Chapter #2 Unit Conversions

Unit Conversion Accidents There have been many serious incidents that have resulted from errors in converting between systems of units. Air Canada Flight 143 (Google it for more details)

Unit Conversion Accidents $125 million Mars Climate Orbiter. Lost in Space. Dp you think there is the potential to make errors in the conversion of units for health care providers?

Conversion Problem Steps Write down the number and unit. Draw lines; a vertical line after the number an unit and horizontal line below the number and unit. Insert a fractional fact to cancel out the original unit. Compare the new unit to the asked for unit a. If the same, you are done. b. If not the same, repeat step 3.

Step 1. Write down the number and unit. 47.2 mg

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg Step 3. Insert fractional fact crossing out original unit

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg Step 3. Insert fractional fact crossing out original unit 47.2 mg 10-3 g mg

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg Step 3. Insert fractional fact crossing out original unit 47.2 mg 10-3 g mg Step 4. Compare new unit to the asked for unit.

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg Step 3. Insert fractional fact crossing out original unit 47.2 mg 10-3 g mg Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

Step 1. Write down the number and unit. 47.2 mg Step 2. Draw lines 47.2 mg Step 3. Insert fractional fact crossing out original unit 47.2 mg 10-3 g = 0.0472 g mg Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

Step 1. Write down the number and unit. 702 cL Step 2. Draw lines 702 cL Step 3. Insert fractional fact crossing out original unit 702 cL 10-2 L cL Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

Step 1. Write down the number and unit. Not a match repeat step #3 Step 1. Write down the number and unit. 702 cL Step 2. Draw lines 702 cL Step 3. Insert fractional fact crossing out original unit 702 cL 10-2 L cL Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

It’s a match, done Step 1. Write down the number and unit. 702 cL Step 2. Draw lines 702 cL Step 3. Insert fractional fact crossing out original unit μL 702 cL 10-2 L cL 10-6 L Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

It’s a match, done Step 1. Write down the number and unit. 702 cL Step 2. Draw lines 702 cL Step 3. Insert fractional fact crossing out original unit 702 cL 10-2 L μL = 7.02 x 106 μL cL 10-6 L Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

Significant Figure Definition Significant figures are the number of numbers read from a measuring device.

What Are Numbers? Numbers are any integers from 1- ∞, and sometimes zero. Zero serves two purposes, it is used as a decimal place holder, a number, or both. How do we determine if a zero is a number or a position holder when determining the number of significant figures for a measurement?

The Zero Test To determine if a zero is a number or a decimal spacer, consider dropping one or more of the zero digits. If dropping a zero changes the value of the measurement, then the zero is a decimal position holder and is not considered to be a number and therefore cannot be counted as a significant figure.

For Example Consider the measurement 100 cm. Dropping the last two zeros changes the value to 1, so the zeros are position holders and not numbers. Since significant figures are numbers by definition, then they are not counted in the significant figure count, thus 100 cm has only one significant figure. Now consider the measurement 100.0 cm. If the last zero is dropped the value of the measurement remains the same. Here the last zero does not space the decimal in this measurement. Since zeros are either decimal position holders, or numbers, then the zero in this case must be a number and counted in the significant figure count since is not a decimal spacer.

Sandwiched Zeros What about the zeros in the center of the measurement of 100.0 cm? Since the last zero is a number and the one at the beginning is a number then the center zeros are sandwiched by two numbers. Sandwiched zeros are always counted as significant figures, thus giving 100.0 cm four significant figures

Zeros Both Numbers and Spacers? For a zero to be counted as a spacer and a number additional information must be given: Common sense to be gained in the laboratory A measuring device so states that they are significant.

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs Reason 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm 1 Zero is a spacer for sure. Additional information required to see if it is a number

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs Reason 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm 1 Zero is a spacer for sure. Additional information required to see if it is a number 3 The last number is not a spacer, since dropping it the value is unchanged. The other zero is sandwiched.

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs Reason 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm 1 Zero is a spacer for sure. Additional information required to see if it is a number 3 3 Zero is sandwiched here

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs Reason 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm 1 Zero is a spacer for sure. Additional information required to see if it is a number 3 3 Zero is sandwiched here 4 Zero is not a spacer. The other zero is sandwiched.

Examples Consider the following list of measurements and determine how many significant figures each measurement contains. Measurements SigFigs Reason 10 cm 10.0 cm 101 cm 101.0 cm 1.00 X 10-3 cm 1 Zero is a spacer for sure. Additional information required to see if it is a number 3 3 Zero is sandwiched here 4 The last zero is not a spacer . The other zero is sandwiched. 3 Only look at the coefficient, these zeros are not spacers

Example How can we express 100 cm to three significant figures?

Example How can we express 100 cm to three significant figures? Use Scientific Notation!

Scientific Notation A way to abbreviate large or small numbers

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number.

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. 4.54

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. 4.54

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. 4.54 X 10

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Step 3, count from the old decimal location to the new decimal location 4.54 X 10

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10. 4.54 X 10

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10. 4.54 X 105 mi

Scientific Notation Examples Convert the following into scientific notation. 454,000 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10. Note: Be sure that the answer contains the same number of significant figures as the starting measurement 4.54 X 105 mi

Scientific Notation Examples Convert the following into scientific notation. b. 0.00283 mi Step 1, place a decimal to the right of the first non-zero number.

