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Scientific Measurement

>> Parts of this equation/concept include:

A - Scientific Notation
B - Significant Figures
>> Calculations Involving Significant Figures
C - Using Units

 

A. Scientific Notation

Many numbers in chemistry are especially large or especially small. Scientific notation is a way to express these values conveniently and the precision to which the values are measured. Scientific notation takes the form of N x 10x, where N is a value between 1 and 10, which can be either positive or negative, and x is an integer. The N part of the value expresses the precision of the measurement. The x part of the value expresses the magnitude or position of the decimal place. One convenient way to remember the direction associated with x is that positive values of x represent large numbers and negative values represent fractions.

>> Example 1

Convert the following values in decimal notation to scientific notation.

  1. 0.0000000002
  2. 66,700,000
  3. 0.000507
  4. 7600

Solution:

  1. It is a fraction, so x is negative. For N to be between 1 and 10, the decimal must be moved behind the 2. That is 10 digits, so the value in scientific notation is 2 x 10–10.
  2. It is a large number, so x will be positive. For N to be between 1 and 10, the decimal must move between the two 6's—7 places. So the value is 6.67 x 107.
  3. 5.07 x 10–4
  4. 7.6 x 103

>> Example 2

Convert the following values in scientific notation to decimal notation.

  1. 7 x 109
  2. 2.4 x 10–5
  3. 7.27 x 10–8
  4. 8.4 x 103

Solution:

  1. a large number, so 7,000,000,000
  2. a decimal, so 0.000024
  3. 0.0000000727
  4. 8400

Calculations involving numbers expressed in scientific notation are easily performed on a scientific calculator. Scientific calculators can be easily identified by the presence of a "log" key. Scientific calculators (and spreadsheet programs like Excell) do not normally express scientific notation as it is written on a page. These devices generally give only the values of N and x. These two values may be separated by a space, may display an "E," or the x may appear as a superscript. You need to be familiar with the operation of your calculator to properly interpret the values. When the answer to a question is expressed in scientific notation, you will be expected to write the answer in the N x 10x format, regardless of the format your calculator uses.

Calculators normally express answers to calculations in decimal (regular) notation, if there is sufficient space. It will probably be valuable to consult the calculator's instruction manual for the method to convert the value given into scientific notation. It is easier than counting zeros and the calculator is less likely to round inappropriately.

Entering values of scientific notation on a scientific calculator requires a special key. This key is usually labeled "E", "EE," or "EXP." There will be a button on the calculator labeled "10x"; do not use this key to enter scientific notation. It has a different function and will often give the wrong answer. Besides, the correct button is easier. To enter a value in scientific notation, follow these steps.

  1. If N is negative, press the +/– key.
  2. Enter the value of N.
  3. Press the scientific notation key (E, EE, or EXP).
  4. If x is negative, press the +/– key (do not use the – function).
  5. Enter the value of x.

>> Example 3

Perform the following calculations.

  1. (1.48 x 10–2)(2.6 x 10–4) =
  2. 4.97 x 10–5 / 8 x 108 =
  3. 5.73 x 108 + 6.1 x 107 =
  4. (5.09 x 105 – 4.9 x 104) / 3.78 x 1010 =

Solution:

  1. 3.848 x 10–6, or 0.000003848 (They are the same number!)
  2. 6.2125 x 10–14
  3. 6.39 x 108
  4. 5.09 x 105 – 4.9 x 104 = 4.6 x 105 4.6 x 105 / 3.78 x 1010 = 1.2169312 x 10–5

 

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B. Significant Figures

The determination of significant figures might also be considered as "rules for rounding." To consistently express the true precision of a measurement or calculation, significant figures must be used. All answers in this and subsequent chapters are expected to be expressed with the appropriate significant figures.

To determine the significant figures in a measurement, report all certain values and one digit that is estimated (for example, estimating between the smallest markings on a ruler). If the value is given as a digital readout, the last digit of the readout is uncertain.

To determine significant figures in a number given as part of a problem, you must first determine whether the value is exact or measured. Exact numbers are either those that were counted or that are definitions. You should consider the context to determine whether a number is exact.

If the number is exact, it has an infinite number of significant figures. Since it is impossible to write down all the significant figures, it is customary to use as few digits as are necessary to accurately represent the number. The exact nature of the number must come from the context in which it is used.

If the number is measured, use the following rules to determine which digits are significant.

  1. All nonzero digits are significant.
  2. All zeros before the first nonzero digit are not significant.
  3. All zeros between nonzero digits are significant.
  4. All zeros after the last nonzero digit and to the right of the decimal are significant.
  5. Zeros to the right of the last nonzero digit and before the decimal are significant if the decimal is shown. If the decimal is not shown they are ambiguous (you can't tell).
  6. If the number is expressed in scientific notation, all digits in the "N" part are significant; no other digits are.

Hint: If the number of significant figures is ambiguous in decimal notation, use scientific notation instead.

>> Example 4

How many significant figures are in the following numbers?

  1. 0.0058
  2. 310.00
  3. 5039
  4. 0.090
  5. 4.670 x 10–6

Solution:

  1. Two. Leading zeros are not significant.
  2. Five. Zeros after the decimal and all in between are significant.
  3. Four. Zeros between nonzero digits are significant.
  4. Two. Leading zeros are not significant; trailing zeros are.
  5. Four. All digits in the "N" part of scientific notation are significant.

