vernier scale 756.0 mmHg 756.5 mmHg Your instructor will demonstrate how to read the Vernier calipers. Make sure to record all measure- ments made with the calipers to a precision of 0.002 cm.​1 1. The Vernier explanation uses Copyrighted material and images which can be found at: http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html ​This material is subject to the terms and conditions of the Open Content License. Terms and conditions of this license are available at http://opencontent.org/opl.shtml Chemistry 141 Grossmont College 2 ​ –23 Propagation of Error: An Error Analysis Activity 2–24 ​Chemistry 141 Grossmont College Procedure Procedure Density is a physical property of matter. It is defined as the ratio of the mass of an object divided by its volume: = d​ m​---- ​ V(​ EQ 2.7) For solids and liquids, density is usually expressed in units of g/cm​3​. It is evident that the mass and volume of an object must be known to determine its density. A cylindrical shaped metal slug will be used. Its mass will be determined by simple weighing. The volume, however, will be determined by three different methods. An estimate of error will be made for each method, and then the method or methods with the highest degree of precision and accuracy will be decided. 1. ​Prepare a data table in your lab book, so that there is space to record the quantities indicated in each part. Be sure to estimate and record the uncertainty of each measurement as you are mak- ing the measurements. 2​ . ​Obtain the metal cylinder assigned to you. Weigh the cylinder; be sure to record the uncertainty of the measurement (± 0.001 g or ± 0.0001 g). Part I: Volume by Vernier Calipers 1. ​Use Vernier calipers to measure the diameter and length of the metal cylinder to the 0.002 cm. Your instructor will demonstrate how to use these calipers. Make your measurements in centi- meters and don’t forget to include the uncertainty of your measurements. This will give a vol- ume measurement in cm​3​. TABLE ​cylinder actual mass length diameter of density of ​2.8 ​cylinder number of cylinder ​Density ​cylinder of cylinder SAMPLE Determination ​ measurement ​by Vernier ​absolute ​Calipers error 2. ​Using the dimensions of the cylinder determine the volume: V= ​ π​r​2​l (​ EQ 2.8) 3. ​Obtain the actual density of your cylinder from your instructor. Part II: Volume by Water Displacement 1. ​Partially fill a 50 mL graduated cylinder with water and read the volume. Don’t forget to record the uncertainty of the volume measurements. You must initially have enough water in the cylin- der to completely cover the metal slug, but be careful that the displaced water does not exceed the 50 mL mark. ​2. ​Tip the graduated cylinder at a sharp angle to the vertical and carefully slide the metal slug into the graduated cylinder. Do not drop the metal slug into the graduated cylinder. Take care to avoid splashing water up the sides of the graduated cylinder, or break the cylinder. Place the Chemistry 141 Grossmont College 2 ​ –25 Propagation of Error: An Error Analysis Activity graduated cylinder back on your bench and record the new volume. Don’t forget to include the uncertainty of the measurement and take the temperature of your water.. TABLE 2.9 ​Density Determination by Vernier Calipers mass volume volume of water water dry cylinder and cylinder SAMPLE measurement absolute error Part III: Volume by Archimedes Principle Archimedes’ principle is a law of physics stating that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces. In other words, a force equal to the weight of the fluid it actually displaces buoys up an immersed object. Archimedes’ principle is an important and underlying concepts in the field of fluid mechanics. This principle is named after its discovered, Archimedes Syracuse. Archimedes’ two-part treaties on hydrostatics, called ​On Floating Bodies,​ states that: Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object. Thus, in short, buoyancy = weight of displaced fluid. The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In other words, Archimedes' Principle says that the apparent weight of an object immersed in a liq- uid decreases by an amount equal to the weight of the volume of the liquid that it displaces. Since 1 mL of water has a mass almost exactly equal to 1g, if the object is immersed in water, the difference between the two masses (in grams) will equal (almost exactly) the volume (in mL) of the object weighed. Knowing the mass and the volume of an object allows us to calculate the density. 1. ​Record dry mass of the metal cylinder. 2 ​ . ​Set up the balance with the balance pan apparatus moved so it is below the platform. 3​ . ​Wrap the string around your metal cylinder and hang it from the hook on the balance. FIGURE 2.3 2–26 ​Chemistry 141 Grossmont College Results and Calculations Part C: Density of Metal Slug by Volume Measurements the beaker with DI water up to within one inch of the top rim. ​5. ​Immerse your cylinder in the water, being careful not to let it touch the walls or bottom. 