# Numbers and Calculations

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1 Numbers and Calculations Topics include: Metric System, Exponential Notation, Significant Figures, and Factor-Unit Calculations Refer to Chemistry 123 Lab Manual for more information and practice. Learn metric prefixes and base units. Numbers and Metric System Science has to assign numbers to physical properties to test theories and repeat experiments Science and all countries except US use International System of Units (SI units) 7 Base Units Base units: Unit Name Unit Symbol Length Meter m Mass Kilogram kg Time Second s Temperature Kelvin K Amount of Substance Mole mol Electric Current Ampere A Luminous Intensity Candela cd Any measurement can be expressed in these units Derived Units Velocity m/s m s -1 Acceleration m/s 2 m s -2 Force kgm/s 2 kg m s -2 N (Newton) Energy kgm 2 /s 2 kg m 2 s -2 J (Joule) Volume m 3 Use other modified but commonly used (older) units if needed Common (older) Units Volume Liter L = dm 3 Length Angstrom A =.1nm = m Pressure Atmosphere atm Energy Calorie cal Volume Milliliter ml 1

2 Volume Cubic centimeter cm 3 = ml Use conversion factors to convert from English to metric 2.54 cm = 1 inch m = 39.4 in 454 g = 1 lb (pound) kg = 2.2 lb 1 L = 1.06 qt (quart) Prefixes Metric System prefixes are based on powers of ten In order to span the range from large (planets) to small (atoms) Prefix Abbreviation Factor Scientific Notation Giga G 1,000,000, mega M 1,000, kilo k 1, hecto h deka da deci d centi c milli m micro µ nano n pico p Examples of length: 1 Mm = Chattanooga-Tampa 10 6 m ~ 620mi 1 km =.62 miles 1 m = meter stick 1 mm = width of paper clip 1 µm = size of particles that are harmful to lungs (cigarette smoke) 1 nm = size of small molecule Exponential Notation Exponents and Operations 1000 = 1x = 1x = 2 x 10 3 Multiply (add exponents) 2

3 (10 3 )(10 3 ) = = 10 6 (10 3 )(10-3 ) = =10 0 = 1 Divide (subtract exponents) (10 3 )/(10 3 ) = = 10 0 = 1 Power (multiply exponents) (10 3 ) 3 = 10 9 (2 x 10 3 )(3 x 10 6 ) = 6 x 10 9 (2 x10 3 )/ (3 x 10 6 ) = 2/3 x =.66 x 10-3 = 6.6 x 10-4 Addition and Subtraction (only if to the same power) (6.2 x 10 3 ) (0.2 x 10 3 ) = 6.0 x 10 3 (3000)/(15) = 3.0 x 10 4 / 1.5 x 10 1 = 2.0 x 10 3 Why use exponents? Atoms are so small that there are huge number of atoms in even small amount of matter Ex 18.0g of water contains 6.02 x molecules! And you don t want to have to write Significant Figures (sig fig) Number written to correctly show the precision of number Mass of Penny Significant Figures Precision 3 g or g or g or g or g or But has only 2 significant figures (zeros to left of first digit don t count for sig fig) Zero to right is significant g (4 sig figs) or (5 sig figs) Zero to left of number is not significant kg (1 sig fig) 310 so write as 3.1 x so write as 3 x

4 Round off 5, 6, 7, 8, 9 round up , 1, 2, 3, 4 round down Significant Figures in Computations + or - no sig fig to right of number with least number of decimal places can not go past the uncertainity in 0.3 place x or / no more sig fig than least precise number x since only 3 sig fig in 5.61 then answer can only be 3 sig fig Scientific Notation Write number as exponent to indicate correct sig fig Write not as 29,979,000,000 cm/s but instead as x cm/s or x cm s -1 Why are metric conversions important NASA Says Metric Mixup Doomed Mars Spacecraft Summary: The \$125 million spacecraft that was destroyed on a mission to Mars was probably doomed by NASA scientists embarrassing failure to convert English units of measurement to metric ones. The Mars Climate Orbiter flew too close to Mars and is believed to have broken apart or burned up in the atmosphere. NASA s Jet Propulsion Laboratory said that its preliminary findings showed that Lockheed Martin Astronautics in Colorado submitted acceleration data in English units of pounds of force instead of the metric unit called Newtons, At JPL, the numbers were entered into a computer that assumed metric units. Ex. How many newtons in 1 pound of force? 4

