Units and Unit Conversions

Transcription

1 Units and Unit Conversions SI Base Units ITEMS IN BOLD TYPE COMMON IN CHEMISTRY Amount of Electric Luminous Mass Length Time Temp substance current intensity Unit name gram meter second kelvin mole ampere candela Symbol g m s K mol A cd SI Prefixes with examples of unit conversions PREFIXES IN BOLD TYPE MUST BE KNOWN BY THE STUDENT Prefix Multiplier Conversion examples Other useful conversions Giga (G) GPa = 10 9 Pa Mega (M) MeV = 10 6 ev Kilo (k) kg = 10 3 g = 1000 g Base 1 Deci (d) dl = 10-1 L = 0.1 L 10 dl = 1 L Centi (c) cm = 10-2 m = 0.01 m 100 cm = 1 m 10 cm = 1 dm Milli (m) mg = 10-3 g = g 1000 mg = 1 g 10 mg = 1 cg Micro (µ) µmol = 10-6 mol µg = 1 g 1000 µg = 1 mg Nano (n) ng = 10-9 g 10 9 ng = 1 g Pico (p) pm = m pm = 1 m SI Derived Units and other Commonly Used Units Measurement Unit Name & Symbol Common conversions Volume Energy Pressure cubic meter m 3 1 L = 1 dm 3 1 ml = 1 cm 3 Liter L 1 μl = 1 mm 3 calorie cal 1 cal = 4.18 J 1 Cal = 1 kcal electron-volt ev 1 ev = J atmosphere atm 1 atm = 760 torr 1 atm = kpa mmhg mmhg 1 atm = = 14.7 psi 1 bar = 100 kpa Time second s 1 min = 60 s 1 hr = 60 min = 3600 s Temperature celsius ºC T K = T C K = F

2 Method of direct substitution or reciprocal insertion 3.69 mg = g [substituted 10-3 for m] 8.00 kg = g [substituted 10 3 for k] 1.23 nm = m [substituted 10-9 for n] 2.54 cm = m [substituted 10-2 for c] 2.25 g = mg [inserted 10 3 m between number and unit] 1.99 L = ml [inserted 10 3 m between number and unit] 1.06 m = cm [inserted 10 2 c between number and unit] Method of unit cancellation for SI units Use a unit conversion factor (a fraction) to multiply by a starting unit to get a different unit. Complete the conversion: 3.69 mg = g 3.69 mg 10-3 g = g NB: result same as by direct substitution mg One can express this variously using exponential and decimal notation. Each is legitimate and acceptable. Some examples follow mg g = g or, in exponential notation = g 1000 mg 3.69 mg g = g 10 3 mg NB: 1000 mg = 1 g means the same as 1 mg = g 10 3 mg = 1 g means the same as 1 mg = 10-3 g Using the above conversion methods, you can confirm the following 1.21 µm = m = m = mm = mm 4.54 kg = 4540 g = g = mg = µg m = 525 mm = mm = mm 25.4 mm = m = m = 2.54 cm = µm

3 Method of unit cancellation for other units Not all useful scientific conversions are based on powers of 10, however. For instance, in different disciplines (and when dealing with different magnitudes) other pressure units are used much more frequently than pascals or bars. Consider that when using the conversion 1 atm = 760 torr or 1 atm = 760 mmhg one cannot simply move decimals or manipulate exponents, one must at some point multiply and/or divide by the number 760. Unit cancellation works well for any units. Complete the conversion: 38 torr = atm 38 torr 1 atm = atm = atm 760 torr Complete the conversion: 75 atm = torr 75 atm 760 torr = torr = torr atm Method of unit cancellation for multi-step conversions When performing unit conversions, you may not know a direct conversion between two units. If you know the conversions relating each to another (third) unit, the equivalence may still be calculated. For instance, most (or many, or hopefully some) people know that 10 mm = 1 cm and 1 mm = 1000 µm Fewer (probably) could tell you that 1 cm = µm. But the above two conversions can be performed sequentially to arrive at the correct result. Problems in scientific calculations routinely involve several sequential conversions. Two, three, and four step conversions are quite common. Complete the conversion: 37.5 µg = mg 37.5 μg 10-6 g _ mg _ = mg = mg = mg μg 10-3 g NB: these two conversion factors can be combined to form 10-6 mg 10-3 µg Complete the conversion: 7.5 mj = kcal 7.5 mj 10-3 J _ cal _ kcal _ = kcal mj 4.18 J 10 3 cal

4 Method of unit cancellation for general problem solving When solving problems in science and engineering, you will often work with material properties and physical constants that are unit conversion factors. Thus, solving problems consists of determining the unit of the desired result and sequentially using conversion factors until the units match the given data. Some are simple and obvious, others less so. Below is a short, non-exhaustive, list of properties and constants that one can use as conversion factors. Speed and velocity (distance per time) Dosage Density (mass per volume) Heat capacity Keep in mind, that when doing general unit conversions (such as: m to mm, kg to g, cm to in) one generally does not round the result. Unit conversion factors are exact by definition (usually) and do not affect the precision of the measurement. When solving problems in general, one would round the final result to match the precision of the least precise data point. One should not round during intermediate steps of a multi-step problem. At the very least one should keep one extra significant figure through intermediate steps (as compared to the number of significant figures of the final result). For instance, if the final result calls for three significant figures, then a minimum of four should be maintained throughout a multi-step calculation. Calculate the distance traveled in 5.5 seconds at a speed of 25 meters per second. 5.5 s 25 m = m = m rounding gives m = 140 m s Calculate the mass of 200 ml of ethanol at a density of 0.79 grams per milliliter. 200 ml 0.79 g ml -1 = 158 g rounding to 2 sf gives 160 g Calculate the mass of mol of water at a molar mass of grams per mole mol g mol -1 = 9.01 g already has 3 sf, no rounding needed Calculate the thermal energy needed to heat some water, given appropriate data J g -1 K g 200 K = J = 83.6 kj Calculate the volume of IV solution needed to give a patient a 325 mg dose from a stock solution containing 1200 mg per liter. 325 mg 1200 mg L -1 = L rounding to 2 sf gives 0.27 L = 270 ml

5 Area and volume units Area (squared) and volume (cubed) units are often useful in calculations. When canceling, the value and unit must each be squared or cubed. To find the lengths of the square or cube, use the square root or cube root. A bacterial colony covers an area of 2875 cm 2. Convert this to square meters cm 2 (10 2 m) 2 = m 2 = m 2 cm 2 Fertilizer is spread over a field at 3.0 g m 2. Convert this to g cm g (10 2 m) 2 = g cm 2 m 2 cm 2 A metal cube has a volume of 12.5 cm 3. What is the length of the sides of the cube? cm 3 = (12.5 cm 3 ) 1/3 = 2.32 cm Can check, = 12.5 Interesting cases Percent means per hundred. Multiplied by a base unit, it gives a centi-unit. Gold is present at 2% in 3 grams of metal pins. What is the mass of gold in the pins? 3 g 2% = 6 cg or 3 g ( ) = g = 6 cg A 5 kg sample has 3% impurity. How many grams of impurity are present? 5 kg 3% = ( ) g ( ) = g = 150 g Similarly parts per values are useful. A contaminant is present at 4.8 ppm in 5.0 g of water. Calculate the total mass of contaminant present. [ppm = parts per million; µ = 10 6 or one-millionth] 5.0 g 4.8ppm = 5.0 g ( ) = g = 24 µg Product units are calculated as the product of two units with no canceling. How many kilowatt-hours of energy are used if a 100 W lamp is lit for 5 hours? 100 W 5 h = 500 W h 500 W h = kw h