Departament de Ciències Experimentals i TecnologiaCiències Naturals. 3 ESO.

CONVERSION FACTORS

. A conversion factor is a fraction whose numerator and denominator are the same quantity expressed in different units. For example, 2.54 cm and 1 in. are the same length, 2.54 cm  1 in. This relationship allows us to write two conversion factors:

We use the first of these factors to convert inches to centimeters. For example, the length in centimeters of an object that is 8.50 in. long is given by

The units of inches in the denominator of the conversion factor cancel the units of inches in the given data (8.50 inches). The centimeters in the numerator of the conversion factor become the units of the final answer. Because the numerator and denominator of a conversion factor are equal, multiplying any quantity by a conversion factor is equivalent to multiplying by the number 1 and so does not change the intrinsic value of the quantity. The length 8.50 in. is the same as 21.6 cm.

In general, we begin any conversion by examining the units of the given data and the units we desire. We then ask ourselves what conversion factors we have available to take us from the units of the given quantity to those of the desired one. When we multiply a quantity by a conversion factor, the units multiply and divide as follows:

If the desired units are not obtained in a calculation, then an error must have been made somewhere. Careful inspection of units often reveals the source of the error.

SAMPLE EXERCISE 1.9

If a woman has a mass of 115 lb, what is her mass in grams? (Use the relationships between units given on the back inside cover of the text.)

Solution Because we want to change from lb to g, we look for a relationship between these units of mass. From the back inside cover we have 1 lb  453.6 g. In order to cancel pounds and leave grams, we write the conversion factor with grams in the numerator and pounds in the denominator:

The answer can be given to only three significant figures, the number of significant figures in 115 lb.

PRACTICE EXERCISE

By using a conversion factor from the back inside cover, determine the length in kilometers of a 500.0-mi automobile race.

Answer: 804.7 km

Using Two or More Conversion Factors

It is often necessary to use more than one conversion factor in the solution of a problem. For example, suppose we want to know the length in inches of an 8.00-m rod. The table on the back inside cover doesn't give the relationship between meters and inches. It does give the relationship between centimeters and inches (1 in.  2.54 cm), though, and from our knowledge of metric prefixes we know that 1 cm  Thus, we can convert step by step, first from meters to centimeters, and then from centimeters to inches as diagrammed in the column.

Combining the given quantity (8.00 m) and the two conversion factors, we have

The first conversion factor is applied to cancel meters and convert the length to centimeters. Thus, meters are written in the denominator and centimeters in the numerator. The second conversion factor is written to cancel centimeters, so it has centimeters in the denominator and inches, the desired unit, in the numerator.

SAMPLE EXERCISE 1.10

The average speed of a nitrogen molecule in air at 25°C is 515   Convert this speed to miles per hour.

Solution To go from the given units,   to the desired units,   we must convert meters to miles and seconds to hours. From the relationships given on the back inside cover of the book, we find that 1 mi  1.6093 km. From our knowledge of metric prefixes we know that 1 km 103 m. Thus, we can convert m to km and then convert km to mi. From our knowledge of time we know that 60 s  1 min and 60 min   1 hr. Thus, we can convert s to min and then convert min to hr.

Applying first the conversions for distance and then those for time, we can set up one long equation in which unwanted units are canceled:

Our answer has the desired units. We can check our calculation using the estimating procedure described in the previous "Strategies" box. The given speed is about 500   Dividing by 1000 converts m to km, giving 0.5   Because 1 mi is about 1.6 km, this speed corresponds to   Multiplying by 60 gives about   Multiplying again by 60 gives   The approximate solution (about  ) and the detailed solution ( ) are reasonably close. The answer to the detailed solution has three significant figures, corresponding to the number of significant figures in the given speed in 

PRACTICE EXERCISE

A car travels 28 mi per gallon of gasoline. How many kilometers per liter will it go?

Answer: 12 km L

Using Density as a Conversion Factor

The conversion factors previously noted convert from one unit of a given measure to another unit of the same measure, such as from length to length. We also have conversion factors that convert from one measure to a different one. The density of a substance, for example, can be treated as a conversion factor between mass and volume. Suppose that we want to know the mass in grams of two cubic inches (2.00 in.3) of gold, which has a density of 19.3 g  cm3. The density gives us the following factors:

Because the answer we want is a mass in grams, we can see that we will use the first of these factors, which has mass in grams in the numerator. To use this factor, however, we must first convert cubic

inches to cubic centimeters. The relationship between in.3 and cm3 is not given on the back inside cover, but the relationship between inches and centimeters is given: 1 in.  2.54 cm (exactly). Cubing both sides of this equation gives (1 in.)3  (2.54 cm)3 from which we write the desired conversion factor:

Notice that both the numbers and the units are cubed. Also, because 2.54 is an exact number, we can retain as many digits of (2.54)3 as we need. We have used four, one more than the number of digits in the density (19.3 g cm3). Applying our conversion factors, we can now solve the problem:

The final answer is reported to three significant figures, the same number of significant figures as is in 2.00 and 19.3.

