In This Lesson: Scientific Notation and Unit Analysis (Lesson 4 of 6) Today is Tuesday, February 13 th, 2018 Stuff You Need: Calculator Pre-Class: By now you ve probably heard of scientific notation. What s the point?
Today s Agenda Scientific Notation Writing and calculating Unit Analysis Conversions! Where is this in my book? P. 74 and following
By the end of this lesson You should be able to convert numbers to and from scientific notation. You should be able to convert measurements between various units, including those of the metric system.
Scientific Notation Scientific notation is a way for scientists to more easily represent very large or very small numbers. For example, later this year you will learn about the quantity called a mole. A mole equals 602,000,000,000,000,000,000,000 of something. Similarly, an electron weighs very little almost nothing. An electron weighs 0.0000000000000000000000000000 00091 kg
Without Scientific Notation? How much does one mole of electrons weigh? 0.000000000000000000000000000000091 kg x 602000000000000000000000???
Other Scientific Notationworthy Measurements
I get it now what? Numbers written in scientific notation take this form: M x 10 n M is between 1 and 10 (1 or above, less than 10). M is sometimes called the mantissa or significand. n is an integer.
Converting 2,500,000,000. to Scientific Notation Make 2,500,000,000 into scientific notation. Step 1: Put in an understood decimal point. Step 2: Decide where the decimal point must end up so that there is only one nonzero digit to its left (M). Step 3: Count how many places you have to bounce the decimal point to get there (n). Step 4: Rewrite in the M x 10 n form. 9 8 7 6 5 4 3 2.5 x 10 9 2 1
Converting 0.0000579 to Scientific Notation 1 2 3 4 5 Make 0.0000579 into scientific notation. Step 1: Put in an understood decimal point (already done). Step 2: Decide where the decimal point must end up so that there is only one nonzero digit to its left (M). Step 3: Count how many places you have to bounce the decimal point to get there (n). Step 4: Rewrite in the M x 10 n form. 5.79 x 10-5
Practice Try the top half of our Scientific Notation and Unit Analysis sheets.
Scientific Notation Remember that moving the decimal point left makes n go up by one; moving it right makes it go down by one. Now how do we get out of scientific notation?
Converting Out of Scientific Notation You know how 10 0 = 1? Or how anything 0 = 1? When you re given a number in scientific notation and need to convert it to standard notation, you want to make the 10 n part equal to 10 0. If it s positive, like 10 4, move the decimal right. If it s negative, like 10-3, move the decimal left.
Converting Out of Scientific Notation Convert 4.3 x 10 3 to standard notation. 4.3 4300. 1 2 3 Convert 5.9721 x 10-2 to standard notation. 0.059721 5.9721 2 1
Scientific Notation Since you now know how to write in scientific notation, let s talk about how to calculate in scientific notation. We ll start with addition and subtraction.
Addition and Subtraction If the exponents are the same, simply add or subtract the (M) numbers in front and leave the x 10 n part unchanged. You can do this for addition or subtraction. 4 x 10 6 + 3 x 10 6 7 x 10 6 4 x 10 6-3 x 10 6 1 x 10 6
Addition and Subtraction If the exponents are not the same, we have to move a decimal point until they are. Move the decimal on the smaller number! 4 x 10 6 4 x 10 6 + 3 x 10 5 4.00 x 10 6 + 3.00 x 10 5 4.00 x 10 6 + 0.30 x 10 6 4.3 x 10 6
Practice Solve the following in scientific notation: 2.37 x 10-6 0.0237 2.37 x 10-4 -6 + 3.48 x 10-4 + 3.48 x 10-4 3.5037x 10-4 3.50 x 10-4
Practice Let s try combining sig figs and scientific notation with another worksheet (half). Significant Figures and Scientific Notation Upper half
Multiplication and Division For multiplication, just multiply the digits in front (M), then add the exponents (n). Remember that you still need to make sure the number in front (M) is between 1 and 10. 8 x 10 3 x 3 x 10 9 24 x 10 12 2.4 x 10 13 2 x 10 13
Multiplication and Division For division, do the opposite of multiplication. Just divide the digits in front (M), then subtract the exponents (n). 6 x 10 8 / 2 x 10 7 = 3 x 10 1 30
Practice Solve the following in scientific notation: 2 x 10-2 / 5 x 10-6 = 0.4 x 10 4 4.0 x 10 3 4 x 10 3
One final note You may notice your calculator giving you answers like this: 5.665 E 8 5.665 8 In both cases, your calculator is trying to say: 5.665 x 10 8
Practice Let s try combining sig figs and scientific notation with another worksheet (half). Significant Figures and Scientific Notation Lower half
Unit Analysis Now that you re used to doing some conversions of just numbers, it s time to start converting measurements. To do this, we ll need to manipulate numbers and units, both parts of a measurement, in a systematic process known as unit analysis. Unit analysis is also known as: Factor Label Fence Post Unit Cancellation Dimensional Analysis Let s start with learning the units.
