Measurements, Significant Figures and Graphing

Transcription

1 Measurements, Significant Figures and Graphing The objective of this exercise include: To become familiar with some common general chemistry laboratory equipment, its uses and limitations. To learn the difference between precision and accuracy. To learn about types of errors. To learn how to determine or estimate uncertainties in measurements. To learn the rules of significant figures and apply them. Pre laboratory Reading: Brown, LeMay Chapter 1: sections 4, 5 and 6 & Bursten Appendix A: Mathematical Operations Laboratory Manual Appendix A: Treatment and Representing Experimental Data Appendix B: Laboratory Equipment and Use: Calibrated Glassware, The Buret, The Volumetric Pipet, The Graduated Pipet and The Volumetric Flask Background: Measurements are a central part of the science of chemistry. Your textbook contains a large number of scientific facts and several scientific theories. The facts were obtained and the theories are supported by carefully made measurements. In order for experimental results to be meaningful, the experimental chemist must: 1. Record the results of his or her measurements carefully. 2. Repeat the measurement to increase its reliability. Since individual results from a series of repeated measurements will seldom be the same, a best value is obtained by taking the average, or mean, x. 3. Establish the probable limits of uncertainty that can be placed on the measurement. When a measurement is made, one of the goals is to achieve as high a degree of accuracy and precision as possible. Do not confuse precision with accuracy. Precision is defined as how close a series of measurements of the same quantity are to each other or, in other words, how reproducible the results are. The precision of a series of measurements of the same quantity can be quantitatively expressed in terms of the range in the results: Range = Highest Value Lowest Value The smaller the range the better the precision of the data. Accuracy refers to how close the measurement is to the true or accepted value. Whereas precision can be determined from the measurements themselves, without knowledge of the true or accepted value, in order to determine accuracy you must know the true or accepted value. Accuracy is often quantitatively reported in terms of percent error: Percent Error = Error in measurement Accepted value x 100% where Error in Measurement = Experimental Value Accepted Value The sign of the calculated error, either plus or minus, is retained when reporting a percent error, because it indicates whether the result was either too high or too low, respectively, from the true Foothill College-Chemistry 1A, Dr. Larson page 1 Last Modified 8/23/11

2 or accepted value. The smaller the percent error, regardless of the sign, the better the accuracy. Errors associated with making measurements can be divided into two types: systematic and random. A systematic error causes a measurement to always be too low or too high. Systematic errors arise from a faulty measuring device, a defect in the procedure or a consistent mistake in using the device. One example is an improperly zeroed balance. A systematic error affects the accuracy of each measurement. Random errors result in individual measurements that are just as likely to be too high as too low. The magnitude of random error depends on the precision of the measuring device and the skill of the person making the measurement. A small random error results in a high degree of precision. Random errors cannot be avoided, since there is always some degree of uncertainty in every measurement. This means that the average of several repeated measurements will be more reliable than any single measurement. As a final note, although good precision is often an indication of good accuracy, this need not always be the case. It is possible to have low accuracy due to a systematic error, but still have high precision. Each instrument (e.g., ruler, beaker, thermometer, balance, etc.) you use in the laboratory has a precision that determines the uncertainty of measurements, due to random error, taken with that instrument. The precision of a measuring device is usually expressed in terms of a ± value indicating the limitation of the device. The common instruments you will use in General Chemistry can be divided into two types: those that have a graduated scale and can make measurements over a range of values (e.g., ruler, thermometer, balance, graduated cylinder, graduated pipet, beaker) and those that measure a single, fixed volume of a liquid (e.g., volumetric flask, volumetric pipet). The distance between graduation marks on a ruler, thermometer, buret or other glassware may be subdivided into ones, tenths, hundreds or other divisions depending on the precision of the device. A 50 ml graduated cylinder, for example, has graduation marks at each ml. Since the experimenter can estimate between the graduation marks, the volume can be measured and recorded to the tenth of a ml, as illustrated below. A buret, on the other hand, has graduation marks at each one-tenth ml and the hundredth place can be estimated. Therefore, an extra digit to the right is gained when the buret is used and we say that the buret is more precise. Rule of Thumb: For instruments with graduation marks, record the measurement to 1/10 or 0.1 of the SMALLEST division. 50 ml An illustration showing the top part of a 50 ml graduated cylinder containing a liquid is shown here. Let s say that a person records the volume of the liquid in the cylinder as 48.5 ± 0.2 ml. How many significant figures does the measured volume of 48.5 ± 0.2 ml have? The answer is three, the 48 we know for certain and the 5 that was estimated. In a number representing a scientific measurement, the last digit to the right is taken to be inexact (it is estimated) and is counted as a significant figure. The recorded uncertainty of ±0.2 ml indicates the precision of the device. The uncertainty of ±0.2 ml indicates that the volume actually lies somewhere in the range of 48.3 to 48.7 ml. You may read the volume as 48.3 ml, your friend may read it as 48.7 ml; both 40 ml are within the estimated ±error range. Recording a measurement with the correct number of significant figures is critical in order to reflect the precision of the measuring device correctly. How was the uncertainty of ±0.2 ml determined? It was determined based upon the smallest scale division of the instrument and how sure the person making the measurement believed they were about the value of the estimated digit. Usually the uncertainty is within the range of ±1/10 of the smallest scale division up to ±1/2 of the smallest scale division. In this example that would be from ±0.1 ml up to ±0.5 ml and the person making the measurement decided that ±0.2 ml was a reasonable uncertainty. If the uncertainty in a measurement is not stated, then it is assumed to be ± one unit in the estimated digit. For example, a reported value of 48.5 ml would imply an uncertainty of ± 0.1 ml (48.6 ± 0.1 ml). If the uncertainty in the measurement is not ± one unit in the estimated digit, then the recorder has a responsibility to report the actual uncertainty. Note that an uncertainty, by definition, has only one significant figure. In this example, since we are uncertain about the reading in the tenths place (±0.2 ml), it does not make sense to report an uncertainty in any digit further to the right. In the previous graduated cylinder example, the person making the measurement estimated the uncertainty and reported it as an absolute uncertainty. Absolute uncertainties carry the same units as the measurement itself. For some instruments the uncertainty is conveniently given on the device as either an Foothill College-Chemistry 1A, Dr. Larson page 2 Last Modified 9/14/11

