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UNIT 6 - SCIENTIFIC NOTATION AND SIGNIFICANT FIGURES CH1030 Mark Stacey SCIENTIFIC NOTATION Science often has to use very large or very small numbers. This would be very difficult to quickly and neatly write out or to communicate. Scientific Notation is a form that allows very large or very small numbers to be expressed easily. SCIENTIFIC NOTATION Scientific notation uses an exponent of 10 along with a base number to communicate a number. 12000.00 Becomes 1.230000 x 10 4 SCIENTIFIC NOTATION An easy way to convert to scientific notation is to count the number of times the decimal has to be moved to place it at the ones position. 12000.00 4 places left 1.230000 x 10 4 SCIENTIFIC NOTATION For large numbers, this will mean moving the decimal to the left. This results in the 10 in scientific notation having a positive exponent. 12000.00 4 places left 1.230000 x 10 4 SCIENTIFIC NOTATION For small numbers (numbers less than 1), the decimal is moved to the right. In scientific notation, this results in a negative exponent. 0.00000123 6 places right 1.230000 x 10 -6 SI UNITS Chemists worldwide have agreed on a single system of units, the SI Units (Système International d'Unités) so that measurements are understood and shared accurately. However, for many everyday calculations, other units are occasionally used. Published research will generally use SI Units. SI UNITS Most of these units should be familiar from everyday life. Note that the kilogram (kg) is used as the base unit, not the gram (g). This is because a single gram is a very small amount of material, making kilograms more useful in most situations. (1 kg = 1000g) SI UNITS When larger or smaller amounts of things need to be measured, the SI system uses the standard metric prefix system. SI UNITS Units for “derived” measurements have direct and simple translations to their SI components. SI UNITS – CELSIUS/KELVIN The most commonly used non-SI unit is Celsius for temperature in place of Kelvin. Celsius is avoided as its negative values make using them in formulas difficult. However, as the difference in one degree Celsius is equal to one Kelvin, they are equivalent when talking about temperature change (eg: a temperature increase of 5˚C is the same as 5K) SI UNITS – CELSIUS/KELVIN Converting from Celsius and Kelvin is easy: ˚C = K - 273.15 K = ˚C + 273.15 ACCURACY AND PRECISION Accuracy and precision are often used interchangeably in everyday speech. In science, they have specific meanings. Accuracy refers to how close a measured value is to the true value. Precision refers to how consistent a device is at measuring things. ACCURACY AND PRECISION An accurate device obtains values close to the true value. Example: If a liquid is at 25˚C and a thermometer records values between 24 and 26, it is fairly accurate. An inaccurate thermometer could record numbers between 27 and 28, or between 23 and 24. ACCURACY AND PRECISION A precise device obtains consistent results (regardless of how accurate they are to the true value). If a pH meter was used on an acid and returned results of 3.0, 3.1, and 2.9 – it is fairly accurate. If another meter recorded results of 2.5, 3.0 and 4.0, it is much less accurate. Again, accuracy does not reflect how close the values are to the true value (that is accuracy) but is only concern with how consistent the values are. ACCURACY AND PRECISION ACCURACY AND PRECISION Naturally, scientists aim for experimental tools and methods that are both accurate and precise. Scientists state their level of accuracy and precision when reporting their work. This can be shown mathematically or graphically. ACCURACY AND PRECISION Graphs like this show the potential error in their data using “I”-shaped bars. This suggests that the true value is within that range. SIGNIFICANT DIGITS In any measurement, some of the digits of a number are absolutely known, while others are a best estimate. In this example, the value “1” is known for sure, but the next number is estimated by the observer. SIGNIFICANT DIGITS If an observer estimates this ruler to read 1.2m, the “1” is known for certain, while the “2” is uncertain. Numbers we know for certain are called significant digits or figures. SIGNIFICANT DIGITS It is important to take into account how many digits in a measurement are significant. More significant digits in a measurement mean that our answers and conclusions will have more significant digits, and thus be more detailed, nuanced and accurate/precise. SIGNIFICANT DIGITS There are two major methods for indicating significant digits. 