Chapter One, Section Two | |

**OBJECTIVES**

- To become familiar with the SI system for stating the units of measurements.
- To learn how to work with significant figures.
- To learn how to use the factor-label method to convert between different units of measurements.

**I. UNITS**

We use units often in our daily lives and probably don't think a lot about it. We say, "It will take me 30 minutes to get to KSU, if the traffic is not a problem." The word "minutes" is a unit of time. We say that it is 27 miles from KSU to the Omni. The word "miles" is a unit of distance. We say that we want a gallon of milk. The word "gallon" is a unit of volume. We say that we weigh xxx pounds. The word "pounds" is a unit of weight. If we made these statements without using units, the sentences would be meaningless. Similarly in working problems related to science, any number without units is meaningless -- remember that! We do sometimes say that it is 95 degrees outside, and strictly speaking that is a number without units. The units are implied to be the units used in our country, Fahrenheit. We are just lazy and say the number without units. In science, we can't be lazy!

In science, we don't use the same units that we are used to using in our everyday life in the United States of America. We use the *Systeme Internationale* units or SI units. The name is French because the system was developed in France. So we
have to learn a new system of units, which is used by most of the people in the world outside of the U.S.A.

The Fundamental SI Units are as listed below:

Physical Quantity | Name of Unit | Abbreviation |

Mass | kilogram | kg |

Length | meter | m |

Time | second | t |

Temperature | Kelvin | K |

Electric Current | ampere | a |

Amount of substance | mole | mol |

Luminous intensity | candela | cd |

Derived SI units come from combinations of the above such as the following:

area | square meter | m^{2} |

volume | cubic meter | m^{3} |

velocity | meter per second | m s^{-1} |

acceleration | meter per second squared | m s^{-2} |

density | kilogram per cubic meter | kg m^{-3} |

molar mass | kilogram per mol | kg mol^{-1} |

molar volume | cubic meter per mole | m^{3} mol^{-1} |

molar concentration | mole per cubic meter | mol m^{-3} |

The prefixes used with these units are given below and in your book (Table 1-2) and you must memorize them today.

Prefix | Symbol | Exponential Notation |

exa | E | 10^{18} |

penta | P | 10^{15} |

tera | T | 10^{12} |

giga | G | 10^{9} |

mega | M | 10^{6} |

kilo | k | 10^{3} |

hecto | h | 10^{2} |

deka | da | 10^{1} |

deci | d | 10^{-1} |

centi | c | 10^{-1} |

milli | m | 10^{-3} |

micro | mc or µ | 10^{-6} |

nano | n | 10^{-9} |

pico | p | 10^{-12} |

femto | f | 10^{-15} |

atto | a | 10^{-18} |

**II. MEASUREMENT**

When you have seen one measurement you have seen them all is certainly not true. There are three concepts involved:

- Accuracy
- Precision
- Measuring Device

If we want to measure this line: ______________________________

we could get several different answers depending upon the measuring device. One device might give 3.5 inches by one person with measuring device A and 3.56 inches by another person with measuring device B. However when they swap devices, they might not get the same results as recorded initially with A and B. So all three concepts are involved.

**Accuracy and Precision**

The images below neatly shows the difference between accuracy and precision.

Imagine that the above represents the results of a "ring toss" game. The first image is clearly very precise, but not accurate. The second has rings all over the place and hence is neither precise nor accurate. The third is "just right" --
precisely accurate. If both people measure __with the same precision__ then when they swap measuring devices, their results should agree. Each person should get 3.5 inches with device A and each should get 3.56 inches with device B.

**Measuring Device**

We have all seen rulers with different accuracies and scales with different accuracies. Some rulers are marked to the nearest quarter inch and some to the nearest eighth inch. Some scales are marked to the nearest pound and some to the nearest ounce (to use U.S. units). Which are correct? They are both correct, just one is reporting more accuracy than the other. The results of a measurement then depend upon the preciseness of the measurement and the accuracy of the measuring device.