3 absolute uncertainty or a percent uncertainty. You should always check the measuring devices you use to see if the uncertainty is given on the device and, if given, record this uncertainty with your measurement. For example, a graduated pipet may have an absolute uncertainty of ± 0.03 ml written on it. For volumetric flasks and volumetric pipets the absolute uncertainty is often indicated on the instrument. If it is not given, a good rule of thumb for the volumetric flasks and pipets most commonly used in the General Chemistry lab is an uncertainty of ±0.2 ml for flasks and ±0.02 ml for pipets. The volume delivered from a 20 ml volumetric pipet with an uncertainty of ±0.02 ml is recorded as ± 0.02 ml, indicating that the volume is somewhere between ml and ml. If reported as a percent, the uncertainty in this pipet would be ±0.1%. As another example of a percent uncertainty, consider a 200 ml beaker with a uncertainty of ±5% written on the beaker If you measure liquid using this beaker, the measured volume of liquid has an uncertainty of ±5% of 200 ml, or ±10 ml. Thus, if you fill the beaker to the 150 ml calibration line, the measured volume is 150 ml with an absolute uncertainty of ±10 ml. Note that the recorded volume in this case has two significant figures, not three. When using this device it would be incorrect to report the volume measured to a decimal place that is smaller than the tens place since this digit is the inexact digit. For instance, it would be incorrect to indicate a volume of 152 ml, even if you observe that the liquid level is slightly above the 150 ml calibration line; the precision of the device does not allow you to read the volume to the ones place. It should now be clear that the concept of significant figures is directly linked to precision and NOT to accuracy. Consider measuring the mass of a piece of gold several times with a digital balance. You will obtain nearly the same measurement each time, within a small random error in the rightmost digit that is characteristic of the balance. You will have determined the mass of the gold with high precision. However, if the balance is not calibrated correctly, reading either consistently too high or too low (a systematic error), the mass of the gold will be inaccurate. The accuracy of the balance can be checked using standard weights of known mass, something you would definitely want to do if you were buying or selling the gold! Well-trained analytical chemists also have methods for checking the accuracy of their volumetric glassware. Remember also that glassware that is designed to accurately measure volumes is calibrated for a certain temperature. In such cases, the temperature at which the glassware is calibrated is given on the glassware itself and the most accurate measurements will be obtained at the indicated temperature. Glassware that is designed to accurately measure volumes of liquids is calibrated to either contain an amount of liquid measured in the glassware or to deliver an amount of liquid from the glassware You will find that glassware is often marked with the letters TC (to contain) or TD (to deliver). To obtain accurate volume measurements, we must understand the meaning of the notation TC and TD. For example, a 50 ml graduated cylinder that is marked TC and is filled to 25.5 ml CONTAINS 25.5 ml of the liquid. When the liquid is poured out of the cylinder, some of the liquid will adhere to the walls of the cylinder so that less than 25.5 ml is actually poured out, an inaccurate result. If it is essential that the experimenter know the amount of liquid poured out of the measuring device accurately, then a TD device is preferred. A TD device DELIVERS the measured volume. Laboratory Exercise Safety: Remember to always wear your safety glasses while in the chemistry laboratory. Equipment: Each bench top will be given an identifying number (1 through 8) and will have the following equipment set up. 100 ml volumetric flask filled to the mark with water 10 ml volumetric pipet 5 ml graduated pipet 250 ml beaker partially filled with water 100 ml graduated cylinder partially filled with water 10 ml graduated cylinder partially filled with water 50 ml buret partially filled with water Procedure: Using the equipment provided on your lab bench top, complete parts A through C on the data and report sheets that follow. DO NOT change the volumes of water in the glassware. Your instructor will also record the volumes in order to check your answers. Foothill College-Chemistry 1A, Dr. Larson page 3 Last Modified 9/14/11