4. ​Fill FIGURE 2.4 the mass on the scale. ​7. ​Subtract submerged mass from dry mass and record the difference. Determination by Archimedes’ Principle 8. ​temperature mass mass ​Calculate o ​ f of dry submerged ​the c​ ylinder of ​volume w ​ ater 6. ​Read TABLE 2.10 ​Density cylinder SAMPLE ​ of the cylinder ​ measurement ​using the density ​absolute ​of water ​error at the recorded temperature. ​9. ​Calculate the density of the cylinder. Results and Calculations Part C: Density of Metal Slug by Volume Measurements Part I: Volume by Vernier Calipers 1. ​Use your length, diameter and mass measurements and their uncertainty values to calculate the relative error and percent relative error in your length, diameter and mass measurements. As illustrated in example 2.7 Calculating Absolute Error. ​2. ​Determine the volume and density of the cylinder. Chemistry 141 Grossmont College 2 ​ –27 Propagation of Error: An Error Analysis Activity the uncertainty in each measurement. ​4. ​Then calculate the relative error and percent relative error in your measurements. TABLE 2.11 ​Density Determination by Vernier Calipers 5. ​mass length diameter volume density ​Calculate o ​ f of of of dry of cylinder cylinder cylinder ​the c​ ylinder cylinder ​percent 3. ​Calculate SAMPLE error ​measurement ​in your ​ density determination. percent relative absolute error relative error ​ error Part II: Volume by Water Displacement 1. ​Use your volume and mass measurements and their uncertainty values to calculate the relative error and percent error in your volume and mass measurements. ​2. ​Determine the density of the cylinder and its uncertainty value. Then calculate the relative error and percent relative error in your measurements.. TABLE 2.12 ​Density Determination by Water Displacement 3. ​mass volume volume volume density ​Calculate o ​ f of water water of dry cylinder cylinder ​the c​ ylinder and ​percent c​ ylinder SAMPLE error in ​measurement ​your density ​ determination. percent relative absolute error relative error ​ error Part III: Volume by Archimedes’ Principle the volume of the cylinder using the density of water at the recorded temperature. ​2. ​Next, calculate the density of the cylinder. ​3. ​Then calculate the relative error and percent relative error in your measurements. . TABLE 2.13 ​Density Determination by Archimedes’ Principle 1. ​Calculate mass of dry cylinder mass submerged cylinder mass difference volume cylinder from mass difference density of cylinder SAMPLE measurement ​ percent relative absolute error relative error ​ error 4. ​Finally, calculate the percent error in your density determination. 2–28 ​Chemistry 141 Grossmont College References Create a table summarizing your major data. TABLE 2.14 ​Density of Cylinder using different methods Vernier Caliper SAMPLE ​ Which method is more accurate when determining the volume of the metal slug the calipers, water displacement, or Archimedes’ Principle? Be sure to use your data to support your answer. Which method is more precise when determining the volume of the metal slug the calipers, water displacement, or Archimedes’ Principle? Be sure to use your data to support you answer. References ​Masterton and Slowinski: ​Elementary mathematical preparation for General Chemistry.S ​ aunders Publishing, 1974. Shoemaker, Garland, and Steinfeld: ​Experiments in Physical Chemistry.​ McGraw-Hill, 1974. Kratchvil, H.: ​Chemical Analysis. B ​ arnes and Noble, 1969. Volume Displacement Archimedes’ Principle density of cylinder percent error Chemistry 141 Grossmont College 2 ​ –29 Propagation of Error: An Error Analysis Activity 2–30 ​Chemistry 141 Grossmont College Post Lab Questions Post Lab Questions 1. ​Compare the density of the metal slug using each method in this experiment. a. ​Are they the same or different? b. ​Which method resulted in the largest percent error? Why? c. ​Which method resulted in the smallest percent error? Why? Chemistry 141 Grossmont College 2 ​ –31 Propagation of Error: An Error Analysis Activity 2–32 ​Chemistry 141 Grossmont College Density: A Study in EXPERIMENT 3 ​ Precision and Accuracy (Part A) Introductio n In this lab, you will be investigating the concepts of precision and accuracy. You will be doing an experiment in which you will be measuring the density of some glass beads. Although you will be learning a little bit about density measurements, the primary goal of the lab is for you to come to grips with the separate concepts of accuracy and precision. In any scientific investigation, when the results are reported, it is standard practice that the investigating scientist carefully consider both the issues of accuracy and precision. Since these two concepts are often confused, we will begin with a careful definition of each. Precision: ​A measure of the amount of random variation in the measurement of data. Accuracy: ​A measure of how far an experimental result is from the true or correct value. Whenever a scientist makes measurements, there will always be some random variation in the values recorded. If one were to use a stop water to measure the time, t, it took an object to fall a certain distance, one might record data such as the following: 1.49 s, 1.48 s, 1.53 s, 1.48 s, 1.50 s, 1.47 s, 1.52 s, 1.52 s, 1.46 s The random variation does not necessarily reflect an error on the part of the person doing the measurements, but rather it may reflect the limit of the precision of the time measuring device (and the ability of the person controlling the stopwatch to hit the button at the right time). The precision of the experiment is a measure of the size of the random variation in the experiment. In this experi- ment, you will learn to calculate the standard deviation of the measurement: the most common accepted statistical measure of precision. Another way of thinking about precision is as a measure of the amount of random error in an exper- iment. Any experimental error is considered to be random if its result could make the calculated or measured value either too high or too low. Any potential error in an experiment which could have a predicted effect on the result, making it either definitely too high or too low, would be considered a systematic error (see below). Chemistry 141 Grossmont College 3 ​ –33 Density: A Study in Precision and Accuracy (Part A) For the same experiment as described above, if the exact height from which the mass was dropped was