2. 12 g/mL. Because a milliliter is equivalent to a cubic centimeter, the first thing to do is to convert all the length measurements to centimeters: 1 cm, 1.5 cm, and 0.5 cm. Then multiply the converted lengths to get the volume:
3. 0.616 g/mL. Simply plug the given values for volume and mass into the density formula and solve for density:
Using Conversion Factors
So what happens if you need to convert between one set of units and another, perhaps from a non-SI unit to an SI unit or between SI units? Well, first, you need to understand what a conversion factor is. Second, you need to know how to set up a conversion problem and solve it.
A conversion factor is a ratio represented by a fraction that is equal to 1. It simply uses your knowledge of the relationships between units to convert from one unit to another. When using a conversion factor, you’re not actually changing anything about the physical quantity a measurement represents. You’re simply changing the units in which that quantity is reported. For example, if you know that there are 2.54 centimeters in every inch (or 2.2 pounds in every kilogram or 101.3 kilopascals in every atmosphere), then converting between those units becomes simple algebra.
Remember: All the numbers and measures you encounter in chemistry represent physical quantities of matter. When using conversion factors, you’re simply representing that physical quantity in another way, using a different unit. A meter stick is always going to be the same length, whether you say it’s 1 m long or 100 cm long. Peruse Table 2-4 for some useful conversion factors. And remember that if you know the relationship between any two units, you can build your own conversion factor to move between those units.
Table 2-4 Conversion Factors
* One of the more peculiar units you’ll encounter in your study of chemistry is mm Hg, or millimeters of mercury, a unit of pressure. Unlike SI units, mm Hg doesn’t fit neatly into the base-10 metric system, but it reflects the way in which certain devices like blood pressure cuffs and barometers use mercury to measure pressure.
Warning: Chemistry teachers are sneaky. They often give you quantities in non-SI units and expect you to use one or more conversion factors to change them to SI units – all this before you even attempt the “hard part” of the problem! So with that in mind, expect to use conversion factors throughout the rest of this book.
Here is a simple example illustrating the use of the conversion factor.
Examples
Q. An absent-minded professor named Steve measures the mass of a sample to be 0.75 lb and records his measurement in his lab notebook. His astute lab assistant, who wants to save the prof some embarrassment, knows that there are 2.2 lb in every kilogram. The assistant quickly converts the doctor’s measurement to SI units. What does she get?
A. The sample’s mass is 0.34 kg.
Notice that something very convenient happens because of the way this calculation is set up. In algebra, whenever you find the same quantity in a numerator and in a denominator, you can cancel it out. Canceling out the pounds (lb) is a lovely bit of algebra because you don’t want those units around, anyway. The whole point of the conversion factor is to get rid of an undesirable unit, transforming it into a desirable one – without breaking any rules. Always let the units be your guide when solving a problem. Ensure the right ones cancel out, and if they don’t, go back and flip your conversion factor.
Remember an algebra rule: You can multiply any quantity by 1, and you’ll always get back the original quantity. Now look closely at the conversion factors in the example: 2.2 lb and 1 kg are exactly the same thing! Multiplying by
Q. A chemistry student, daydreaming during lab, suddenly looks down to find that he’s measured the volume of his sample to be 1.5 cubic inches. What does he get when he converts this quantity to cubic centimeters?
A. The volume is 25 cm3.
Rookie chemists often mistakenly assume that if there are 2.54 centimeters in every inch, then there are 2.54 cubic centimeters in every cubic inch. No! Although this assumption seems logical at first glance, it leads to catastrophically wrong answers. Remember that cubic units are units of volume and that the formula for volume is
Now consider the dimensions of the cube in centimeters:
Practice Questions
1. A sprinter running the 100.0 m dash runs how many feet?
2. At the top of Mount Everest, the air pressure is approximately 0.330 atmospheres, or a third of the air pressure at sea level. A barometer placed at the peak would read how many millimeters of mercury?
3. A league is an obsolete unit of distance used by archaic (or nostalgic) sailors. A league is equivalent to 5.6 km. If the characters in Jules Verne’s novel 20,000 Leagues Under the Sea travel to a depth of 20,000 leagues, how many kilometers under the surface of the water are they? If the radius of the Earth is 6,378 km, is this a reasonable depth? Why or why not?
4. The slab of butter that Paul Bunyan slathered on his morning pancakes is 2.0 ft wide, 2.0 ft long, and 1.0 ft thick. How many cubic meters of butter does Paul consume each morning?
Practice Answers
1. 330 ft. Set up the conversion factor as follows:
2. 251 mm Hg.
3.
The radius of the Earth is only 6,378 km, and 20,000 leagues is 17.5 times that radius! So the ship would’ve burrowed through the Earth and been halfway to the orbit of Mars if it had truly sunk to such a depth. Jules Verne’s title refers to the distance the submarine travels through the sea, not its depth.
4. 0.11 m3. The volume of the butter in feet is
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