>> Calculations Involving Significant Figures

The rules for rounding after doing a significant figure calculation depend on the type of mathematical operation. Regardless, rounding uses the "weakest link" principle, and the final value will not be more precise than values that are used to do the calculation. Always do the calculations using all digits; then round that answer using the significant figure rules. Calculators do not understand significant figures and will give too many or too few. Rounding is your job.

For addition and subtraction, the answer should have the same number of decimal places as the value with the fewest decimal places.

>> Example 5

Report answers to the correct number of decimal places.

  1. 0.742 + 0.0259 =
  2. 4.00 – 3.5 =
  3. 1.1 x 107 + 9.11 x 106 =

Solution:

  1. 0.7679 = 0.768. The first value only had three decimal places, whereas the second had four. The last digit, 9, rounds up.
  2. 0.5. There is only one decimal place in the second answer. In addition and subtraction it is common to gain and lose significant figures.
  3. To determine decimal places it would be better to convert the values to decimal notation. 11,000,000 + 9,110,000 = 20,110,000. In the scientific notation of the first value, only the 1s are significant, whereas the second value has significant digits further to the right. Consequently, the answer can only go to the millions decimal place and the answer is 20,000,000. So that the reader can determine how many of these digits are significant, the answer should be reported in scientific notation = 2.0 x 107.

For multiplication and division the answer should have the same number of significant figures as the value with the fewest significant figures.

>> Example 6

Perform the calculation and report the answer with appropriate significant figures.

  1. (0.969)(0.0078) =
  2. 8.950/0.040600 =
  3. (6.0 x 105)(6.195 x 10–7) =
  4. (4.80 x 1013)/(8.00 x 104) =

Solution:

  1. 0.969 has three significant figures; 0.0078 has two significant figures. The answer, 0.0075582, should have two significant figures. Therefore the answer is 0.0076.
  2. 8.950 has four significant figures; 0.040600 has five significant figures. The answer should then have four significant figures = 220.4.
  3. 6.0 x 105 has two significant figures; 6.195 x 10–7 has four significant figures. The answer is 0.37.
  4. Both values have three significant figures. Therefore the answer should have three significant figures. The answer is 6.00 x 108.

For mixed operations (both addition and multiplication), use each rule in the same order as the mathematical operations.

>> Example 7

Perform the following operations and report the answer to the appropriate significant figures.

  1. (0.428 + 0.0804) / 0.009800 =
  2. (31.6 24.78) + (0.569 6.64) =

Solution:

  1. The addition is first 0.428 + 0.0804 = 0.5084. Only 0.508 is significant, but rounding is a last step, so calculate 0.5084/0.009800 = 51.87755. The sum/numerator had three significant figures and the denominator had four, so the answer should have three = 51.9.
  2. In this example the multiplication comes first (31.6 24.78) = 783.048.
    Three digits are significant or it is significant to the decimal.
    (0.569 6.64) = 3.77816. Three digits are significant.
    The sum of the preceding answers is 786.82616. Since the first value is only significant to the decimal, the answer should only be reported that far and the answer is 787.

>> Example 8

Report the answers to Example 3 in the scientific notation section to the appropriate significant figures.

Solution:

  1. 3.8 x 10–6
  2. 6 x 10–14
  3. 6.4 x 108
  4. 509,000 – 49,000 = 4.60 x 105 / 3.78 x 1010 = 1.22 x 10–5

 

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C. Using Units

All numbers in chemistry come with units, normally SI units. SI units use prefixes to change the magnitude of the number. You should memorize the relationship between the common prefix and the unit without the prefix. Since most people find whole numbers easier to work with than fractions, it would probably be easiest to learn the relationships that way. If u stands for the base unit (e.g., g, L, or m), the most common relationships are

1 Mu = 1,000,000 u
1 ku = 1,000 u
100 cu = 1 u
1000 mu = 1 u
1,000,000 u = 1 u

Remember that SI units are symbols, not abbreviations. Therefore no period is associated with the unit symbol.

When doing addition and subtraction calculations involving units, the units must be the same for each value added or subtracted. That unit will also be the unit of the answer. If the values are not the same, the units of one value should be changed to the units of the other.

>> Example 9

1.234 m + 35.4 cm =

The units must be the same. Since 1.234 m = 123.4 cm, the solution is 123.4 cm + 35.4 cm = 158.8 cm. (Note that both values, once in the appropriate units, go to the tenth decimal place, so the answer is also to the tenth decimal place, according to the significant-figure rules.) Alternately, since 35.4 cm = 0.354 m, the solution is 1.234 m + 0.354 m = 1.588 m. Since 1.588 m = 158.8 cm, both answers are the same and both answers are equally correct!

When doing multiplication and division, units are treated in the same way that variables are in an algebraic equation.

>> Example 10

  1. (0.316 g/mL)(21 mL) =
  2. (1.8646 cm)(1.28 cm) =

Solution:

  1. 6.6 g. Two significant figures; the milliliters cancel.
  2. 2.39 cm2. Three significant figures; cm cm = cm2.

 

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