known, the equations of motion from the basic physics course (assuming the acceleration due to gravity to be 9.80 m/sec​2​) could be used to calculate the theoretical time it should take the mass to fall. In this case, the correct theoretical value could be compared to the results. For example, if the “true” or “theoretical” value for the time it should take the mass to drop were 1.45 seconds, the accuracy of the experiment could be calculated. In this experiment, you will measure the accuracy as the %-error of the measurement. In general, scientists are always able to measure the random error of an experiment. There are situ- ations in which there is no known “true” or “accepted” value for a measurement. In this case, the scientist may not be able to calculate the accuracy as a %-error. Normally, in such a situation, the scientist will do a very similar experiment to the one to be reported, only doing the very similar experiment on slightly different case in which the “true” value is known. The scientist will then cal- culate the %-error in this case as a check on the validity of the measurement they are reporting. Another way of thinking of accuracy is to think of accuracy as a measure of the systematic error in an experiment. For example, in the mass-dropping experiment described above, if one were to anticipate the mass hitting the floor, and press the button just a little before the mass hit, that would definitely make the time measured too small. This would be an example of a systematic error. If the person doing the experiment were to consistently make such an error, it would effect the accuracy (and therefore the %-error), but not the precision (and therefore not the standard deviation). Theory For a set of measurements of a variable x, the standard deviation is calculated using the following equation: σ ​= Σ ​---------------------- ​( ​ x ​ – ​ x ​)​2 n​ –​ 1​ (EQ 3.1) Where ​σ ​is the symbol for standard deviation (sometimes the letter s is used to represent the stan- dard deviation). The symbol ​x ​represents the individual measurements, while the symbol ​x repre- sents the average of the measurements in question. In this equation, ​n ​is the number of measurements. For example, one could calculate the standard deviation of the measurements of time listed above. The calculation is shown below: TABLE 3.1 t (s) ​t –​ ​t (​ s) ​( ​t ​– ​t ​)2​ (s​2​) 1.49 –0.004 0.000016 1.48 –0.014 0.000196 1.53 +0.036 0.001296 1.48 –0.014 0.000196 1.50 +0.006 0.000036 1.47 –0.024 0.000576 1.52 +0.026 0.000676 1.52 +0.026 0.000676 1.46 –0.034 0.001156​3–34 C ​ hemistry 141 Grossmont College Theory = 13.45 t​ ∑ (​ ​t​ ​– ​t ​)​ = ------------- ​ = 1.494 s ​ 9​ 0.004824 ​ s​2 ​(EQ 3.2) 2​ i Therefore, the standard deviation is: = 0.004824 σ​ ---------------------- ​ = 9–1​ 0.0245 s ​ (EQ 3.3) The average and the standard deviation are combined to yield the result that the time it took the mass to fall is in the form t = ​t ​± ​∆​t:​ t= ​ 1.494 ​± ​0.025 s ​(EQ 3.4) Note that both the average value and the standard deviation should end with the same number of digits past the decimal. Also note that the units label follows the standard deviation, not the average value. The accuracy of the time measurement may be calculated as well. The equation for %-error is as follows: = % error ​ measured ----------------------------------------------------------------------------------- ​ theoretical ​ value ​× ​100 % ​ value ​ value – theoretical ​ (EQ 3.5) If the measured value is below the accepted or theoretical value, then there will be a negative %- error. Be sure to show the positive or negative sign (e.g. +5.67% or -5.67%) when reporting your answer. What does the sign tell you about the %-error? For the time experiment described above, then, the %-error is calculated (assuming the true value to be 1.45 s) as: = 1.494 ----------------------------- ​ – % error ​ 1.45 ​ 1.45 ​× ​100 % = 3.0 % ​ (EQ 3.6) In general, a %-error is only reported to one or at most two significant figures. In this experiment, you will be measuring both the mass and the volume of some glass beads. The density will be measured using the well-known equation: = d​ m​---- ​ v​(EQ 3.7) After calculating both the mass and the volume of some glass beads a number of times, you will calculate the average values and the standard deviations of both values. You will then calculate the density, the standard deviation of the density, and the %-error from the true value for the density of the glass beads. What about the precision of your density measurement? In other words, you will be measuring the standard deviation of the mass measurement as well as of the volume measurement. The precision of the mass and volume measurements are determined when the standard deviation is measured. The question is how to use these results to estimate the size of the random error (precision) in the density when it is calculated by dividing the mass by the volume. Chemistry 141 Grossmont College 3 ​ –35 Density: A Study in Precision and Accuracy (Part A) One way to do this would be to simply calculate the random error of the two measurements used to calculate the density and add the two errors. There are two problems with this approach. First, one cannot add apples to oranges. In other words one cannot add the error in the mass (with units grams) to the error in the volume (with units of milliliters). A partial solution to this problem would be to calculate the percentage of the two errors and adding them. In other words, if the standard deviation of the mass measurement was 5.2% of size of the average mass, and the standard devia- tion of the volume measurement was 8.4% of the size of the average volume, one could conclude that the random error in the density measurement was 5.2 + 8.4 = 13.6%. The problem with this solution is that it overestimates the error. There is a significant probability that the error in the volume will have the opposite effect on the density as the error in the mass mea- surement. In other words, the errors can cancel. The correct statistical measurement of the com- bined error due to two measurements (​x a​ nd ​y)​ being used to calculate a secondary value (​z)​ is given by the following equation: (EQ 3.8) The estimated error, ​∆​z,​ can be calculated as follows, (EQ 3.9) Where z is the calculated value and ​∆​z is the estimated error in the calculated value. In this formula, and are the standard deviation in the measured values ​x ​and ​y.