High School Chemistry/Using Measurements

The metric system is a decimal system. This means that making conversions between different units of the metric system are always done with factors of ten. Let's consider the English system – that is, the one that is in everyday use in the US as well as England – to explain why the metric system is so much easier to manipulate. For instance, if you need to know how many inches are in a foot, you only need to remember what you at one time memorized: 12 inches = 1 foot. But now you need to know how many feet are in a mile. What happens if you never memorized this fact? Of course you can look it up online or elsewhere, but the point is that this fact must be given to you as there is no way for you to derive it out yourself. This is true about all parts of the English system: you have to memorize all the facts that are needed for different measurements.

Lesson Objectives

 * Understand the metric system and its units.
 * Convert between units.
 * Use scientific notation in writing measurements and in calculations.
 * Use significant figures in measurements.

The Metric System
In the metric system, you need to know (or yes, memorize) one set of prefixes and then apply them to each type of measurement. Then if a larger measurement is needed, such as kilometers, but you have used a meter stick, you only need to move the decimal to convert the units.

If you have measured the distance as 60.7 meters, what is the length in kilometers?

Solution: 60.7 meters = 0.0607 kilometers since there are 1,000 meters in 1 kilometer.

Not only can you easily convert from kilometers to meters, but conversions, such as liters to cubic meters, are also easy. Try converting from cubic feet to gallons! All metric system conversions simply require the moving of the decimal and/or adding zeros. You don't even need a calculator. On the other hand, if you had to convert from miles to inches, not only would you have to remember all of the conversion factors, but you would probably also need a calculator to make the conversion.

Metric Prefixes
The metric system uses a number of prefixes along with the base unit. To review: the basic unit of mass is a gram (g), that of length is meter (m), and that of volume is liter (L). When the prefix centi is place in front of gram, as in centigram, the measure is now $$\tfrac{1}{100}$$ of a gram. When milli is placed in front of meter, as in millimeter, the measure is now $$\tfrac{1}{1000}$$ of a meter. Common prefixes are in Table 2.2:

Unit Conversions
Making conversions in the metric system is relatively easy: you just need to remember that everything is based on factors of ten. For example, let's say you want to convert 0.0856 meters into millimeters. Looking at the chart above, you can see that 1 millimeter is 10−3 meters; another way to say this is that there are 1000 millimeters in one meter. You can set up a mathematical expression as follows:


 * $$0.0856\,m \times \frac{1000\,mm}{1\,m} = ?\,mm$$

When you solve this equation, you first want to see which units to divide out. In this case, you notice meters appear in both the numerator and denominator, so you will be able to cancel them.


 * $$0.0856\,\underline{m} \times \frac{1000\,mm}{1\,\underline{m}} = ?\,mm$$

Now all that is left to do is multiply 0.0856 by 1000. To do this, you are just going to move the decimal point three places to the right:


 * $$0.0856\,m \times \frac{1000\,mm}{1\,m} = 85.6\,mm$$

Convert 153 grams to centigrams.

Solution:

$$153\,\text{grams} \times \frac{100\,\text{centigrams}}{\text{gram}} = 15300\,\text{centigrams}$$

Scientific Notation
Scientific notation is a way to write very large or very small numbers (Figure 2.4); the decimal point is moved so that there is one digit in the unit's position and all of the decimal places are held as a power of ten. This is important in chemistry because many of the measurements we make either involve very large numbers of atoms/molecules or very tiny measurements, such as masses of electrons or protons. For example, consider a number such as 839,000,000. While this number is not too difficult to write out, it is more conveniently written in scientific notation. Written in scientific notation, this number becomes: 8.39. The "108" means that ten is multiplied by itself eight times: 10×10×10×10×10×10×10×10. As you can see, writing 108 is much easier!



We can also use scientific notation to write very small numbers. Take a number such as 0.00000481. It is easy to make mistakes in counting the number of zeros in this number. Also, many calculators only let you enter in a certain number of digits. When we write this in scientific notation, it is important to notice that the measurement is less than one, therefore, the exponent on 10 will be negative: this number becomes 4.81. In this case, the decimal point was moved six places to the right.

It is important that you know how to perform calculations using numbers written in scientific notation. For example, the following problem shows two numbers with exponents being multiplied together:


 * (2.90)(1.60) =?

To solve this problem, you would multiply the terms (2.90 and 1.60) like you normally would; then you would add the exponents:

103 × 106 = 109
 * 2.90 × 1.60 = 4.64

Therefore, combining these values gives the answer 4.64.

Significant Figures
The tool that you use determines the number of digits that will be in a measurement. For example, if you say an object has a mass of "5 kg", that is not the same as saying it has a mass of "5.00 kg" since you must have measured the masses with two different tools – the two zeros in "5.00 kg" would not be written if the tool that was used could not measure to two decimal places. Even though the mass seems to be the same, the uncertainty of the measurement is not. When you say "5 kg", that means you have measured the mass to within ± 1 kg. The actual mass could be 4 or 6 kg. For the 5.00 kg measurement, you have measured the mass to within ± 0.01 kg, so the actual mass is between 4.99 and 5.01 kg.

Using Significant Figures in Measurements
How do you know how many significant figures are in a measurement? General guidelines are as follows:


 * Any nonzero digit is significant (4.33 has three significant figures).
 * A zero that is between two nonzero digits is significant (4.03 has three significant figures).
 * All zeros to the left of the first nonzero digit are not significant (0.00433 has three significant figures).
 * Zeros that occur after the decimal are significant. (40.0 has three significant figures. The zero after the decimal point tells us that the value was measured to the tenths place).
 * Zeros that occur without a decimal are not significant (4000 has one significant figure since the zeros are holding the 4 in the thousands position).

How many significant figures are in the number 1.680?

Solution:

There are three nonzero digits and one zero appears after the decimal point. Therefore, there are four significant figures.

How many significant figures are in the number 0.0058201?

Solution:

There are 4 nonzero digits and 1 zero between two numbers. Therefore, there are 5 significant figures. The first three zeros are not significant since they are simply holding the number away from the decimal point.

Lesson Summary

 * The metric system is a decimal system; all magnitude differences in units are multiples of 10.
 * Unit conversions involve creating a conversion factor.
 * Very large and very small numbers are expressed in exponential notation.
 * Significant figures are used to express uncertainty in measurements.

Review Questions

 * 1) Convert the following linear measurements:
 * (a) 0.01866 m = _______________ cm
 * (b) 2156 mm = ______________ m
 * (c) 15.38 km = ________________ m
 * (d) 1250.2 m = ________________ km
 * 1) Convert the following mass measurements:
 * (a) 155.13 mg = ________________ kg
 * (b) 0.233 g = _________________ mg
 * (c) 1.669 kg = ________________ g
 * (d) 0.2885 g = ________________ mg
 * 1) Write the following numbers in scientific notation:
 * (a) 0.0000479
 * (b) 251,000,000
 * (c) 4260
 * (d) 0.00206
 * 1) How many significant figures are in the following numbers?
 * (a) 0.006258
 * (b) 1.00
 * (c) 1.01005
 * (d) 12500

Vocabulary

 * scientific notation
 * A shorthand way of writing very large or very small numbers. The notation consists of a decimal number between 1 and 10 multiplied by an integral power of 10. It is also known as exponential notation.


 * significant figures
 * Any digit of a number that is known with certainty plus one uncertain digit. Beginning zeros and placeholder zeros are not significant figures.