Significant Figures: A Brief Tutorial

Transcription

1 Significant Figures: A Brief Tutorial Mr. Berkin *Please note that some of the information contained within this guide has been reproduced for non-commercial, educational purposes under the Fair Use provisions of U.S. Copyright Law. The author holds no claim to such material and no endorsement is intended or implied. Reproduction and dissemination of this guide in part or in whole (for any purpose) is strictly prohibited without the prior written consent of the author (Mr. Berkin). Mr. Berkin Significant Figures: A Brief Tutorial Page 1 of 8

2 Significant Figures (Sig Figs): Significant figures speak to the accuracy in one s calculations. Assuming that there has not been any mechanical error or human error, the greater the number of sig figs, the more exact your data will be. As such, it is always important to make sure laboratory equipment is properly calibrated and reset (zeroed) prior to taking each new reading. Also, ensure you subtract the tare weight (AKA the unladen weight, is the weight of an empty vehicle or container). In the lab, the gross weight (AKA the total weight) will likely be the weight of a chemical (the net weight) within a container (probably a watch glass). In the prior description, the weight of an empty watch glass would be the tare weight. Gross weight = Net weight + Tare weight It is important to understand that the number of significant figures is directly tied to the tool through which measurements are made. Weighing the same object using different types of scales (assuming they differ in accuracy) will produce different results (please see below). Moreover, calculations with numbers having different degrees of accuracy will produce different results (often substantially). The same object could have given these readings: Analytical balance: Weight = ± g Pan balance: Weight = 4.2 ± 0.1 g Most of the time sig figs are written without the ± value. In such instances, it is generally understood that the last digit contains at least 1 unit of uncertainty and that the value given for it is estimated. Students should not be confused by this concept. Although the last digit is estimated, it is still a significant figure as it derived from a measurement (a measured value) as opposed to an estimate. Using a meter stick to measure the length of a desk to the nearest tenth of a centimeter is an example of a measurement. Stating that the distance between Miami and Washington DC is 1000 miles would be an estimate. A measurement of 1000 miles (or 1000 ± 1 miles or miles) would be written in scientific notation as x 10 3 (4 sig figs). However, an estimate of 1000 miles would be written in scientific notation as 1 x 10 3 (1 sig fig); wherein, the zeros following the last non-zero digit are ignored (treated as placeholders ). To be most accurate, an estimate (as defined here) or a measured value of 1000 may contain one to four sig figs. The additional zeros are then termed placeholders. Unless explicitly told otherwise, treat all zeros at the end of an estimated value (following the last non-zero digit) to be placeholders. Treat all measured values as having the degree of certainty that is at least 1 unit and equal in magnitude to the last digit shown for the value (e.g. treat a measurement of as ± ; which, has 8 sig figs). For reference, a value such as 2900 ± 100 would only have 2 sig figs (see the rules below for further information). Mr. Berkin Significant Figures: A Brief Tutorial Page 2 of 8

3 Determining Significant Figures: Rules (Non-Calculations) ++ : 1. All non-zero digits are significant. 2. Zeros between non-zero digits are significant. 3. Zeros to the left of the first non-zero digit are not significant. 4. Zeros that are located both at the end of a number and to the right of the decimal point, are significant. 5. A number ending in zeros, that are located to the left of the decimal point, may or may not be significant. ++ These rules apply to values that are not exact (as in an exact measurement). They have no degree of uncertainty. Such values are valid to an infinite number of significant figures. Stoichiometric coefficients fall into this category. This is also true for conversion factors, such as 12 inches in 1 foot. Another example is that of (exactly) 29 students in a classroom This concept will be particularly important for calculations involving sig figs. Mr. Berkin Significant Figures: A Brief Tutorial Page 3 of 8