​ Therefore, the final result for the calculated value is z = ​z ± ​ ​∆​z​. Experimental Procedure Part I: Measuring the mass of the glass beads 1. ​Using a balance, measure the mass of forty glass beads eight different times. It is most conve- nient to measure the beads in a weighing boat. Be sure to measure different beads each time to ensure that you are getting a random selection of the beads in the jar. 2​ . ​Record the eight values of the mass in a table in your lab book, being sure to get as much precision out of the balance as possible. 3​ . ​After recording the values, determine the average mass of the beads. 4​ . ​Go on to calculate the standard deviation of the mass measurements. 5​ . ​In addition, report the average ± the standard deviation. 6​ . ​Be sure to report the units and the number of significant figures properly. Show all your calculations. Part II: Measuring the volume of the glass beads 1. ​Check out a buret from the stockroom and fill a buret about one-third to one-half full with deionized water. ​2. ​Carefully record the volume of water in the buret. If you are not sure how to record the volume measurement from a buret accurately to two places past the decimal, ask your instructor for help. 3–36 ​Chemistry 141 Grossmont College Experimental Procedure z​ = -- ​x​ or ​z ​= ​xy ​ σ​ 2​ y​ ∆​z ​= ​z ​⎛ ​⎝ ​ ----- ​x​x⎞​ ​⎠ 2​ ​+ ​⎛ ​⎝ σ​ ​ y ​ ----- ​yy​ ⎞​ ​⎠ ​ σ​x σ​ add forty glass beads to the buret and record the volume again. ​4. ​Repeat this process for a total of eight additions of forty beads, recording the final volume each 3. ​Now, time. Record the results in a table in your lab book. ​5. ​When you are done, drain the water from the buret and dump the wet beads into the container provided marked “wet glass beads”. ​6. ​Calculate the volume of forty beads for each of the eight cases, and continue on to calculate the standard deviation of the volume measurements. ​7. ​Report the average of the volume of forty beads ± the standard deviation in the correct format. Part III: Calculating the density and the error in the density 1. ​First, use the average mass and average volume from above to calculate the density of the glass beads. ​2. ​Next, calculate the %-error of your measurement, assuming that the correct value for the density of the glass beads is d = 2.35 g/mL. ​3. ​In addition, calculate the random error in the density using Equation 3.9 for calculating the accumulated uncertainty for a calculation involving two measurements. ​4. ​Finally, record the density of the glass beads from your measurements as d = d ± ​∆​d. Chemistry 141 Grossmont College 3 ​ –37 Density: A Study in Precision and Accuracy (Part A) 3–38 ​Chemistry 141 Grossmont College Post Lab Questions Post Lab Questions 1. ​Is the size of your random error big enough to explain the difference between your measured value of density and the expected value of 2.35 g/mL? Explain. 2. ​Based on your answer to question #1, would you need to invoke some sort of systematic error to explain the difference between your calculated density and the expected value of 2.35 g/mL, or can all the error be assumed to be due to random error? Explain. 3. ​Give 4. ​Give two examples of random error in this experiment. two examples of systematic error in this experiment. In each case, would the proposed systematic error make the calculated density too high or too low when compared to the correct value? Chemistry 141 Grossmont College 3 ​ –39 Density: A Study in Precision and Accuracy (Part A) 3–40 ​Chemistry 141 Grossmont College Density: A Study in EXPERIMENT 4 ​ Precision and Accuracy (Part B) Validit y In this part of the statistics lab, we will consider two additional aspects of statistical treatment of data which are very important to scientists. The first is validity. ​Validity ​is a measure of how well cause and effect are correlated. Testing claims of the validity of a cause and effect relationship between two variables is perhaps the most basic part of what scientists do. If the effect of a new drug in treating a particular disease is being tested, the drug must be tested on a set of patients as well as a control group which receives a “placebo.” The obvious question is whether or not there is a significant difference in symptoms of disease between those who took the drug and those who were given a placebo. Validity is a measure of whether two different results are truly different statistically. For example, a scientist could study the colon cancer rates of those who eat Wheaties and those who do not. Let us imagine that the colon cancer rate of those who eat Wheaties is 24.5 per thousand, while those who do not eat Wheaties have a cancer rate of 24.0 per thousand. Is the scientist justified in reporting that eating Wheaties can increase your likelihood of getting cancer? The answer is almost certainly no!!! The two different results almost certainly do not differ enough to statistically justify conclud- ing there is a relationship between