Scientific Notation Examples Convert the following into scientific notation. b. 0.00283 mi Step 1, place a decimal to the right of the first non-zero number. Note: Be sure that the answer contains the same number of significant figures as the starting measurement 2.83 mi

Scientific Notation Examples Convert the following into scientific notation. b. 0.00283 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Note: Be sure that the answer contains the same number of significant figures as the starting measurement 2.83 X 10 mi

Scientific Notation Examples Convert the following into scientific notation. b. 0.00283 mi Step 1, place a decimal to the right of the first non-zero number. Step 2, place X 10 after the number. Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10, unless the number is less than one, if so, then negative power Note: Be sure that the answer contains the same number of significant figures as the starting measurement 2.83 X 10-3 mi

Example How can we express 100 cm to three significant figures? Use Scientific Notation!

Example How can we express 100 cm to three significant figures? Use Scientific Notation! 1.00 X 102

DENSITY What is heavier 5 pounds of lead or 5 pounds of feathers? What takes up more space, 5 pounds of lead or 5 pounds of feathers?

ROUNDING When measurements are combined to provide information, can the resultant information be of a higher quality than the measurements?

ROUNDING When measurements are combined to provide information, can the information be of a higher quality than the measurements? No, information provide by combining measurements cannot have an accuracy, or precision greater than the least precise measurement that provided the information.

Why Round After a Calculation Since information provided by combining measurements cannot have a higher quality than the measurements providing the information, then answers to problems must be rounded to give the same quality as the measurement with the least quality. Rounding rules are designed to give answers the desired quality. They are posted on the course web site and restated on the following slide.

ROUNDING RULES Rounding is the process of providing results that have the same quality as measurements with the least quality. Since there are different mathematical methods of combining measurements, then different rounding rules are required to provide sensible results of measurement combinations.

Addition and Subtraction Round the calculated answer so that it contains the same number of decimal places as the measurement with the least number of decimal places.

Addition and Subtraction Example: 61.772 cm + 4.0 cm 65.772 cm Round to one decimal place 65.8 cm

Multiplication and Division Round the calculated answer so that it contains the same number of significant figures as the measurement with the least number of significant figures. In other words, if the measurement with the least number of significant figures contains two significant figures, then the rounded answer should contain two significant figures.

Multiplication and Division Example: 61.772 cm X 4.0 cm 247.088 cm Round to two significant figures 66 cm

Logarithms Round the calculated answer so that it contains the same number of decimal places as the measurement with the least number of significant figures. In other words, if the measurement with the least number of significant figures contains two significant figures, then the rounded answer should contain two decimal places.

Anti-logarithms Round answer so that the number of significant figures matches the number of decimal places as the measurement with the least number of decimal places. In other words, if the measured number contains three decimal places, then the answer should be rounded so that it contains three significant figures.

DENSITY What is heavier 5 pounds of lead or 5 pounds of feathers? Both the same. This is an old riddle to confuse density with weight What takes up more space, 5 pounds of lead or 5 pounds of feathers?

DENSITY What is heavier 5 pounds of lead or 5 pounds of feathers? Both the same. This is an old riddle to confuse density with weight What takes up more space, 5 pounds of lead or 5 pounds of feathers? Feathers, since they are less dense.

g/mL, g/cm3, (for solids and liquids), or g/L for gases DENSITY UNITS g/mL, g/cm3, (for solids and liquids), or g/L for gases Grams is a measure of mass using a laboratory balance, while mL, cm3 and liters are volume measurements. Cm3 is found be measuring the length, width, and height of a regularly shaped object and multiplying these measurements together. The volume measurement of mL is found from a graduated cylinder shown on the next slide. Archimedes principle is used to determine the volume on an irregularly shaped object, also on the next slide.

Archimedes Principle Archimedes was the scientist for king Hiero II of Sicily. King Hiero ordered a new crown and provided the goldsmith with several pounds of gold. When the crown was finished, King Hiero suspected the goldsmith of substituting some of the gold with silver and keeping the rest of the gold for himself. The King requested that Archimedes determine if his crown was pure gold or not. Since Archimedes knew the density of pure gold, the trick was to determine the density of the crown. If the crown’s density is the same as pure gold, then the gold smith did not substitute silver for some of the gold. The challenge was how to find the volume of the royal crown, which is illustrated on the next slide.

Archimedes Principle We can determine the volume of irregularly shaped objects by displacement. How can we determine the volume of a gas? Gases fill whatever container they are placed in. So it’s the volume of the container !

DENSITY PROBLEM SOLVING STRATEGY Use the four step unit analysis. Organize the measurements to give density units. Sample Problems Calculate the density of a 4.07 g sample of rock that displaces 1.22 mL of water. Calculate the density of a 4.22 g sample of wood that measures 2.0 cm by 1.33 cm by 3.56 cm. Mercury has a density of 13.6 g/mL. Find the mass of 125 mL of mercury. Water has a density of 1.00 g/mL. Find the volume, in liters, of a 3.22 kg sample of water. What does an object do in water with A density greater than water? A density less than water? A density equal to water?

English/Metric Conversions Definitions 2.54 cm = in 946 ml = qt 454 g = lb cm3 = mL Please Remember Definitions are not measurements and do not contain significant figures

Sample English/Metric Conversion Problems Convert 708 pounds to kilograms. Convert 50.0 liters to gallons. Convert the density of water to pounds per gallon. How many cubic meters are contained in 33 liters? The density of aluminum is 2.70 g/mL. Find the thickness of aluminum foil that measures 2.0 cm by 5.66 cm that has a mass of 1.23 g.

The End