Technique B Significant Figures_1_1_1

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Technique B: Significant Figures

Section 1: Purpose of Technique

The number of significant figures of a quantity is the count of digits required to maintain precision. The number of significant figures is very important. Proper use of significant figures gives an immediate estimate of how precisely the quantity has been measured.

For example, a bathroom scale shows a value of 201.3 pounds for a person’s weight. When converting from pounds to kilograms, you divide by a conversion factor of 2.20462 pounds per kilogram. Modern calculators show a result of 91.30825267 kg, but not all of these digits are needed (or desired). You should keep only the digits necessary to describe the precision of the original measurement but in the new units. Using the rules for significant figures and mathematical operations that follow, the result should be reported as 91.31 kg.

Section 2: Rules for Significant Figures

Determine the number of significant figures in a measurement:

1. All nonzero digits are significant.

Examples: 4.79 has three significant figures.

2779 has four significant figures.

856.429 has six significant figures.

2. Zeros in the middle of a number are significant.

Examples: 1.049 has four significant figures.

4.50809 has six significant figures.

780.12 has five significant figures.

3. Leading zeros (at the very beginning of the number) are not significant.

0.032 has two significant figures. (It may be written as 3.2 × 10^-2).

0.0000411 has three significant figures. (It is also 4.11 × 10^-5.)

If you are not sure of how many significant figures are present, convert the number to scientific notation. Numbers written in scientific notation are in the form of a coefficient multiplied by 10 raised to an exponent. The number of significant figures is the number of digits in the coefficient.

Examples: 0.00579 has three significant figures (may be written as 5.79 x 10^-3)

0.048023 has five significant figures (may be written as 4.8023 x 10^-2)

0.000024 has two significant figures (may be written as 2.4 x 10^-5)

4. Zeros at the end of a number are significant if the number contains a decimal point.

Examples: 1.30 has three significant figures.

4.00 has three significant figures.

350. has three significant figures. (The decimal point at the end is the indication that the zero is significant.)

5. Zeros at the end of a number which does not contain a decimal point may or may not be significant (it is unclear). When in doubt, assume they are not significant.

The number 2500 could have been measured to the nearest ± 100 or to the nearest ± 10 or to the nearest ± 1. Since it is not indicated, you are forced to assume the worst case: the greatest uncertainty. 2500 has two significant figures. (2.5 × 10³)

140 has two significant figures. (1.4 × 10²)

If you are recording a number such as “2500”, it is best to include the error estimate so there is no ambiguity.

Section 3: Rules for Rounding

If a calculated number has too many digits, it needs to be rounded to the appropriate number of significant figures. The rules for rounding are as follows:

1. Increase the last retained digit by one if the next digit is greater than or equal to 5.

Examples: 4.248 rounded to three significant figures. It rounds to 4.25

347.9 rounded to three significant figures. It rounds to 348

7815.246 rounded to six significant figures. It rounds to 7815.25

2. Decrease the last retained digit by one if the next digit is less than 5.

Examples: 94.24571 rounded to three significant figures. It rounds to 94.2

1.50802 rounded to five significant figures. It rounds to 1.5080

624804 rounded to two significant figures. It rounds to 620000

NOTE:

There are several ways of dealing with rounding when the next digit after the last retained digit is 5. The convention we are using is covered by Rule 1 (above). However, your instructor may want you to use a different one.

Section 4: Determining the number of significant figures in a calculation

A calculation cannot be more precise than the least precise measurement used to do the calculation. (The least precise measurement limits the precision of the result.)

1. Multiplication and Division:

The number of significant figures in the result should be the same as that in the number with the least significant figures.

3.42 ÷ 1.3 = 2.630769 Rounded result = 2.6

The reported result of 2.6 has two significant figures. This is because the number “1.3” in the original calculation has 2 significant figures (the fewest in this calculation).