eating Wheaties and getting colon cancer. The problem of determining validity of a result is especially difficult in the biological sciences, and even more so in the medical sciences. For example, consider the following hypothetical study. A group of subjects was surveyed and it was discovered that people in the army have a 30% higher lung cancer rate than those not in the army. This 30% difference is certainly statistically valid. Con- clusion: being in the army causes lung cancer. Wrong!!! What this study fails to do is to adjust the results for smokers. In fact, those in the army have a 40% higher rate of smoking. It was not being in the army which caused cancer, it was smoking. The conclusion is that anyone doing a scientific study must very carefully consider all the relevant variables which could conceivably effect a given result. Once all the variable have been controlled for, the results must still be checked for statistical validity. In other words, is there a valid correlation between a change in a given variable and the result measured. Chemistry 141 Grossmont College 4 ​ –41 Density: A Study in Precision and Accuracy (Part B) The ​t T ​ est The most common statistical test for whether a scientific measurement of an effect is valid is the ​t ​test. For a given set of data, one being the test, the other being the control, the question is whether the average value measured is statistically different. Is there a valid effect? To provide an example, consider the following data: TABLE 4.1 measurement # height of plant using just water height of plant using “Mighty Grow” 1 58 cm 64 cm 2 62 cm 55 cm 3 53 cm 58 cm 4 61 cm 66 cm 5 54 cm 56 cm 6 57 cm 62 cm ​average ​57.5 cm 60.2 cm Conclusion: “Mighty Grow” makes the plants grow faster. Not so fast! We must apply the ​t t​ est. Look at Equation 4.1 =​ x​ –​ t ​ ---------------- ​ 1 ​s ​ p ​4–42 ​Chemistry 141 Grossmont College (​ EQ 4.1) Where is the average of the first set of data, is the average of the second ​Nt​ he ​2 ​are two the sets number of data. of The measurements pooled standard for each deviation set of is data, given and by ​sp​ Equation is the pooled 4.2. ​ standard set of data, deviation ​N​1 ​and of (EQ 4.2) The value of ​t ​is calculated and compared to a ​t t​ able. If it is greater than the relevant t​ ​value in the table, then the difference between the two measurements is valid. A table of ​t v​ alues is included. For example, from the data in Table 4.1 on page 42, one can calculate the ​t ​value to be: TABLE 4.2 ​Set #1 Set #2 0.25 14.44 20.25 27.04 20.25 4.84 12.25 33.64 12.25 17.64 0.25 2.24 x​ -------------------- ​ N ​ N ​ 2​ N ​ 1 1​​ 2 ​+ ​N 2​ x​ 1​ x​ 2​ s ​p = ​ ∑ -----------------------------------------------------------------------​ ( ​x​ ​ – ​N x​ ∑ ​( ​xi​ –​ ​x1​ )​​ 2 ​= 65.50 ​∑ (​ ​x​i ​– ​x​2 )​​ 2 ​= xi​ –​ ​x2​ ​)​2 i1 ​ 2+ ​ 1 1 )​​ ​ ∑ ​ – ​( ​x​ ​ – ​x​ )​ 2 ​( ​x​ –​ ​x​ )​ ( + ​N ​ 2​ i2 ​ 2​ 2​ i 1​ 2​ 99.84 Validity =​ --------------------------------- ​ s ​p ​ 65.50 ​ 6 + ​(EQ 4.3) (EQ 4.4) Now, checking the ​t ​table, the number of degrees of freedom is ​N​–1. Critical values for t (two- tailed). Use these for the calculations of confidence intervals. Since we had six measurements, the number of degrees of freedom is five. At the 90% confidence level, with 5 degrees of freedom, ​t = ​ 2.015. Since our value for ​t w ​ as 1.14, there is not a valid cor- relation between use of the fertilizer and plant height. If we had used the 50% confidence level, ​t ​= 0.727, and the result would be valid. In other words, at a 50% confidence level, there is at least a small statistical effect of using the fertilizer. Q test Scientists use the ​Q t​ est​, an empirical evaluation, to determine whether a measurement should be rejected. In statistics, the ​Q t​ est ​is used for identification and rejection of outliers. This test should be used sparingly and never more than once in a data set. To apply a ​Q t​ est for bad data, arrange the data in order of increasing values and calculate ​Q a​ s defined: (EQ 4.5) If ​Qcalculated ​ > ​ then reject the questionable point. ​ ​Qtable ​ = 4.07 ​ = ​ x​ –​ x​ -------------------- ​ N ​ N ​ + 6 99.84 – 2 ​ t ​ ---------------- ​ 1 ​s ​ p ​ 2 ​ N ​ 1 1​​ 2 = 60.2 -------------------------- ​ + ​N ​2 ​ 6 ​ = 1.14 ​ 6​ TABLE 4.3 ​T-test Table –57.5 ------------ ​ 6 ​ · 6​ +​ 4.07 ​ Degrees of Freedom 50% 60% 70% 80% 90% 95% 98% 99% 99.5% 99.8% 99.9% 1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 127.3 318.3 636.6 2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60 3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92 4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610 5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869 6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959 7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041 9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781 10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587 11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437 12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.428 3.428 3.930 4.318 Q​ = --------------- ​ range ​ TABLE 4.4 ​Q-test gap Table Number of Values 3 4 5 6 7 8 9 10 Q​90% ​0.941 0.765 0.642 0.560 0.507 0.468 0.437 0.412 Q​ ​ 95% ​0.970 0.829 0.710 0.625 0.568 0.526 0.493 0.466 Chemistry 141 Grossmont College 4 ​ –43 Density: A Study in Precision and Accuracy (Part B) EXAMPLE 4.5 ​Evaluate the following data using the Q test:. TABLE 4.6 ​A student does five trials on a substance to find its molar mass. Trial # Molar Mass (g/mol) 1 152.9 2 148.0 3 153.5 4 154.2 5 154.5 The student applies the proceeding steps: 1. ​First, arrange the values in increasing order: 148.0 g/mol, 152.9 g/mol, 153.9 g/mol, 