Of what is this a map? Those other countries are Myanmar (Asia) and Liberia (Africa).
The SI Units Le Système Internationale Physical Quantity Name Abbreviation Mass kilogram kg Length meter m Time second s Temperature Kelvin K Electric Current ampere A Quantity mole mol Luminous Intensity candela cd
Aside: A Butt Load? In winemaking, a butt is a traditional unit of measurement for volume. It s equal to 126 gallons (or two hogsheads). Yeah, there are a lot of weird measurements out there. With that said, if you have a buttload of something, well you have 126 gallons.
Aside: Defining a Unit Something to consider: How did we decide what a kilogram weighs? (or any of the other units) What I mean is, did we use an object and say this is a kilogram and compare everything else to it? Well? Yes! The American Kilogram video
Aside: Why Don t We Use Metric? Pirates? Pirates may be the reason why the US does not use metric article
Aside: George Carlin If we have mileage, yardage, and footage, why don t we have inchage?
Metric System Units (KNOW THIS) Length: Meter (m) Or centimeters, millimeters, kilometers. Mass: Gram (g) Or milligrams, kilograms. Volume: Liter (L) Or milliliters. Note: cm 3 (or anything 3 ) is a measure of VOLUME.
SI Prefixes Common to Chemistry Prefix Unit Abbr. Exponent In other words Kilo k 10 3 1000 of base Hecto h 10 2 100 of base Deka da 10 1 10 of base [BASE] - 10 0 - Deci d 10-1 10 in base Centi c 10-2 100 in base Milli m 10-3 1000 in base Micro 10-6 10 6 in base Nano n 10-9 10 9 in base Pico p 10-12 10 12 in base
Aside: Scale of the Universe Exactly how big are these units? If an ipod nano were really a nanometer, how big would it be? Scale of the Universe.lnk
Metric Conversions Usually 1000 100 10 1 0.1 0.01 0.001 0.000001 0.000000001 0.000000000001 kilo. hecto. Deka. (unit). deci. centi. milli micro nano pico King HenryDoesn t1.00 m m 10.0 DrinkChocolateMilk dm 100. cm maybe not possibly? g Example: L Convert 100 centimeters to meters. Step 1: Find your starting unit. Step 2: Find your ending unit. Step 3: Move the decimal point one digit to the left each time you move left. Move the decimal point one digit to the right each time you move right. Step 4: Solve: 100 cm = 1 m.
Metric Conversions 1000 100 10 1 0.1 0.01 0.001 0.000001 0.000000001 0.000000000001 kilo. hecto. Deka. (unit). deci. centi. milli micro nano pico m g Example: 0.0000000000052 0.000000000052 0.00000000052 L DL 0.0000000052 L 0.000000052 dl cl 0.000052 ml 0.052 μl nl 52. pl Convert 52 picoliters to Dekaliters. Step 1: Find your starting unit. Step 2: Find your ending unit. Step 3: Move the decimal point one digit to the left each time you move left. Move the decimal point one digit to the right each time you move right. Step 4: Solve: 52 pl = 0.0000000000052 DL (5.2 x 10-12 )
Metric Conversions 1000 100 10 1 0.1 0.01 0.001 0.000001 0.000000001 0.000000000001 kilo. hecto. Deka. (unit). deci. centi. milli micro nano pico m g L Write the following in regular notation and scientific notation: How many mm in a m? 1000 mm 1 x 10 3 mm How many cm in a m? 100 cm 1 x 10 2 cm How many m in a km? 1000 m 1 x 10 3 m How many cm in a km? 100,000 cm 1 x 10 5 cm How many nm in a pm? 0.001 nm 1 x 10-3 nm
Practice Now try the bottom half of our Scientific Notation and Unit Analysis sheets. Use the first column for scientific notation stuff (if you feel confident in it). Also try Metric Conversions Worksheet. Challenge: #6 and #7.
Useful Conversions 1 mi = 1.6093 km 1 L = 1.0567 qt 1 lb = 453.59 g 1 ml = 1 cm 3 1 in = 2.54 cm 1 kg = 2.2046 lb F = (9/5) C + 32 C = (5/9)( F 32 )
Unit Analysis What we ll next be doing is converting measurements of one unit (miles, for example) to another unit (kilometers, maybe). First, write down what you know. Then, here s the setup: starting measurement target unit given unit This part is your conversion factor. THE NUMERATOR AND DENOMINATOR MUST BE EQUIVALENT!!!1 Lastly, multiply the whole top part together, multiply the whole bottom part together, and divide the top by the bottom.