4 Data and Report: Lab Bench Number: Part A: This part can be completed together with the other people at your lab bench. For each piece of glassware given complete the following table. Make sure you include units where applicable! Some spaces may be left empty if the information is not given or not applicable for a particular device. In some cases, the uncertainty may be given on the glassware, in other cases you will have to either estimate the uncertainty or refer to the background reading for this exercise. Remember, you must include a unit on your uncertainty. Data Table I: Information About Some Glassware Used in Chemistry Measuring Device 250 ml beaker 50 ml buret Temperature at which the glassware is calibrated. Is it To Contain (TC) or To Deliver (TD)? Is the glassware graduated or designed to measure a fixed volume? For graduated glassware give the smallest scale division. What is the uncertainty (precision) of the glassware (± x)? 10 ml volumetric pipet 100 ml volumetric flask 100 ml graduated cylinder 10 ml graduated cylinder 5 ml graduated pipet Questions: 1. Which of the given measuring devices would you use to do the following: a. Measure approximately 150 ml of a liquid? b. Measure 50.0 ml of a liquid? c. Deliver ml of a liquid? d. Deliver between 0.00 and 5.00 ml of a liquid? Foothill College-Chemistry 1A, Dr. Larson page 4 Last Modified 9/14/11

5 2. Why is the temperature at which laboratory glassware is calibrated important? More specifically, how does the temperature affect the volume measurements? Part B: Complete this part individually. You will compare your results with the results of the other people at your lab bench when finished. 1. Report the volume of water in each of the following devices to the correct number of significant figures. Include the uncertainty in your measurement. a. 100 ml volumetric flask filled exactly to the calibration mark: b. 250 ml beaker: c. 100 ml graduated cylinder: d. 10 ml graduated cylinder: 2. Record the reading for the water level in the 50 ml buret to the correct number of significant figures. Include the uncertainty in your measurement. (Remember that in this case you are recording the level of the water and that the scale increases numerically downward. You are not recording the actual volume of water in the buret. This can seem strange at first. Reading about the buret in Appendix B of the Lab Packet will help you understand.) Part C: Compare your measurements in Part B with the other people at your lab bench. 1. One of the measurements should be the same for everyone at your lab bench. Which one is it and why? 2. The other measurements made should vary slightly within your group. Why do they vary? Foothill College-Chemistry 1A, Dr. Larson page 5 Last Modified 9/14/11

6 Part D: Graphing In an experiment, Bud N. Chemist measured the mass of a clean, dry beaker. He then used a buret to measure water into the beaker and reweighed the beaker plus water. He repeated this, measuring out additional water and recording the total mass of water plus beaker each time. The results of his measurements are given below: Temperature of Water: 20.0 C Total Volume of Water Added (ml) Total Mass of Beaker Plus Water (g) Using suitable graph paper, make a graph of total mass of beaker plus water (y-axis) versus total volume (x-axis) of water in the beaker following the rules for good graphing given in Appendix A. Plan the range of your axes so that you can extrapolate to find the y-intercept. Do not forget to label your axes (with units) and title your graph. Draw the best straight line through your four points being sure to extrapolate to the y-intercept. Staple your graph to the end of this lab when finished. a. Using two points near the ends of your line that are not your data points determine the slope of the line. (Remember, the slope of the line will have units of Y units/x units). Show your calculation with units for the slope below. Use correct significant figures in your answer. Slope of line = b. From the graph read and record the y-intercept of the straight line (do not forget the units). Y-intercept 2. Both the slope of the line and the y-intercept have physical meaning in this simple exercise. a. What physical property of water does the slope of the line represent? (Hint, let the units of the slope help you.) b. What does the y-intercept represent? Foothill College-Chemistry 1A, Dr. Larson page 6 Last Modified 9/14/11