154.2 g/mol, 154.5 g/mol 2. ​Find the difference between the suspicious value and the value closest to it: 152.9 g/mol - 148.0 g/mol = 4.9 g/mol 3. ​Calculate the difference between the highest and lowest value: 154.5 g/mol - 148.0 g/mol = 6.5 g/mol 4. ​Obtain the quotient, Q, by dividing step 2’s answer by step 3’s answer: --------------- ​ Q ​= ​ 5. ​Finally, range ​ gap ​ 4.9 = ----------------- 6.5 ​4–44 C ​ hemistry 141 Grossmont College ​(EQ 4.6) compare the value of Q​90% found in Table 4.4 . ​ Q​90% tells use the maximum values we can have for a 90% confidence. If the calculated result in step 4 is greater ​ than the value for Q​90%​, we can reject the suspicious value. For this example, Q​90% for the five trials is 0.642 and ​ 0.75 is greater than 0.642. Hence, the value for trial 2, 148.0 g/mol, can be rejected. EXAMPLE 4.7 ​Evaluate the following data using the Q test: 0.189, 0.169, 0.187, 0.183, 0.186, 0.182, 0.181, 0.184, 0.181, 0.177 Arrange in increasing order: 0.169, 0.177, 0.181, 0.181, 0.182, 0.183, 0.184, 0.186, 0.187, 0.189 Outlier is 0.169. Calculate ​Q:​ (EQ 4.7) With 10 dence. --------- ​ g​--------- ​ g​ = --------------- ​ gap ​= ​( ​------------------------------------- ​ mol ​ mol ​ = 0.75 ​Q ​ range ​ ( ​observations at 0.177 ​ – ​ 0.169 ​ ) 0.189 ​ – ​ 0.169 ​ 90% confidence, Q​calculated < ​ Q​table​. Therefore keep 0.169 at 90% confi- = 0.008 )​ ------------- ​ = 0.020 ​ 0.400 The “Experiment” The “Experiment” Perform a ​t ​test to see if there is a statistically valid relationship between number of bean seeds sprouted per 100 and exposure to UV light. TABLE 4.8 Experiment # # sprouted without UV irradiation # sprouted with UV irradiation 1 87 71 2 72 64 3 88 80 4 81 69 5 69 70 6 78 70 7 80 72 ​Average 1. ​Calculate the average for each set of data and fill in the blank in the table above. Then do the calculations to find ​t f​ or the two sets of data. Show your calculations in your lab book. ​2. ​What is the number of “degrees of freedom” for the data above? ​3. ​Find the value of t from Table 4.3 on page 43 using your number of degrees of freedom at the 90% confidence level. Compare to your ​t calculated above. According to your result, is there a significant difference in the seed-sprouting rate for seeds irradiated with UV light? If not, is the difference significant at the 50% confidence level? If yes to the 90% confidence level, what is the highest confidence level at which the result is valid according to the table? Least Squares Analysis Of Data Least squares analysis of data is a statistical method for determining the best fit straight line to a set of data. There is hardly any more common thing for a chemist to do than to fit a set of data to a straight line, be it in kinetic studies, absorbance/concentration studies and so forth. Chemists almost invariably use a canned program from excel or other software to determine the slope and intercept of the best straight line fit to a set of data. In this experiment, you will actually do a least squares analysis of a set of data by hand. The theory and equations of least squares analysis is provided in an attachment to this lab write-up. You will be doing a simple experiment to determine the density of a solution using least squares analysis of data. Experimen t Using a 50.0 mL graduated cylinder, measure the mass of the cylinder empty as well as five sets of volume and mass data for the same cylinder and a solution provided. The volumes should be about 10, 20, 30, 40, and 50 mL. Measure both mass and volume with as much precision as the data allows. Record the data in your lab book. That is it! Calculation s 1. ​Perform a least squares fit to the five pairs of data, assuming that the volume is the independent (​x-​ axis) data and the mass is the dependent (​y​-axis) data. Your analysis of the data should include finding both the slope and the intercept (see Equation 4.8 and Equation 4.9), as well as Chemistry 141 Grossmont College 4 ​ –45 Density: A Study in Precision and Accuracy (Part B) the uncertainty in both numbers (see Equation 4.10, Equation 4.11, and Equation 4.12). Record the slope as ​m ​= slope ± error in slope and the intercept as ​b = ​ intercept ± error in intercept. 2​ . ​What is the physical interpretation of the slope of your graph? Does it agree with the correct answer (look it up) within the uncertainty? What is the %-error? Is a systematic error required to explain your %-error? (explain) 3​ . ​What is the physical interpretation of the intercept of your graph? Does this value agree with the correct value within the uncertainty you determined? Calculate your %-error. Is a systematic error required to explain your %-error? 4​ . ​Now, make a graph of your data and do a least squares fit to the same data using a canned pro- gram such as Vernier or Excel, available on the computers and compare to the values you got by hand. ∑ m ​= ​ -------------------------------------------- ∑ x​ ​ ​– ​x )​ ( ​y​ –​ ​( ​x​ ​– ​x )​ ​ y ​) ​= (​ i i i 2​ ​x​ y​ ​ ∑ -----------------------------------------------------​ – ​∑ i i –​ xi​ 2​ ​ i i ∑​n ​x​ )​​ ( ​------------------- ​ ​x​ ​n ​ ∑ ------------------------​ ∑ y​ ​ ​ i 2 (EQ 4.8) b= ​ y –​ ​mx ​(EQ 4.9) s​y (EQ 4.10) (EQ 4.11) (EQ 4.12) 4–46 ​Chemistry 141 Grossmont College = y​ ​ – ​( ​------------------∑ -------------------------------------------------------------------------------------------------------​ ∑​N ​y​ )​​ –​m​ ∑ x​ ​ ​ – ​( ​------------------- ​∑​N x​ ​ )​​ N –​ 2 ​s​ =​ --------------------------------------- ​ ∑​x​ ​s​ ​– ​( ​------------------- ​∑​N x​ ​ ​)​ 1​2 ​ i 2​ 2 ​i 2 ​y s​b ​= s​y -----------------------------​N 1​ ​ – (​ ∑​∑ x​ ​x​ ​ ------------------- ​ )​2 i 2 ​i​ Conductivity and Net