5 force is mass times acceleration? N = 1 lb? N = 1 lb (454g/ lb)( 1kg / 1000g)( m/s 2 )( N/kgms -2 ) = 4.45 N Factor-Unit Method (Dimensional Analysis) Include unit with number and use conversion factors to change units 1. Set up problem 2. Write conversion factors so units cancel 3. Do math to obtain answer 4. Write in scientific notation (check significant figures) Ex. How many inches in 1.2 miles?? in = 1.20 mi( 1760 yd/mi)( 3ft/ yd)( 12 in/ ft) numerator/denominator = in = 7.60 x 10 4 in Ex. How many cm in 1.2 km?? cm = 1.20 km ( 10 3 m/ km)(10 2 cm/ m) = 1.20 x 10 5 cm Ex. How many cm in exactly yard? 2.54 cm = 1 in? cm = (3 ft/ yd)(12 in/ ft)(2.54 cm/ in) exact numbers do not limit sig fig = cm Ex. 10 km/hr is how many m/s? work on numerator and denominator (top and bottom) separately? m/s = 10 km/hr ( hr/60 min) (min/ 60s) ( 10 3 m/ km) = 2.8 m/s Density Problem Density = Mass/ Volume 5

6 Density used as conversion factor values can be found in tables Mg density = 1.74 g/cm 3 H 2 O density = 1.0 g/cm 3 Ex. What is mass of 2.3m 3 of Mg (Magnesium)?? kg = 2.3m 3 (10 2 cm/m) 3 (1.74 g/cm 3 ) (10-3 kg/g) = 4.0 x 10 5 kg More Factor Unit Problems Ex. How many grains of sand at Daytona Beach? assuming 1 grain has volume of 1 mm 3 and beach is 10 mi long ~ 16 km 100 ft wide ~ 30 m 30 ft deep ~ 9.1 m Volume = 16km (10 3 m/km) x 30m x 9.1m = 4.4 x 10 6 m 3 Grains = (4.4 x 10 6 m 3 ) (10 3 mm/m) 3 (grain/mm 3 ) = 4.4 x grains m 3 mm 3 grains be sure to cube number and units Below are calculations to convert English units to metric? m = 10 mi (5280 ft/ mi)(12 in/ft)(2.54 cm/in)(m/ 100 cm) = m = 1.6 x 10 4 m? m = 100 ft (12in/ ft)(2.54 cm/ in)(m/ 100cm) = = 3.0 x 10 1 m? m = 30 ft (12in/ ft)(2.54 cm/ in)(m/ 100cm) = ~ 9.1 m Answer above may not be exact but shows how you can approximate complicated numbers Ex. A child swallowed a whole bottle of Tylenol (Acetaminophen) Tylenol s toxins damage the liver 6

7 ( The treatment is Acetylcysteine ( Must give the correct dosage of the antidote Acetylcysteine (140 mg Acetylcysteine/ kg body weight) so what would correct dose of Acetylcysteine be for 40lb child?? g antidote = (40 lb)(454 g/ lb)(kg/ 1000 g)( 140 mg antidote/ kg)(g/ 10 3 mg) = g of acetylcysteine antidote Ex. How molecules of water in a cup of water (8 oz)? molecules = ( 8 oz H 2 O)( 1lb/ 16 oz)(454 g/1 lb)(mol/ 18 g)(6.02 x /mol) = 7.6 x Other big numbers are small by comparison People on earth ~ 6 x10 9 ~ 6 billion Stars in our galaxy ~ 4 x ~ 400 billion Final math note Logarithms 7

8 Logarithm is number that when (10 or e ) is raised to that power gives desired number. The log of 1000 is 3 because 10 3 = The log of 2000 is because = 2000 The ln of 1000 is because e = 1000 Common logarithm is log ex. log (10 3 ) = 3 Natural logarithm is ln ex. ln (e 3 ) = 3 e is special number = A logarithm has two parts the characteristic and the mantissa. The mantissa tells what the number is and the characteristic tells the power of ten or how big the number is. For example the log of 1288 is log(1288) = which means x10 3 = 1.288x10 3 with 3 = characteristic and.110 = mantissa Working a problem with log (when multiplying numbers you add logs) (56 x 23) = x = = = 1288 Working with same problem with ln (56 x 23) = e x e = e = e = 1288 Prior to use of calculators, logarithms were routinely used for calculations. This is mostly taken care of for you now, but should understand. Calculator has the functions: ln x and e x and these are opposite operations ex: ln ( e 2 ) = 2 log x and 10 x and these are opposite operations ex: log (10 2 ) = 2 the ln of a number is approximately times the log of same number ex: log (1000) = but ln (1000) = and = (2.3026) (3.000) 8

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