Unit Conversion Using the Factor Label Method Objective:
Develop a proficiency in converting the units for a system using the factor label method. Introduction:
In lab this year many of the experiments involve measuring some property of a substance (quantitative analysis) or combining known quantities of materials in a chemical reaction (synthesis). In both types of lab work you will need to be able to express amounts in a variety of units.
Often beginning scientists will record only numbers in their lab notebook. This is a terrible idea and not allowed in any chemistry lab you will ever perform. All numbers you record will have some units associated with them. The unit is in fact more important than the number.
During a family vacation Dennis tired of the ride, asked his mother how much longer they had to drive. Luckily the family was nearing their destination and the mother told Dennis they had seven miles to go. He immediately counted, “1, 2, 3, 4, 5, 6, 7” and announced loudly. “Seven. We are there now. Stop the car.” Dennis will drive you nuts.. Mom looked in the review mirror at him and answered, “Seconds are not the same units as miles.” We know that time and distance are not the same units, but they are related. In this case they are related by how fast the car is traveling, the speed. Typical units for speed are miles per hours (mi/hr). In this case Dennis’ family was traveling 60 mi/hr. Mom thought, “What number should Dennis count up to for these last seven miles to pass?” Mom needed to convert miles into seconds. She might have thought something like this. 60 mi/hr is close to 1 mile/minute, and in a minute my son can count to eighty. So she did the math:
counts counts mile miles 560 min 1 80 1 min 1 7 × × =
For good measure Mom told Denis that in this case seven miles was the same as counting to six hundred. Dennis didn’t quite make it to six hundred, but this unit conversion did kill some time for the bedraggled Mother. Today in lab we will investigate several examples of unit conversion using a process called the factor label method. This is a technique that will be used with regularity in this and other science courses.
1. Proportional Reasoning:
The figure below is a useful example of proportional reasoning. The height of a tall tree can be measured by knowing the length of the shadow cast by the tree and a relationship (proportion) between tree height and shadow length.
In the case above, a small tree of 6.2 ft is found to have shadow of 1.4 ft. This sets a proportional relationship: shadow ft tree ft shadow ft tree ft 4 . 1 2 . 6 or 4 . 1 2 . 6 =
The above proportionality between tree and shadow will remain constant for a given angle of the sun and can be used to find the height of the taller tree in the following way:
shadow ft tree ft shadow ft tree ft X 4 . 1 2 . 6 8 . 11 =
Solving for the height of the tree gives:
shadow ft tree ft shadow ft tree ft X 4 . 1 2 . 6 8 . 11 × =
The conversion factor 6.2 ft tree = 1.4 ft shadow is the proportionality that converts 11.8 ft of shadow to a tree height. Additionally, it should be noted that when you do the math above the units on the left hand side are the same as the units on the right hand side.
tree ft shadow ft tree ft shadow ft tree ft X 52.3 4 . 1 2 . 6 8 . 11 × = =
For all unit conversions we will use a conversion factor similar to 6.2 ft tree = 1.4 ft shadow. Taking advantage of some proportionality is the hallmark of unit conversion.
2. An analogy:
Unit conversion is a matter of convenience: a way of expressing a property
(measurement) in a new (more convenient) unit. The important thing to remember is that the property is unchanged during the unit conversion. You might think of a unit change in the following way. A pole to hang clothes in a closet needs to be a certain length to fit in the closet. The color of the pole has nothing to do with the length. In the same way that you can paint a pole and not change its length, you can also convert the units used to describe its length and not change the pole’s length. Whether the pole is blue or red, 0.00076 miles or 48 inches it will fit in a four foot closet. Blue might look better than red in the room and 48 inches might be easier to measure than 7.6 x 10−4 miles when cutting. You get to choose which quantities are most convenient for you.
3. Unit Transformations:
Today’s lab will focus on how to change units. The conversion is accomplished with one or more equalities (unit conversions). In the picture above it would be 48 inches = 7.6x10− 4 miles. Here is the interesting math. Just as 1
7 7
= , the division of two equal quantities is always unity, meaning that with units our relationship between inches and
miles is also unity. 1
10 6 . 7 48 4 = × − miles inches
. The reciprocal relationship is also unity,
1 48 10 6 . 7 4 = × − inches miles
. Any equality that can be expressed as a ratio will equal unity. The units in the ratio have an important effect. First remind yourself that multiplication by unity does not effect a measurement: ft 4 ft
7 7
4 × = . However multiplication by unity for an expression with units is important: in
ft in
ft 48
1 12
4 × = . The conversion factor 12 in = 1 ft affects the units but does not alter the measurement property of length. It simply expresses length in different units.