EXPERIMENT 5 ​ Ionic Equations 2​ i i2 ​ 2 i 2​ m Backgroun d In this experiment, electrical conductivity will be used as a way of determining the number of free ions present in a substance and to use this information to draw conclusions regarding the type of bonding present in the substance. Types of Bonding There are three basic types of bonds. Two of which are ionic and covalent bonds. Ionic bonds are formed between elements with very different electronegativities. ​Electronegativity ​is the ability of an atom to draw electrons towards itself in a chemical bond. Generally, ionic compounds form between metals and non-metals and are identified by the transfer of an electron from the metal to the non-metal to form charged ions which are held together by electrostatic interactions known as ionic bonds. Covalent bonds form between elements with similar electronegativities. Generally, covalent bonds form between non-metals are characterized by the sharing of electron pairs between the atoms. Polar bonds are formed between two elements with different electronegativities, but which still share the electrons albeit unevenly. Polar covalent bonds are a hybrid of an ionic and a covalent bond. Nonpolar bond covalent bonds form between two elements with similar electroneg- ativities and thus share electrons fairly evenly. Electrolytes versus Non-electrolytes Substances may be classified by their electrical conductivity. ​Electronic conduction ​is a type of electrical conductivity that occurs in metals where charge is carried by electrons. In ​ionic conduction ​the charge is carried by ions. Substances which can conduct electricity are called ​electrolytes​. When ionic compounds are melted or dissolved in water, they form mobile ions that are able to conduct electricity. Polar covalent compounds such as acids and bases will sometimes disso- ciate or break apart in aqueous solution to form ions as well. There are two types strong electrolyte and weak electrolytes. Substances which do not conduct electricity are called ​non-electrolytes​. Examples of non-electrolytes are covalent molecules such as sucrose or table sugar (C​12​H​22​O​11​) and acetone (CH​3​CO- Chemistry 141 Grossmont College 5 ​ –47 Conductivity and Net Ionic Equations CHconduct ​3​). These substances are non-electrolytes because they are not composed of ions and cannot an electric current. Ionic compounds in their crystalline form are also considered to be nonelectrolytes because even though they are composed of ions, the ions are not able to move freely through the crystal and therefore are not able to conduct electricity. Strong electrolytes ​dissociate ~100% in aqueous solution. In chemical equations where ions are included (total and net ionic equations) write the ionized components, since they are the major spe- cies present in the solution. Examples of these are shown below: + Br​- ​ HBr ​(aq) + ​ ​→ ​H​3​O​+ ​(aq) ​ ​ H​2​O (l) (EQ 5.1) (aq) ​ + NO​ (EQ 5.2) -​ 3​ (aq) ​ HNO​3 (aq) ​+ H​2​O (l) ​ ​→ ​H​3​O​+ ​(aq) ​ + HSO​ - ​ (EQ 5.3) H​2​SO​4 (aq) + ​H3​​ O​+ ​(aq) ​ 4​ (aq) ​ ​ H​2​O ​(l) → ​ Weak electrolytes ​are substances that are able to conduct electricity, but conduct poorly. Examples of weak electrolytes are molecular substances that dissociate to a small extent such as weak acids and bases, and ionic compounds that have limited water solubility. They are ​slightly ionizable substances​. In chemical equations where ions are included (total and net ionic equations) write the complete weak acid formula, not the ions. The ions are only a minor component in the total solution. Examples are shown below: + CH​ CO​ +​ CH​3​CO​2​H ​(aq) + ​ H​2​O ​(l) H​ ​ 3​O​ (aq) ​ + F​- ​ HF ​(aq) + ​ ​H​3​O​+ ​(aq) ​ ​ H​2​O (l) (EQ 5.5) (aq) ​ + SO​ +​ HSO​4​- ​(aq) + ​ H​2​O ​(l) H​ ​ 3​O​ (aq) ​ (EQ 5.4) -​ 2​ (aq) ​ 3​ (EQ 5.6) 2- ​ 4​ (aq) ​ (EQ 5.7) NH​3 (aq) ​+ H​2​O ​(l) ​NH​4​+ ​(aq) ​+ OH​- ​(aq) ​ As you complete the experiment, notice the difference in conductivity between strong and weak acids and bases. What does this tell you about the relative degree of dissociation? What are the principal species present in the solution? How do we write these substances in ionic equations? Writing Chemical Reactions Reaction can be classified into two basic types: oxidation-reduction and double displacement. 1. ​Oxidation-Reduction (Redox) Reactions: Electrons are transferred from one reactant to another. Most simply oxidation is the loss of electrons and reduction is the gain of electrons. a. ​Combination reactions where reactants combine to form a new substance: A + E​→​AE ​(EQ 5.8) b. ​Decomposition reactions where reactants brake apart into new substances: AE ​→​A + E (​ EQ 5.9) c. ​Single Replacement Reactions involve the reactivity of an element is related to its tendency to lose or gain electrons; that is, to be oxidized or reduced. Generally speaking: A ​(s) ​+ BC ​(aq) → ​B ​(s) ​+ AC ​(aq) ​(EQ 5.10) ​ 5–48 ​Chemistry 141 Grossmont College Background where A is the more active element and replaces B in the compound. 