43.10 × 9.22713 = 397.6893 Rounded result = 397.7

The reported result of 397.7 has four significant figures. This is because the number “43.10” in the original calculation has 4 significant figures (the fewest in this calculation).

7.58432 * 18.54 * 6.542 ÷ 42.5 = 21.64452 Rounded result = 21.6

The reported result of 21.6 has three significant figures. This is because the number”42.5” in the original calculation has 3 significant figures (the fewest in this calculation)

2. Addition and Subtraction:

The number of decimal places in the result should be the same as in the number with the fewest decimal places. The digits in bold are the important ones to look at.

4.381 31.5563

+131.4 - 1.410

135.781 30.1463

Round to nearest 0.1 Round to nearest 0.001

Report result as 135.8 Report result as 30.146

3. Logarithms and Exponents:

A logarithm is made of two parts, the characteristic and the mantissa. The characteristic is everything before the decimal point in a logarithm. The mantissa is everything after the decimal point in a logarithm. Unless the base of the log is specified differently, it is assumed to be the common log of base 10.

Example 1: log (345.2) = 2.5381 which means that 10^2.5381 = 345.2

In this example, the number “2” in the logarithm is the characteristic and “.5381” is the mantissa.

Example 2: log (0.0024) = -2.62 or, 10^-2.62 = 0.0024

In this example, the number “-2” is the characteristic and “.62” is the mantissa.

With regard to significant figures, the number of significant figures in the mantissa should be the same as the number of significant figures of the original number. In Example 1, "345.2" and ".5381" have the same count of four significant figures. In Example 2, "0.0024" and "0.62" have the same count of two significant figures.

Converting quantities into logarithms is commonly found when working with pH and concentrations. The mathematical definition of pH is: pH = - log (concentration of H⁺ in moles per liter). Moles per liter is abbreviated as ‘M’.

Example: Determine the pH of a solution containing a concentration of H⁺ of 2.56 × 10^-4 M, the calculation is:

pH = - log (2.56 × 10^-4 M) = - (-3.591760035) which rounds to 3.591

In this example, the original concentration (2.56 × 10^-4 M) has 3 significant figures (shown in bold). The mantissa of "3.591" has 3 significant figures.

4. Physical Constants, Discrete Numbers, and Mathematical Constants.

Physical Constants

Final results should reflect the uncertainty due to experimental error, not from calculations. Many physical constants have been measured to great precision so you have many more significant figures than you require. For example, Avogadro's Number (6.0221409 x 10²³) or the speed of light (2.99792 x 10⁸ meters/second) have been measured to great precision. If you are using these constants in a calculation, and are hand-entering the numbers in your calculating device, then include at least two more significant figures in the constant than the quantity with the fewest significant figures in your calculation.

Example: If you are calculating how many atoms are present in a mass of 1.25 x 10^-5 grams of lead (Pb), using the atomic mass of 207.19 grams/mole, the calculation should be set up as:

\(\frac{1.25 \times 10^{-5} \text {grams } P b}{1} \times \frac{1 \text { mole } P b}{207.19 \text { grams } P b} \times \frac{6.0221409 \times 10^{23} \text { atoms } P b}{1 \text { mole } P b}=3.63321 \times 10^{16}\) atoms \(P b\)

Note that the units cancel except for atoms Pb. The rounded result should have three significant figures, so the final result is 3.63 x 10¹⁶ atoms Pb.

In this example, the atomic mass of Pb is a reference value presented with five significant figures, two more than the minimum of three significant figures. Avogadro's Number of 6.0221409 x 10²³ should be entered in the calculation with at least five significant figures (more significant figures will not affect the calculated result after rounding).

Discrete Numbers

Measurements of discrete objects can be considered to have infinite precision. For example, there are precisely 12 eggs in a dozen (i.e., 12.00000…).

Example: If you are calculating a molecular weight that contains 2 copper atoms, the mass of the copper atom (63.55 g/mol) can be multiplied by 2 with no loss of significant figures.