4. Metric Unit Conversion:
A mass of a sample can be expressed in several units of mass. In looking at the 125 gram sample to the right there seems to be an obvious relationship between grams and
kilograms and a less obvious relationship to ounces. Grams and kilograms are mass measurements in the metric or SI (System Internationale)system of measure. The metric system is designed so that units doffer by factors of ten in magnitude. In the case of grams and kilograms there are three factors of ten difference (1000 g = 1 kg). In fact, k means 1000. A long list of metric names and their decimal equivalents are given below. Work problems 1, 2, and 3 on the worksheet.
Prefix: Symbol: Magnitude: Meaning (multiply by): Yotta- Y 1024 1 000 000 000 000 000 000 000 000 Zetta- Z 1021 1 000 000 000 000 000 000 000 Exa- E 1018 1 000 000 000 000 000 000 Peta- P 1015 1 000 000 000 000 000 Tera- T 1012 1 000 000 000 000 Giga- G 109 1 000 000 000 Mega- M 106 1 000 000 kilo- k 103 1000 hecto- h 102 100 deka- da 10 10 - - 100 1 deci- d 10-1 0.1 centi- c 10-2 0.01 milli- m 10-3 0.001 micro- u (mu) 10-6 0.000 001 nano- n 10-9 0.000 000 001 pico- p 10-12 0.000 000 000 001 femto- f 10-15 0.000 000 000 000 001 atto- a 10-18 0.000 000 000 000 000 001 zepto- z 10-21 0.000 000 000 000 000 000 001 yocto- y 10-24 0.000 000 000 000 000 000 000 001
5. Unit Conversion Where the Base Unit Changes:
Changing the base unit of a measurement is extremely important in chemistry. A great example is the change between volume and mass. A procedure may ask the researcher to add 1.7 grams of 2-methyl-pentane(C6H14). The researcher sees that 2-methyl-pentane is a liquid with a density of 0.77 g/mL. The density is a conversion from volume to mass. In this case 0.77 g C6H14 = 1.0 mL C6H14. In this case you should express the quantity of C6H14 in the units that are most easy for you to measure. If measuring volume is easier for our researcher, then she should convert 1.7 grams into a volume. Which conversion ratio should she use?
14 6 14 6 14 6 14 6 77 . 0 0 . 1 0 . 1 77 . 0 H C g H C mL or H C mL H C g
Both ratios are unity but only one is useful for the conversion. Examine how they both work below.
(
)
(
6 14)
14 6 14 6 14 6 14 6 2 14 6 14 6 14 6 14 6 2 . 2 77 . 0 0 . 1 7 . 1 ?_______ 3 . 1 0 . 1 77 . 0 7 . 1 ?_______ H C mL H C g H C mL H C g H C mL mL g H C mL H C g H C g H C mL = × = = × =Clearly the bottom conversion between grams and mL is a useful way for obtaining a result in units of volume. Obtaining the correct units in a factor label problem is not a mindless exercise, but rather a result of a careful examination of the relationship among the units.
Density is a conversion that is context specific. You must know what substance the density is for. Some conversions are not context specific 1000 g = 1 kg is always true, it does not matter what is being weighed. In a problem involving unit conversions
identifying the problem’s specific and universal relationships is useful. Work problems 4 and 5 and the collaboration problem.
6. Percentages are conversion factors.
The following is often found for solutions, “Vineger is a solution that is 3% acetic acid by volume”. This indicates that 100 mL vinegar = 3 mL acetic acid. This relationship would allow the conversion between acetic acid and vinegar, using either conversion factor: acid acetic mL vinegar mL or vinegar mL acid acetic mL 3 100 100 3
The ocean is 3.5 % salt by mass. That is equivalent to saying 100 g of ocean water will give you 3.5 grams of salt. You may have tasted this relationship at the beach this summer.
7. Unit Conversions of Area and Volume:
Neither area nor volume are base measurements, they are combined measurements: area = (distance)2 and volume = (distance)3. In the image below a cube is shown.
The volume of this cube in cm3 is a big number and it would be better to express the volume in m3. The important thing to remember is that each distance (length, width and height) must be converted from cm to m to make this conversion. There are a few equalities between cm and m:
(
)
3(
)
3 3 3 3 3 3 6 3 10 1 1 100 1 100 1 100 1m= cm m = cm m = cm m = × cmThere is one that is absolutely not unity and therefore is not a conversion factor:
3 3
100
1m ≠ cm
There are three of you in this lab today that will use the above inequality as a conversion factor on the first exam. Do Not let it be you.
The volume problem above in the following way making a unit conversion on each cm distances of the cube.
3 3 5 3 512 . 0 100 1 100 1 100 1 cm 10 x 5.12 ?________ m cm m cm m cm m m = × × × =
It is fun canceling each cm with one of the original cm3, but it is perfectly fine to use any of the other correct conversion factors above such as:
3 3 6 3 3 5 3 512 . 0 10 1 1 cm 10 x 5.12 ?________ m cm m m = × × =
Complete Problems 8, 9, and 10 on the worksheet.