20.345 20.345 Drop all non-significant digits down lower Underlining (or over-lining) the last significant digit. Both are acceptable, but please stick to one system or the other. SIGNIFICANT DIGITS All measured values should have a specific number of significant digits. Generally, when using a device we record all significant digits and one non-significant digit. However, there are two types of numbers that have infinite significant digits. SIGNIFICANT DIGITS An exact number, or whole number, has infinite significant digits. For example, if a number of people were to be recorded to be 41, this could be treated as 41.000000…. with infinite zeroes. This is because a person is exactly one person. This is known to absolute accuracy. SIGNIFICANT DIGITS Conversion Factors also have infinite significant digits. This is because 1 cm is equal to exactly 10mm. So we can write the conversion factor 10 cm/mm with infinite significant digits. Simply put, converting from one unit to another should not make your measurement any more or less accurate/precise. SIGNIFICANT DIGITS - RULES When using measurements in calculations, we often need to know how many significant digits a number contains. There are 5 basic rules for determining the number of significant digits: SIGNIFICANT DIGITS - RULES 1) Absolute numbers have infinite significant figures. 2) All non-zero numbers are considered significant. 1.05 0.0110 SIGNIFICANT DIGITS - RULES 3) Interior zeros (zeros surrounded by nonzero numbers) are always significant. 1.05 50.012 0.01010 4) Leading zeros (zeros with no non-zero number to the left of them) are not significant. 0.01010 0.00000012 SIGNIFICANT DIGITS - RULES 5) Trailing zeroes (zeroes to the right of a number) are considered significant. 0.0100 2.30030 134.00 If you have a non-significant zero to the right of your number, you should remove them, or rewrite in scientific notation to remove them. Instead of 240 (where the 0 is nonsignificant), rewrite to 2.4 x 102 SIGNIFICANT DIGITS – ADD/SUBTRACT When adding or subtracting. The final answer contains the same number of decimal places as the input number that has the fewest decimal places. 6.01 + 0.334 = 6.344 = 6.34 (2 dp) (3 dp) (2 dp) SIGNIFICANT DIGITS – MULTIPLY/DIVIDE When multiplying or dividing, the answer will have the same number of significant figures as the component number with the fewest significant figures. 5.5 x 10.4 = 57.2 = 57 (2 sd) (3sd) (2sd) SIGNIFICANT DIGITS – MULTI-STEP PROBLEMS In multiple-step problems, retain one nonsignificant digit (if any) between steps and only round off at the very end. This avoids making the answer more inaccurate by rounding after every single step. SIGNIFICANT DIGITS – MULTI-STEP PROBLEMS Example 1: 3.489 x (5.67 – 2.3) 3.489 x (3.37) 11.758 12 [2 dp – 1 dp] [4 sd x 2 sd] SIGNIFICANT DIGITS – MULTI-STEP PROBLEMS Example 2: 19.667 – (5.4 x 0.916) 19.667 – (4.95) 14.7206 14.7 [2 sd x 3 sd] [3 dp – 1 dp] DERIVED UNIT - DENSITY Density is a useful unit in that it compares an object’s mass with volume. D=m/V D = desnity, m = mass, V = volume DERIVED UNIT - DENSITY Density does not have a set unit to describe it. In SI units, density will typically be given in kg/L (kilograms per litre), but any unit of mass and volume can be used (kg/cm3, g/mL, etc). However, you cannot compare densities accurately unless they are in the same units. DERIVED UNIT - DENSITY Example 1: What is the density of a 5.0g mas that is 10 cm3 in size? d = m /V = 5.0 g / 10 cm3 d = 0.5 g/cm3 DERIVED UNIT - DENSITY Example 2: What is the mass of 5.0 mL of a liquid with a density of 3.0 g/mL? d=m/V m=d*V m = 3.0 g/mL * 5.0 mL m = 15 g DERIVED UNIT - DENSITY Example 3: If a liquid has a density of 0.50 g/cm3, what volume of it has a mass of 17 g? d=m/V V=m/d V = 17 g / 0.50 g/cm3 V = 34 cm3 DERIVED UNIT - DENSITY The SI Units are based on water. 1 g of pure water has a volume of 1 mL or 1 cm3 This means that the density of pure water is 1.0 g/mL