SAMPLE EXERCISE 1.11

What is the mass in grams of 1.00 gal of water? The density of water is 1.00 g mL.

Solution Before we begin solving this exercise, we note the following:

1. We are given 1.00 gal of water.2. We wish to obtain the mass in grams.3. We have the following conversion factors either given,

commonly known, or available on the back inside cover of the text:

The first of these conversion factors must be used as written (with grams in the numerator) to give the desired result, whereas the last conversion factor must be inverted in order to cancel gallons. The solution is given by

The units of our final answer are appropriate, and we've also taken care of our significant figures. We can further check our calculation by the estimation procedure. We can round 1.057 off to 1. Focusing on the numbers that don't equal 1 then gives merely  in agreement with the detailed calculation.

PRACTICE EXERCISE

a. Calculate the mass of 1.00 qt of benzene if it has a density of 0.879 g mL.

b. If the volume of an object is reported as 5.0 ft3, what is the volume in cubic meters?

Answers: (a) 832 g; (b) 0.14 m3

EXERCICES

1. Convert 4567.36 micrograms to kilograms.

2. Convert 475 miles to kilometers.

3. What is the volume in litres of 64.567 pounds of ethanol at 20 °C?

4. The label on a can of cat food tells you there are 0.94 lb of cat food per can with 0.15% calcium. If there are three servings per can, how many grams of calcium are in each serving?

5. When 2.3942 kg of the sugar glucose are burned (combusted), 37,230 kJ of heat are evolved. What is the heat of combustion of glucose in J/g? (Heat evolved is described with a negative sign.)

6. The diameter of a proton is 2 × 10‒15 meters. What is this diameter in nanometers?

7. The mass of an electron is 9.1093897 × 10‒31 kg. What is this mass in nanograms?

8. There are 4.070 × 106 lb of sulphuric acid used to make Jell-O each year. Convert this to kilograms.

9. A piece of Styrofoam has a mass of 88.978 g and a volume of 2.9659 L. What is its density in g/mL?

10. The density of blood plasma is 1.03 g/mL. A typical adult has about 2.5 L of blood plasma. What is the mass in kilograms of this amount of blood plasma?

11. A piece of balsa wood has a mass of 15.196 g and a volume of 0.1266 L. What is its density in g/mL?

12. A ball of clay has a mass of 2.65 lb and a volume of 0.5025 qt. What is its density in g/mL?

13. The density of water at 0 °C is 0.99987 g/mL. What is the mass in kilograms of 185.0 mL of water?

14. The density of water at 3.98 °C is 1.00000 g/mL. What is the mass in pounds of 16.785 L of water?

15. . Sound travels at a velocity of 333 m/s. How long does it take for sound to travel the length of a 100-yard football field?

16. Pain signals are transferred through the nervous system at a speed between 12 and 30 meters per second. If a student drops a textbook on her toe, how long will it take for the signal, travelling at a velocity of 18 meters per second, to reach her brain 6.0 feet away?

17. An electron takes 6.2 × 10‒9 seconds to travel across a TV set that is 22 inches wide. What is the velocity of the electron in km/hr?

18. The mass of the ocean is about 1.8 × 1021 kg. If the ocean contains 0.041% by mass calcium ions, Ca 2+ , how many tons of Ca 2+ are in the ocean? (There are 2000 pounds per ton.)

19. When you are at rest, your heart pumps about 5.0 litres of blood per minute. Your brain gets about 15% by volume of your blood. What volume of blood in liters is pumped through your brain in 1.0 hour of rest?

20. The mass of the ocean is about 1.8 × 1021 kg. If the ocean contains 1.076% by mass sodium ions, Na+, what is the mass in kilograms of Na+ in the ocean?

21. While you are at rest, your brain gets about 15% by volume of your blood. If your body contains 5.2 L of blood, how many litres of blood are in your brain at rest?…how many quarts?

22. While you are doing heavy work, your heart pumps up to 25.0 L of blood per minute. Your brain gets about 3-4% by volume of your blood under these conditions. What volume of blood in litres is pumped through your brain in 125 minutes of work that causes your heart to pump 22.0 L per minute, 3.43% of which goes to your brain?

23. The mass of the ocean is about 1.8 × 1021 kg. If the ocean contains 1.076% by mass sodium ions, Na+, what is the mass in kilograms of Na+ in the ocean?

24. While you are at rest, your brain gets about 15% by volume of your blood. If your body contains 5.2 L of blood, how many litres of blood are in your brain at rest?…

25. The average heart rate is 75 beats/min. Each beat pumps about 75 mL of blood. How many liters of blood does the average person’s heart pump in a week?

26. While you are doing heavy work, your heart pumps up to 25.0 L of blood per minute. Your brain gets about 3-4% by volume of your blood under these conditions. What volume of blood in litres is pumped through your brain in 125 minutes of work that causes your heart to pump 22.0 L per minute, 3.43% of which goes to your brain?

27. We lose between 0.2 and 1 liter of water from our skin and sweat glands each day. For a person who loses an average of 0.89 L H2O per day in this manner, how many quarts of water are lost from the skin and sweat glands in 30 days?