Measurements, Sig Figs and Graphing Chem 1A Laboratory #1 Chemists as Control Freaks Precision: How close together Accuracy: How close to the true value Accurate Measurements g Knowledge Knowledge g Power Error: What really happens Accuracy = Precision There is always uncertainty 1
About those errors Systematic errors: Consistent error in the same direction due to a bias in the data collection like an improperly calibrated balance That can t be right!!! Random errors: Unpredictable fluctuations in a measurement Occur in different directions from the true value Understanding uncertainty: Significant Figures All laboratory measurements have uncertainty associated with the last digit of the value. 8.24 or 8.25 cm? 52.26 or 52.27 ml? We have to keep track of the uncertainty to know how accurate and reliable our results are. We identify the number of significant figures or digits in each measurements and use simple rules for tracking sig figs in our calculations 2
Our Uncertain Equipment For measuring volume: Graduated containers: Cylinders Beakers Burets Pipets Volumetric containers Pipets Flasks Calibration temperature; accuracy To Contain (TC) vs To Deliver (TD) Reading volumes Graduated Cylinders Reading what goes into it Burets Reading what went out of it Meniscus 1.42 ml Uncertain digit 1 0.00 ml --- 1 2 3 3.67 ml --- 4 5 6 7 0 3
Recording measurements Recording a measurement with the correct number of significant figures is necessary to show the precision of the measuring device correctly. Unless stated otherwise, the uncertainty in a number is taken to be ± one unit in the estimated digit. What are the measurements with correct # of sig figs? 10 ml graduated cylinder 100 ml graduated cylinder 50 ml buret Ruler: in and cm Working with measurements Key Steps: 1. Record the measurements carefully. 2. Repeat the measurement to increase its reliability, then calculate the average or mean as the best value 3. Establish the probable limits of uncertainty Precision Range = Highest Value Lowest Value Accuracy: Deviation from true value Accuracy is often reported as percent error: Percent Error = Experimental Value Accepted Value x 100% Accepted value Keep track of significant figures. Do not write a reading of 50.00 ml as 50 ml or as 50.000 on your worksheet. 4
Estimating the Last Digit For instruments marked with a scale, you get the last digit by estimating between the marks. Mentally divide the space into 10 equal spaces, then estimate what the last digit should be. The total number of digits in a measurement, including the last estimate are called the significant figures or sig figs The last digit is just our best estimate of the true value. It tells us that the actual value is between 6.30 and 6.32 ml 100 ml graduated cylinder 10 ml graduated cylinder 52.5 ml 6.31 ml Both of these measurements have 3 sig figs Significant Figures The non-placeholding digits in a reported measurement are called significant figures. Some zeros in a written number are only there to help you locate the decimal point. Significant figures tell us the range of values to expect for repeated measurements. The more significant figures there are in a measurement, the smaller the range of values. Therefore, the measurement is more precise. 12.3 cm has 3 significant figures and its range is 12.2 to 12.4 cm. 12.30 cm has 4 significant figures and its range is 12.29 to 12.31 cm. 5
Zeros & Significant Figures All non-zero digits are significant. 1.5 has 2 significant figures. Interior zeros are significant. 1.05 has 3 significant figures. Trailing zeros after a decimal point are significant. 1.050 has 4 significant figures. Leading zeros are NOT significant. 0.001050 has 4 significant figures Zeros at the end of a number without a written decimal point are ambiguous and should be avoided by using scientific notation. 150,000 is ambiguous How Many Sig Figs? How many significant figures are in each of the following numbers? 3.15 x 10 16 0.007510 10200. 150.108 6
Significant Figures Review When multiplying or dividing measurements with significant figures, the result has the same number of significant figures as the measurement with the fewest number of significant figures. 5.02 89,665 0.10 = 45.0118 = 45 3 SFs 5 SFs 2 SFs 2 SFs. When adding or subtracting measurements with significant figures, the result has the same number of decimal places as the measurement with the fewest number of decimal places. 5.74 + 0.823 + 2.651 = 9.214 = 9.21 2 dec. pl. 3 dec. pl. 3 dec. pl. 2 dec. pl. Multiplication/Division with Addition/Subtraction When doing different math operations on measurements, identify the significant figures for any intermediate answer, then do the remaining steps. Round only at the end for Mastering Chemistry problems. Follow the standard order of operations. Please Excuse My Dear Aunt Sally. n - 3.489 (5.67 2.3) = 2 dp 1 dp 3.489 3.37 = 12 4 sig figs 2 sig figs 2 sig figs 7
Sig Figs and Logarithms The logarithm (or log) of a number to a given base is the power or exponent to which the base must be raised in order to produce that number. If x=10 y ; y = log 10 (x) For example: the log 10 of 100 is 2, because 2 is the power to which 10 must be raised to get 100: 10 2 = 100, so log 10 100 = 2. Characteristic and mantissa of a logarithm Characteristic = number on left of decimal= power of 10 Mantissa = number on right of decimal g # of sig figs Log (3.000) = 0.4774 4 sig figs Log (3.00) = 0.477 3 sig figs Antilogs: # of sig figs still comes from mantissa Antilog (5.89) = 10 5.89 = 7.8 x 10 5 2 sig figs Exact Numbers vs. Measurements Sometimes you can determine an exact value for a quality of an object. Often by counting. Pennies in a pile. Sometimes by definition There are exactly 1000 meters in 1 kilometer. There exactly 2.54 cm/in From integer values in equations. In the equation for radius of a circle (r=d/2), the 2 is exact Exact or exactly implies an infinite number of sig figs If we use an exact number in a calculation, we don t consider it when we determine the number of sig figs 8
Your Mission in lab Today Test your measuring skills; pay attention to accuracy of the equipment and record measurements with correct # of sig figs Each bench top will be given an identifying number (1 8, write it down) 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 Examine each piece of equipment and fill out Data Table I, including units. Write neatly!! Part B: Each person must individually record the volume of water in each piece of equipment in the spaces provided. Do not change any of the volumes of water. Your Continuing Mission Part C: Compare your measurements in Part B with the other people at your bench top and answer the questions Part D: Using the data provided and graph paper, make a graph (by hand) of total mass of water plus beaker (yaxis) versus total volume (x-axis) of water in the beaker following graphing rules given in Appendix A. Plan your axes so you can extrapolate the y-intercept. Perform the calculations and answer the questions based on your graph. Complete follow-up questions before leaving lab 9
Handy Things to Know Range: Highest value Lowest value Provides estimate of precision Accuracy: Experimental Value Accepted Value x 100% Accepted Value Estimated uncertainty: ± 1 unit of estimated digit If volumetric glassware is not marked with the uncertainty, estimate ± 0.2 ml for volumetric flasks and ± 0.02 ml for a volumetric pipet. Why such a big difference? What does that ± 0.02 ml mean for a 20 ml pipet? For Next Week Tuesday: Safety Quiz must pass with 85% or better Completed Measurements assignment due at beginning of lab No PreLab, but review Nomenclature Worksheet ; we will work on it in class. Thursday: Completed PreLab for Density of 7-Up lab due at beginning of lab period Nomenclature Worksheet will be due at beginning of lab 10