Unit Analysis Examples How many shoes do you have if you have 30 pairs? We know that there are 2 shoes in every pair, so that will be our conversion factor. 30 pairs 2 shoes 1 pair = 60 shoes 30 pairs = 60 shoes
Wait what? Weirded out by that cancellation of units thing? Try the same problem using algebra and you might recognize the process: y ( 2 ) ( 60 xy (30 x) = ) = 60 y 1 x 1 x
Unit Analysis Examples 2 km = m We know that there are 1000 m in every km, so that will be our conversion factor. 2 km 1000 m 1 km = 2000 m 2 km = 2000 m
Unit Analysis Examples Convert 7 miles to kilometers (1.61 km in every mi). We know that there are 1.61 km in every mi, so that will be our conversion factor. 7 mi 1.61 km 1 mi = 11.27 km 7 mi = 11.27 km (or 10 km in sig figs)
Unit Analysis Examples Suppose you re running a 5K charity race. How many miles are you running? We know that there are 1.61 km in every mi, so that will be our conversion factor. 5 km 1 1.61 mi km = 3.106 mi 5 km = 3.106 mi (or 3 mi in sig figs)
Unit Analysis Examples 115 lbs = mg (453.59 g to 1 lb) We know that there are 453.59 g in every lb, so that will be our first conversion factor. 115 lbs 453.59 g 1 lb 1000 1 mg g = 52,162,850 mg With this, we ve gotten to grams. Now we need to convert again using our new given unit. 115 lbs = 52,162,850 mg (or 52,200,000 in sig figs)
Unit Analysis Examples 71 mph = km/min (1.61 km in every mi) We know that there are 1.61 km in every mi, so that will be our first conversion factor. Also, the per in miles per hour means that the hour part is in the denominator. 71 mi 1 hr 1.61 1 km mi 1 60 hr min = 1.91 km/min Now we need to convert again to get the time unit. 60 mi/hr = 1.91 km/min (or 2.0 km/min in sig figs)
Unit Analysis Examples Convert 204.3 kpa to torr (760 torr in 101.3 kpa). We know that there are 760 torr for every 101.3 kpa, so that will be our conversion factor. 204.3 kpa 760 torr 101.3 kpa = 1532.75 torr 204.3 kpa = 1532.75 torr (or 1500 torr in sig figs)
Unit Analysis Examples On October 6, 1912, a munitions company measured Washington Senators pitcher Walter Johnson s fastball speed at 122 feet per second. Done prior to the spread of the automobile, no one really measured things in mph. So, how fast was that pitch in miles-per-hour? http://clickamericana.com/wp-content/uploads/swiftest-pitchers-1913.jpg
Convert 122 ft/sec to miles/hour Remember, the per in feet per second means that the second part is in the denominator. 122 ft 1 sec 1 5280 mi ft 3600 1 sec = 83.2 mph Some disclaimers: The last segment of the fence can be broken into two if you re less comfortable with the amount of seconds in a minute. Johnson was later measured to throw at 134 feet per second, which converts to 91.4 mph. All of this is further convoluted by the distance at which the speed was measured. Today, radar guns measure pitch speed 50 ft from the plate (just out of the pitcher s hand), so the fastest pitch ever measured is likely a Nolan Ryan pitch thrown in 1974, adjusted to 108.1 mph. hr
Unit Analysis Examples Convert 0.325 in 3 to mm 3 (1 in = 2.54 cm). We know that there are 2.54 cm in every inch, but that s not the same as cubic inches/centimeters. A cubic inch is 2.54 cm by 2.54 cm by 2.54 cm, or 2.54 3 cm 3. 0.325 in 3 2.54 3 cm 3 10 3 mm 3 = 5325.80 mm 3 1 in 3 1 cm 3
Tips for Remembering Metric Equivalents Now that you know fence post, you could still use that old King Henry whatever thing OR You can use the King Henry whatever thing to simply get the exponent that relates the two prefixes. In other words
Metric Conversions Let s consider the starting unit to be 10 0 (or 1). Next, count off exponents in each direction, like this: 1000 100 10 1 0.1 0.01 0.001 10 0 0.000001 10 3 0.000000001 10 6 0.000000000001 10 9 kilo. hecto. Deka. (unit). deci. centi. milli micro nano pico m g L Suppose you get a problem like this and don t know how many mm are in a pm: 52 mm 10 9 pm 10 mm = 0 5.2 x 1010 pm
Tips for Checking Work Make sure the top (numerator) of the conversion factor equals the bottom (denominator). Make sure your answer makes sense. If a milliliter is smaller than a liter, then converting to milliliters should end up with a larger number. When converting rates (miles per hour, kilometers per second), do the problem in two steps. For example, if it s mi/hr to km/sec, convert miles to kilometers, then convert hours to seconds. (or hours to minutes to seconds)
Time for some practice Unit Analysis Practice Problems Tough ones include #1 and #4.
Predicting Units? You can also use this cancellation method to find units of an unknown. Example: 5000 g 4.184 J g C 50 C = 1,046,000 J ΔH = 1,046,000 J
Predicting Units? Try another example. Solve for C: 1200 J 1234 g C (50 C) 1200 J 61700 g C C 1200 J ( 61700 g C ) C ( 0.0194 J g C ) C