2. ​Double Displacement Reactions (aka Ion Exchange Reactions) two aqueous solutions are mixed together to produce a precipitate, slightly ionizable substance, or a gas. AB ​(aq) + 5.11) a. ​Precipitate — formation of an insoluble compound. Solubility rules are ​ CD ​(aq) →​ ​ AD ​(?) + ​ BC ​(?) (EQ ​ given below: TABLE 5.1 ​Solubility Rules Soluble Ionic Compound Exception alkali metals, ammonium (NH​ Compounds Containing ​ b. ​Gas +​ 4​ ), nitrates (NO​3​-​), chlorates (ClO​3-​​ ), — bubbles or effervescence. Common gases include hydrogen gas, H​2 (g)​, oxygen gas, O​2 (g)​, hydrogen sulfide, H​2​S ​(g)​, ammonia, NH​3 (g)​, carbon dioxide, CO​2 (g)​, and sul- fur dioxide, SO​2 (g)​. When these compounds are “formed” they are unstable and decompose into gases and water: H​2​CO​3 (aq) → ​H​2​O ​(l) + 5.12) ​ ​ CO​2 (g) (EQ ​ H​2​SO​3 (aq) → ​H​2​O ​(l) + 5.13) ​ ​ SO​2 (g) (EQ ​ “NH​4​OH” ​(aq) → ​H​2​O ​(l) + ​ ​ NH​3 (g) ​(EQ 5.14) c. ​Slightly ionized substance — heat usually accompanies the formation of water, H​2​O, acetic acid, HC​2​H​3​O​2​, or any other slightly ionized compound (weak acids, weak bases). Writing Ionic Equations When you write ionic equations, you need to show the principal species present in the solutions. By testing the conductivity of a variety of solutions you can determine whether the principal species none perchlorates (ClO​4​-​) none acetates (CH​3​CO​2​-​) aluminum (Al​3+​) and silver (Ag​+​) chlorides (Cl​-​), bromides (Br​-​), ​ iodides (I​-​) Ag​+​, mercury(I) (Hg​2​2+​), and Pb​2+ fluorides (F​-​) alkaline earth metals and lead(II) (Pb​2+​) sulfates (SO​4​2-​) Ca​2+​, Sr​2+​, Ba​2+ ​insoluble Ag​+​, Hg​2​2+​, and Pb​2+ ​slightly soluble carbonates (CO​ Insoluble Ionic Compound Exception ​Compounds Containing ​ 2-​ ), phosphates (PO​4​3-​), 3​ oxalates (C​2​O​4​2-​), chromates (CrO​42-​ ​ ), ​silicates (SiO​4​2-​) alkali metals, NH​4​+ oxides (O​2-​), hydroxides (OH​-​) alkali metals, NH​4​+ ​soluble Ca​2+​, Sr​2+​, Ba​2+ ​slightly soluble sulfides (S​2-​) alkali and alkaline earth metals, NH​4​+ Chemistry 141 Grossmont College 5 ​ –49 Conductivity and Net Ionic Equations are ions or undissociated or undissolved particles. For each of the substances test for electrical con- ductivity and determine the principal and minor species present in the solution. EXAMPLE 5.2​Write the balanced conventional equation for the reaction of nitric acid and magnesium acetate. Remember that the conventional equation shows all species as neutral compounds. 2 HNO​3 (aq) ​+ Mg(C​2​H​3​O​2​)​2 (aq) ​→ ​Mg(NO​3​)​2 (aq) ​+ 2 HC​2​H​3​O​2 (aq) ​(EQ 5.15) EXAMPLE 5.3​Write the total ionic equation for the reaction. ​Remember that the total ionic equation shows all species as they appear in solution. + 2 NO​ - ​ + Mg​2+ ​ + 2 C​ H​ O​ - ​ → ​Mg​2+ ​ + 2 NO​ - ​ + 2 HC​ H​ O​ (EQ 5.16) 2 H​+ ​(aq) ​ 3​ (aq) ​ (aq) ​ 2​ 3​ 2​ (aq) ​ (aq) ​ 3​ (aq) ​ 2​ 3​ 2 (aq)​ the net ionic equation for the reaction. ​The net ionic equation shows only the species that undergo reaction. No spectator ions. + 2 C​ H​ O​ - ​ → 2 ​HC​ H​ O​ (EQ 5.17) ​ Net Ionic Equation: 2 H​+ ​(aq) ​ 2​ 3​ 2​ (aq) ​ 2​ 3​ 2 (aq) ​ Remember to simplify the coefficients EXAMPLE 5.4​Write when necessary. + C​ H​ O​ → ​HC​ H​ O​ (EQ 5.18) ​ H​+ ​(aq) ​ 2​ 3​ 2​- ​(aq) ​ 2​ 3​ 2 (aq) ​ EXAMPLE 5.5​Complete the following table: TABLE 5.6 Ions ​ Conductivity ​ Minor species in ​ Major species present in solution Few, ​5–50 C ​ hemistry 141 Grossmont College (None, ​ Many) ​ solution ​Before Rxn: ​HNO​3 (aq) ​Good Many H​+ ​(aq)​, NO​3​- ​(aq) ​n/a Before Rxn: ​Mg(C​2​H​3​O​2​)​2 (aq) ​Good Many Mg​2+ ​(aq)​, ​ , NO​ C​2​H​3​O​2​- ​(aq) ​n/a After Reaction Good Many Mg​2+ ​(aq)​ ​ , -​ 3​ (aq)​ HC​ H​ O​ 2​ 3​ H​+ ​ 2 (aq) ​ , C​ H​ O​ (aq)​ 2​ 3​ -​ 2​ (aq) ​Note that water is omitted as a major species since it a solvent. Notice that the conductivity is good before and after the reaction due to the presence of ions in the solutions before and after the reaction. Acetic acid is a poor conductor and only partially ionizes to give minor species. Procedure Procedure Part 1 - Conductivity Classification Test and record the conductivity of each substance and solution listed below using the method demonstrated by your instructor. Unless otherwise noted waste will go in the inorganic waste container. Then, after noting the range of conduc- tivities measured, classify each as having essentially no ions, a few ions, or many ions. For each substance record also the major and minor species present in the sample. Your data table may be similar to the one shown below: TABLE 5.7 Substance S​ AMPL E​ Bond Type (Polar covalent, Ionic) Ions (None, Conductivity ​ Major Few, Many) species present in solution Observations Minor species in solution ​ a. ​HBr ​(aq) , Br​- ​ polar covalent, fully ionized Good Many H​3​O​+ ​(aq)​ (aq) ​n/a All three bulbs lit. b. ​HF (​ aq) , F​- ​ +​ polar covalent, partially ionized Poor Few HF ​(aq) H​ ​ 3​O​ (aq)​ (aq) Two bulbs lit weakly. ​c. ​KCl ​(s) ​ionic, nonionized None None KCl ​(s) ​n/a No light. d. ​ ​Deionized water ​e. ​Tap water ​f. ​Sucrose C​12​H​22​O​11 (s) ​g. ​C​12​H​22​O​11 (aq) ​h. ​NaCl ​(s) ​i. ​NaCl ​(aq) ​j. ​0.1 M HgCl​2 ​k. ​0.1 M HCl ​l. ​0.1 M NaOH ​m. ​0.1 M NH​3 ​n. ​0.1 M NaCl ​o. Methanol, CH​3​OH ​(l) (organic waste) ​p. ​CH​3​OH ​(aq) CH​3​CO​2​H ​(l) r. ​ ​CH​3​CO​2​H ​(aq) (organic waste) ​q. ​Glacial acetic acid (Add water slowly to glacial acetic acid and record how the con- ductivity changes.) ​s. ​0.1 M CH​3​CO​2​H ​t. ​KClO​3 (s) (Test in a ​ (Heat crucible using a Bunsen burner.) Chemistry 141 Grossmont College 5 ​ –51 Conductivity and Net Ionic Equations Part 2 - Effect of Solvent crucible.) ​u. ​KClO​3 (l) ​molten Test the conductivity of the following solvents and mixtures. Put all waste in the organic waste container. Your data table may be similar to the one shown below: TABLE 5.8 Ions ​ Substance Conductivity ​ (None, ​ Few, ​ Many) Part 3 - Correlating Chemical and Conductivity Behavior Be sure to use similar amounts of each solid and acid. You will compare the conductivity of the acids with their reac- tion rates. Compare the rates of reaction (fast, medium, slow, or no reaction) of: carbonate, CaCO​3 (s)​, with 6 M acetic acid and 6 M hydrochloric acid; ​2. ​zinc with 6 M acetic acid and 6 M hydrochloric acid. 1. ​calcium Part 4 - Observing Ch...
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