**III. SIGNIFICANT FIGURES**

Now suppose that person 1 measures the above line with device A and reports a length of 3.5 inches and person 2 measures the length of this line __________with device B and reports a length of 1.45 inches. What is the sum of the lengths of the two lines? First we must be certain that the accuracy of A is really only in the tenths of inches and the accuracy of B is really to the hundredths of inches. We also have to be certain that each person is measuring with the same precision. Assuming that both are true, then how do we determine the sum? Well now we are into a new topic: significant figures. The number of significant figures of a measurement depend upon the accuracy of the measuring device (we will assume that the people doing the measurements are precise in their measurement). When measurements are made with devices of different accuracy, we need to be very careful in combining the measurements to make sure that we consider the proper use of significant figures.

There are some rules to follow to allow you to get the computation done correctly considering significant figures. I summarize them as follows:

- Non-zero integers always count as significant figures.
- Leading zeros and zeros used to locate the decimal point are not significant. Thus the number 0.002 has only one significant figure.
- Zeros that arise as a part of the measurement are significant. Thus the number 0.005025 has four significant figures.
- What about the number 600? The number of significant figures is ambiguous unless we use scientific notation in which case the number might be represented as 6.00 x 10
^{2}if we mean it to have three significant figures, or 6.0 x 10^{2}if the number has only two significant figures. - Certain values such as those that arise from definitions are exact. For example, there are 1000 ml in one liter and the number 1000 is exact which means that it is not used in determining the final number of significant figures in a product, sum,
quotient, or subtraction process. Values obtained by counting are also exact, for example the H
_{2}molecule contains exactly 2 atoms. A dozen is exactly 12 of something. - The result of an addition or subtraction should be reported to the same number of decimal places as that of the term with the least number of decimal places.
- The answer to a multiplication or division should be reported to the same number of significant figures as is possessed by the least precise term used in the calculation.
- Rounding is done after all of the calculation is complete. If the figure following the last number to be retained is less than 5, all the unwanted figures are discarded and the last number is left unchanged: 3.7644 is 3.76 to three significant figures. If the number following the last number to be kept is 5 or greater then the last number is rounded up by one: 6.2504 is 6.3 to two significant figures. 6.348 is also 6.3 to two significant figures.

Going back to our original problem. One measurement was 3.5 inches and the other measurement was 1.45 inches. Adding the two we might report a distance of 4.95 inches, but this is three significant figures which is too accurate a number considering the two measuring devices. We can only report an answer to two significant figures, so we report a combined distance of 5.0 inches. Note that we rounded up and we reported our result with a zero after the decimal and that indicates that the zero is significant.

**IV. DIMENSIONAL ANALYSIS or The FACTOR-LABEL METHOD**

Often we have to convert from one system of units to another. For example, you may be wondering how much I *really* weigh since I told you earlier that I weigh 90.8 kgm. The method of unit conversion is called either the unit factor method or
dimensional analysis. In order to make conversions we must know some conversion factors. Two of the more useful are that there are 2.54 cm in one inch and that there are 453.6 gms in one pound. Let's consider the gram/pound conversion.

We have the equality:

453.6 g = 1 pound

So divide both sides by 453.6 gms and we have

453.6 g/453.6 g = 1 = 1 pound/453.6g

So since 1 pound divided by 453.6g is equal to one, multiplying by that ratio will not change the value of the numerical value of the number -- just the unit. So let's give it a try.

90.8kg x 1000gm/kg = 90800 g where we used another definition ratio (1000g/1k). Now use the pound conversion:

90800g x 1 pound/453.6 g = 200.2 pounds. But what about sig figs? We only have 3 sig figs in the given number of 90.8 kg so we say that I weigh 200. pounds where I put a "." after the number 200 which indicates that all three digits are significant.
Alternatively we could say that I weigh 2.00 x 10^{2} pounds.

Another common conversion we Americans need to make is between Fahrenheit and Celsius. Here we can't use a factor-lable method directly because the different zero point for the two methods. Instead we have to use:

^{o}F = ^{o}C x 9/5 + 32

and

^{o}C = (^{o}F - 32)*5/9

In learning this type of material it is crucial to practice, practice, practice.

- Go to this web site for help in unit conversions..
- Go to this web site for more help with significant figures.
- Go to this site to practice working with significant figures.
- Go to this web site to practice converting volumes.
- Go here to practice converting masses.
- Go here to practice converting lengths.
- WebCT

After you have studied this material and practiced some problems, take quiz Two. If you score at least 80 on the test then you are ready to continue to the next section.

*Web Author: Dr. Leon L. Combs*
*Copyright ©2001 by Dr. Leon L. Combs - ALL RIGHTS RESERVED*