4 Examples: Rule 1: [ x 10 4 ] (5 sig figs) 126 [1.26 x 10 2 ] (3 sig figs) Rule 2: [ x 10 2 ] (5 sig figs) [ x 10 1 ] (5 sig figs) Rule 3: 0.12 [1.2 x 10-1 ] (2 sig figs) [1.2 x 10-2 ] (2 sig figs) [1.2 x 10-3 ] (2 sig figs) [1.2 x 10-4 ] (2 sig figs) [ or x 10 0 ] (5 sig figs) [4 x 10-6 ] (1 sig fig) Rule 4: [ or x 10 0 ] (5 sig figs) [ x 10 2 ] (6 sig figs) *Note the increase in accuracy within the following examples: 3.0 (2 sig figs) 3.00 (3 sig figs) (4 sig figs) (5 sig figs) Rule 5: 200 [2 x 10 2 ] (1 sig fig) [4 x 10 6 ] (1 sig fig - assume the value is not exact or a direct measurement) ± 1 [ x 10 6 ] (7 sig figs) *Note how the same value of 120 g is displayed differently in scientific notation as the accuracy (and level of confidence) changes. This is inherently based upon the accuracy of the measurement and as such, the tool used to take the measurement. As seen below, it is the accuracy, not the numeric value, which determines the number of sig figs (and how the measurement is displayed in scientific notation): 120 g (accuracy of ± 10 g) [1.2 x 10 2 ] (2 sig figs) 120 g (accuracy of ± 1 g) [1.20 x 10 2 ] (3 sig figs) 120 g (accuracy of ± 0.1 g) [1.200 x 10 2 ] (4 sig figs) Note: On your calculator, you may find x 10 written using E. For example, an estimate of may be written in scientific notation as 4 x 10 6 or as 4 E+6. Also, for reference, exponential notation differs from scientific notation as shown through this example: 120 could be written as 120 x 10 0, 12.0 x 10 1, or 1.20 x 10 2 using exponential notation, but can ONLY be properly written as 1.20 x 10 2 when using scientific notation. Sig fig rules are inherently incorporated into scientific notation; particularly, all of the digits expressed before the exponential term are significant. Mr. Berkin Significant Figures: A Brief Tutorial Page 4 of 8

5 Significant Figures - Mathematical Operations*: *Always be mindful that exact values are good for an infinite number of sig figs. All calculations employ the same principal that the accuracy of the final answer can be no greater than the [accuracy of the] least accurate measurement. Calculations Using Addition & Subtraction: The final answer can have no more decimal places than the least accurate measurement. Round to the last decimal place. It is highly advised that you line up your decimal points when performing calculations. Examples: g H 2 O (solvent) g salt (solute) g solution (6 places after the decimal point) (4 places after the decimal point) (2 places after the decimal point) (on calculator) (rounded ++ to 2 places in the answer 4 sig figs final answer) 19.2 g g g SUM = g Final Answer = 147 g (The last weight was only known to the nearest 1 g.) ++ Note: For values of five through nine, always round up (overestimating) and for values of zero through four, always round down (underestimating). Round off numbers ONLY at the end of calculations. This will prevent error from being carried through your operations. Mr. Berkin Significant Figures: A Brief Tutorial Page 5 of 8

6 Calculations Using Multiplication & Division: The final answer can have no more significant figures than the number being multiplied or divided with the lest number of significant figures (the least accurate measurement). Examples: (8 significant figures) x (5 significant figures) (on calculator) (rounded to 5 significant figures final answer) Cost of copper in a penny that is made of pure copper. Mass of a penny is 2.51 grams. The price of pure copper is 67 cents per pound g ( ) = lb lb ( ) = = 0.37 Notice that the value of 0.37 is the final answer when the calculations are properly rounded. This is a result of the factor of 67 cents/pound (containing 2 sig figs) and is a measurement that is not exact it is inherently understood that the true cost of copper lies on a continuum and unlike an independent conversion factor (like that of 12 inches/foot), this value is dependent and subject to change. This is also clearly different than the example of 29 being an exact number when exactly 29 students are counted in a classroom. Density of an object with a mass of g (6 sig figs) and a volume of 25.0 cm 3 (3 sig figs). Density = = g cm -3 = 1.16 g cm (13 sig figs) x 55.1 (3 sig figs) = = = 2.09 x (3 sig figs) Mr. Berkin Significant Figures: A Brief Tutorial Page 6 of 8

7 Recording Data In The Lab: Always remember to record one extra decimal place beyond the markings of the device used to measure the quantity of material in question. The extra decimal place is known as our estimated number. However, please remember that this is not an estimate (as previously defined) and is considered to be a significant figure. When in between the markings of a tool, try to divide the space between the nearest marking surpassed (one below) and the next one (one greater) into ten equal parts. Next, attempt to reasonably estimate the location. Please see the example below. Looking at this measurement, you should be able to determine that the nail is 2.85 cm. Note too that most people will measure the value to fall between 2.84 cm and 2.86 cm (± 0.01 cm). Reading a graduated cylinder s meniscus can sometimes be a bit more tricky due to the degree of concavity (or convexity with liquid mercury) and ensuring that your eye is level with the meniscus. Please note the following diagram. Mr. Berkin Significant Figures: A Brief Tutorial Page 7 of 8

8 Measurement & Parallax: Parallax is the apparent shift in the position of an object when that object is viewed from different locations. This can be a source of error within an experiment. As such, it is important to always take measurements and use measuring equipment dictated by their intended use. Always read ruler measurements where your eye is perpendicular to the surface of the ruler. Take readings on a graduate cylinder with your eye at the level of the meniscus. Measurement may be too long: Measurement may be too short: Mr. Berkin Significant Figures: A Brief Tutorial Page 8 of 8