7 3. Compare your slope value to the accepted value given in the CRC Handbook of Chemistry located in the laboratory. Calculate the percentage error in your experimental value. Show the calculation below. This gives the accuracy in Bud s results. Accepted Value: Calculate the percent error: Follow-Up Questions: Be complete in your answers. Show units in set-up and answers and report all answers to the correct number of significant figures. 1. Give the absolute uncertainty (implied precision) as a (± value) that is indicated in the following recorded values: a ml b g c. 1.30x10 3 atm 2. A 50 ml beaker has an uncertainty of ±10% written on it. What is the absolute uncertainty for this beaker? 3. Do the following calculations. a cm x 2.8 cm = b g g = c. 5.19x10 2 cm cm cm = d. [(2.841 x 10 4 ml) (1.2 x 10 3 ml)] x g/ml = e. Convert 925 C to Kelvin = 4. Express the results of the following to the proper number of significant figures. a. log (6.19) = b. antilog ( 7.01) = 5. A block of metal with dimensions 5.2 cm x 2.1 cm x 4.6 cm has a mass of g. Calculate the density of the metal. 6. A sample of a certain compound weighing g is found to contain g of carbon and g iodine. The only other element present is hydrogen. What is the percent by mass of hydrogen in the compound? Foothill College-Chemistry 1A, Dr. Larson page 7 Last Modified 9/14/11

8 7. The accepted value for the melting point of pure aspirin is 135 C. You measure the melting point of aspirin and obtain 132 C, 133 C, 139 C and 140 C in four separate trials. Your partner finds 138 C, 137 C, 138 C and 139 C. Remember to show your work for the following questions: a. Calculate the range in the measurements for both you and your partner. Range in Your Measurements = Range in Your Partner s Measurements = b. Calculate the average of your measurements and of your partner s measurements. Your Average = Your Partner s Average = c. Calculate the percent error in your average and in your partner s average. Percent Error in Your Average = Percent Error in Your Partner s Average = d. Whose results are most precise? How did you decide? e. Whose results are most accurate? How did you decide? f. Does either data set appear to have a systematic error? Explain how you decided. Foothill College-Chemistry 1A, Dr. Larson page 8 Last Modified 9/14/11

9 Pre-laboratory Exercise (2 pages): Complete the following and turn it in at the start of lab lecture. Your answers are expected to be complete and based upon the background information provided in this experiment. 1. Define the term precision: 2. Define the term accuracy: 3. Give the following formulas: (You must memorize these.) a) The formula for calculating the range of a series of measurements of the same quantity. b) The formula for calculating the percent error of a measurement. c) Which of the above, range or percent error, is used to quantitatively describe: i) accuracy? ii) precision? d) What does a negative percent error indicate about a measured value? Foothill College-Chemistry 1A, Dr. Larson page 9 Last Modified 9/14/11

10 4. The mass of a piece of gold is measured twice, once using a top loading balance and once using an analytical balance, with the following results: Top loading g Analytical g a) Which of the two measurements is MOST PRECISE? (i) top loading (ii) analytical (iii) cannot determine without more information b) Which of the two measurements is MOST ACCURATE? (i) top loading (ii) analytical (iii) cannot determine without more information 5. Define the term random error: a) What does the magnitude of random error depend upon? b) Consider an average value obtained from repeated measurements of the same quantity. Does random error primarily affect the precision of the average, the accuracy, or both? 6. Define the term systematic error: a) What factors can cause a systematic error? b) Consider an average value obtained from repeated measurements of the same quantity. Does a systematic error primarily affect the precision of the average, the accuracy, or both? 7. Give the number of significant figures in each of the following: g cm 1.60 x 10 9 m 52.0 C 8. Round each of the following to 3 significant figures: (Use scientific notation where needed to avoid ambiguity.) g m cm 3133 ml Foothill College-Chemistry 1A, Dr. Larson page 10 Last Modified 9/14/11