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AND SIGNIFICANT FIGURES

Chemistry deals with VERY LARGE and VERY SMALL . An example of a VERY LARGE is the number of water molecules in a glass of water (approx. 10,000,000,000,000,000,000,000,000). An example of a VERY SMALL number is the number of pounds that a single electron weighs (0.0000000000000000000000000000020). It becomes extremely tedious to write out all the zeros required for these very large and very small numbers. It is difficult to keep track of all the zeros, and won't display a number that long. By using scientific notation, we can avoid these problems of very large and very small numbers.

In scientific notation, numbers are expressed in the form: M x 10n, where M is a number between 1 and 10 and n is an integer. Thus, we would write the number of water molecules in a glass of water as 1 x 1025 molecules and we would write the weight in pounds of a single electron as 2.0 x 10-30 pounds. The factor of 1025 means that the point should be moved 25 places to the right from (1.), and the factor of 10-30 means that the decimal point should be moved 30 places to the left from (2.0).

EXAMPLES

common notation scientific notation

0.00257 <------> 2.57 x 10-3 65,000 <------> 6.5 x 104 100 <------> 1 x 102

Keep in mind that if a number in scientific notation has a negative exponent, that number is less than one and if the number has a positive exponent, that number is greater than ten.

SMALL numbers <------> negative exponents LARGE numbers <------> positive exponents

In chemistry, most of the numbers we deal with represent measurements of some kind. So when we look at a reported number we are interested to know how many of the digits reported we can be absolutely certain of. Leading zeros are considered "insignificant", i.e. they are "just" place-holders for the decimal point and do not represent measured digits. Trailing zeros are USUALLY "insignificant", but not always.

Consider the following example:

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4 5

You are measuring the length of an object with a centimeter ruler, and you determine that the length falls between 4.3 cm and 4.4 cm. You approximate that the actual length is 4.36 cm and record that measurement. You have recorded the measurement to three significant figures. The "4" and the "3" are "certain" digits, there is no guesswork involved in their determination. The "6" is an estimated digit - it is an - however, it is still significant because it gives meaningful information about how far between the two markings the measurement actually lies. Recording any more digits would be ABSOLUTELY MEANINGLESS! To record the length as 4.36294 cm, for example, would be extremely misleading. THE GENERAL RULE IS THAT THE SIGNIFICANT FIGURES (OR SIGNIFICANT DIGITS) CONSIST OF ALL OF THE CERTAIN DIGITS PLUS THE FIRST ESTIMATED DIGIT.

The measurement 4.36 cm has 3 significant figures.

Let's convert this to meters: 4.36 cm (1 m/100 cm) = 0.0436 m The measurement 0.0436 m has 3 significant figures.

Let's convert this to micrometers (also called microns): 0.0436 m (106 m/1 m) = 4.36 x 104 m The measurement 4.36 x 104 m, or 43600 m has 3 significant figures.

Let's convert this to kilometers: 0.0436 m (1 km/1000 m) = 4.36 x 10-5 km The measurement 4.36 x 10-5 km, or 0.0000436 km has 3 significant figures.

NOTE: Converting to different metric units DOES NOT CHANGE the number of significant figures in the measurement!!!!!

The following rules are used to determine the number of significant figures in a reported measurement.

** Nonzero digits are significant. ** Captured zeros are significant (20.5 g -> 3 s.f.). ** Leading zeros are not significant (0.00025 m -> 2 s.f.). ** Trailing zeros are significant only if there is an explicit decimal point in the number (0.0250 m -> 3 s.f.; 250,000 m -> 2 s.f.; 2.50 x 105 m -> 3 s.f.) ** Exact numbers such as counting numbers (e.g. there are 27 students in the class) or numbers true by definition (e.g. there are 100 cm in 1 m or 12 inches in 1 foot) can be thought of as having an infinite number of significant figures (100.0 cm/1 m, 12.0 inches/1 foot) and thus will never limit the number of significant figures in a calculation. ** Note that scientific notation displays only significant figures in the mantissa! Leading zeros will not occur in a number between 1 and 10. And non-significant trailing zeros must be dropped.

One of the advantages of using scientific notation is that (except for exact numbers) only significant digits are retained. This makes it much easier to count significant figures!

0.00025 m ---> 2.5 x 10-4 m ---> 2 s.f. 250,000 m ---> 2.5 x 105 m ---> 2 s.f. ---> 2.50 x 105 m ---> 3 s.f. 20.5 g ---> 2.05 x 101 g ---> 3 s.f.

When performing calculations, use the following rules:

** Addition and Subtraction: Keep the same number of decimal places in the answer as in the measurement with the fewest decimal places. Example: 23.573 g + 0.15 g = (23.723 g) = 23.72 g Since the two measurements are to the thousandths place and the hundredths place, respectively, your answer should be reported to the hundredths place.

** Multiplication and Division: Keep the same number of significant figures in the answer as in the measurement with the fewest significant figures. Example: 5.25 g/mL x 0.50 mL = (2.625 g) = 2.6 g Since the two measurements have three significant figures and two significant figures, respectively, your answer should be reported to two significant figures.