Significant Figures, Measurement, and Calculations in Chemistry

Transcription

1 Significant Figures, Measurement, and Calculations in Chemistry Carl Hoeger, Ph.D. University of California, San Diego SigFig 1 Part 1: Measurements, Errors, and Significant Figures Carl Hoeger, Ph.D. University of California, San Diego SigFig 2 1

2 Measurements Science is about making observations and measurements; how do we convey to someone just HOW good/sure of our data we are? All quantitative measurements are made using some sort of device (analog or digital). Need value, units, and error. All data collection has errors associated with it: Determinate Errors: aka systematic errors; have a definite direction and magnitude and have an assignable cause. Indeterminate Errors: aka random errors or noise; errors that simply arise from uncertainties in a measurement, sloppy techniques, anomalous physical and environmental factors; these errors are typically untraceable. Determinate errors can be eliminated; Indeterminate errors can not. SigFig-3 Measurements (cont.) Consider the measurement of a length of bar: Is it 8.3, 8.32, or 8.323? Or 8.322? What are the units? Error? Depends! Calibration Multiple measurements to give means and standard deviations: ( ) x i x x i x = ; s ; CV s 100 N = N 1 = x SigFig-4 2

3 Accuracy and Precision Accuracy: closeness of measurement to the true value Precision: closeness of individual measurements to one another SigFig-5 Significant Figures-1 Two types of numbers exist: measured and exact numbers. Measurements or determined values contain significant figures and must convey to the reader the certainty of your measurement. This is where significant figures (SigFigs) comes in. Exact values (exact numbers) have an unlimited number of significant figures and never detract from a measurement or its subsequent modification or use. SigFig-6 3

4 Exact Values Exact numbers can either be cardinal (counted) or defined. Cardinal or counted values represent values that, by their very nature or how they are obtained, are indivisible without destroying the identity of the unit. 10 dogs or 85 cars; 123 molecules; 13 silver atoms; 2500 electrons; 2 chloride ions in 1 calcium chloride. Defined values are those for which the value has been exactly defined. 1 in these expressions is EXACTLY 1 1 in = 2.54 cm; 1000 m = 1 km; 1 mg = g; 1 lb = 16 oz All metric to metric or English to English conversions What about constants? c = x 10 8 m/sec? SigFig-7 Measured Values Determined values have SigFigs associated with them. Reported values must convey the level of precision of the measurement to a detached third-party (i.e. an unseen colleague). A measured value is comprised of digits we know for certain plus ONE we are uncertain of (the last digit to the right) ml Consider the graduated cylinder to the left; we know the first two digits with certainty (you can read them directly from the markings) but the last 6 is estimated. Another person might read it as a 5 or a 7, hence it is uncertain. By convention, the last digit is considered to be variable by ±1 unit (unless specified otherwise) SigFig-8 4

5 Significant Numbers/Digits When given a measurement or a determined value, what digits are significant? All non-zero integers always count as sig figs; There is assumed to be a ± 1 difference in the last significant digit shown; Zeros are tricky--depends on the type: Captive zeros: always significant; 1001 cars; cm Leading zeros: never significant; 0010 screws; L Trailing zeros: only significant if a decimal point is present 210. Atoms; mm; Angstroms NOTE: for any number expressed in scientific notation, all digits shown are significant (except leading zeros) SigFig-9 Sig Figs: Calculation Considerations Determined by the type of mathematical operation carried out Multiplication or Division: least precise number determines sf in answer; = (1sf ) 4 Addition or Subtraction: Answer can have no more decimal places than the one with the fewest; ( ) = (1 dp) Logs: No more digits to right of decimal than sf in original number; 2 log = = (3 dp; note : the 1 is not a sf ) Antilogs: No more sf in answer than digits to right of decimal in original number; antilog 2.86 = = = (2 sf ) SigFig-10 5

6 Rounding A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. If you are rounding to the tens, hundreds place or higher, you must put zeroes in the lesser places (the ones place, for example) to indicate to what place you have rounded. Use the following guidelines when rounding off numbers: When the first digit dropped is less than 5, the last digit remains unchanged. When the first digit dropped is 5 or greater, the last digit retained is increased by 1. When rounding, ignore ALL digits after the first digit you are dropping in making your decision. Always round at the end of any set of calculations. Round each to 3 sf: i) 3203; ii) 1.768; iii) ; iv) SigFig-11 Combined Calculations In calculations involving BOTH addition/subtraction AND multiplication/division, significant figure considerations must be noted before and after each calculation involving addition or subtraction. You still round at the end. A simple rule of thumb is as follows: Take note of how many sig figs the multiplication and division steps allow; Take note of how many decimal places the addition and subtraction steps allow; Your answer can have no more decimal places than allowed by the addition/subtraction but no more sig figs than allowed by the multiplication/division. Usual way to do this is to get your answer, set the correct number of sig figs and then lessen the number of decimal places if necessary. SigFig-12 6

7 Scientific Notation When expressing an extreme large number such as the number of atoms in a mole, or a very small number such as the mass of a single atom, scientists use scientific notation. The basic format of scientific notation is M x 10 n, where M is a real number between 1 and 10 and n is a whole number = 1 (note: any number to the 0 power is 1) 10 1 = = 10 * 10 = = 10 * 10 * 10 = = 1 / 10 = = 1 / 10 / 10 = = 1 / 10 / 10 / 10 = SigFig-13 Scientific Notation (con t) For practical reasons it is common to express all numbers that are larger than 1000 or smaller than 0.01 in scientific notation. When converting a number to scientific notation, move the decimal place to the right or to the left until what remains is a number to the left of the decimal place between 1 and 9. Moving the decimal to the right will result in a negative power of ten, to the left in a positive power of ten: Y Y Y Y Y. X X X X X Express 3456 in scientific notation: = x 10 3 Express in scientific notation: = 6.70 x 10-4 SigFig-14 7

8 Power Notation: px values It is not always easy or convenient to use scientific notation, especially when comparing or tabulating long lists of numbers in scientific notation; used primarily when discussing acidity and basicity. Use px values; by definition: px = log X Therefore: X px X = 10 px px X SigFig-15 Units Just as important as the value are the units associated with that value, as they tell us about what we are measuring! Units need to be as detailed as necessary: 12.7 lbs vs lbs of hamburger vs lbs of lean hamburger; 3 atoms vs. 3 sodium atoms vs. 3 sodium ions SI or metric units almost exclusively in chemistry. Fundamental units vs. Derived units: Length: meter (m); Mass: kilogram (kg); Time: second (sec); Temperature: Kelvin (K); Chemical amount: mole (mol) Energy: joule (J); Pressure: atmosphere (atm); Volume: liter (L) Units quite often begin with modifiers ( power prefixes ) to relay size of quantity without having to use scientific notation: 5.32 x 10 3 J or 5.3 kj SigFig-16 8

9 Common Unit Modifiers ( Power Prefixes ) giga - G mega - M kilo - k deci - d centi - c milli - m micro nano - n pico - p fempto - f atto - a yocto - y SigFig-17 Questions? Comments? This presentation has been brought to you through the generous support of the University of California, San Diego s Instructional Improvement Program. If you have questions, comments, or wish to use these presentations in your University s courses, please feel free to send an to Dr. Carl Hoeger at chemcook@gmail.com This is episode SigFig1 ; if you have specific comments or questions regarding this episode please note this in your SigFig-18 9