Mathematical Constants

It generally does no harm to use too many significant figures in your constants, but it is not necessary. If your calculator has a constant such as π or e, then just use the button on your calculator and the number provided.

Example: The area of a circle is given by A= π r². For a radius of 4.247 cm, find the area.

A = 3.14159 × 4.247 cm × 4.247 cm = 56.6648 cm². This rounds to 56.66 cm².

Note that the value of π used has six significant figures, only two more than is needed for the quantity of “4.247 cm” which has four significant figures.

The use of 3.14 as a value of π in the above calculation would be incorrect because it has too few significant figures. The following calculation is provided to show what happens when you use too few significant figures.

Example of incorrect significant figure usage:

A = 3.14 × 4.247 cm × 4.247 cm = 56.6362 cm². This rounds to 56.64 cm².

Although it seems like a minor difference (0.02 cm²) between the two numbers, these differences can propagate through many calculations and result with real differences in results.

Section 5: Determining significant figures for measured numbers

Laboratory measuring equipment such as balances, scales, glassware marks, rulers, and calipers have limits on precision. The number of significant figures in measured quantities should imply the precision in the measurement. Equipment with digital readouts can be usually read as-is. All other equipment may have precision limits declared on them, or may be derived from the precision of the smallest graduation.

Refer to the specific users' manual, experiment or technique for additional guidance, or ask your instructor. Some general examples are provided below for common laboratory equipment.

For balances:

A balance that is precise to the nearest 0.01 grams (or 10 milligrams) will have a number precise to the nearest 0.01 grams. A mass measured on this balance as 5.24 g has three significant figures and is assumed to have an error of +/- 0.01 grams.

A balance that is precise to the nearest 0.1 milligrams will have a number precise to the nearest 0.0001 grams. A mass measured on this balance as 5.2400 has five significant figures and is assumed to have an error of +/- 0.0001 grams.

For rulers:

A ruler that is graduated to 0.1 centimeter (or 1 millimeter) will be measured to the nearest 0.1 cm or 1 mm. The length should be estimated to the nearest graduation. A length of 4.2 cm or 42 mm has two significant figures.

For graduated cylinders:

A 10 ml graduated cylinder that is graduated to 0.1 milliliter has a tolerance of 0.01 ml. It should be measured to the nearest 0.01 ml by estimation. A volume measured by this cylinder as 5.62 ml has three significant figures.

For burets:

A 50 ml buret that is calibrated to 0.1 milliliter can be estimated to the nearest 0.01 milliliter if sufficient distance exists between calibration marks. A volume measured as 15.24 ml has four significant figures.

For transfer pipets:

A 25 ml "Class A" transfer pipet has a specification of +/- 0.03 ml. Properly used, this transfer pipet will deliver 25.00 ml of a solution, which should be recorded with four significant figures. Note: A more formal error analysis may use the specific declared error limits and is beyond the scope of this document.

For volumetric flasks:

A 100 ml Class A volumetric flask has a tolerance of +/- 0.08 ml. Properly used, this flask will contain 100.0 ml of a solution, which should be recorded with 4 significant figures. Note: A more formal error analysis may use the specific declared error limits and is beyond the scope of this document.

Section 6: Other ways of reporting significance

Using significant figures is the most common way to present quantities with an estimate of the uncertainty in the last digit. It is a little more complicated to calculate statistically valid error range estimates.

However, sometimes specific error estimates are presented along with the number. Here are some results from metals analysis of a National Institute of Standards and Technology (NIST) standard reference material (Buffalo River Sediment, Standard Reference Material 2704):

Selected element	Weight %	Error
Aluminum	6.11	+/- 0.16
Calcium	2.60	+/- 0.03
Carbon	3.348	+/- 0.016
Iron	4.11	+/- 0.10
Magnesium	1.20	+/- 0.02

The declared numbers on the standard reference material have a minimum number of significant digits, and that the error estimates are explicitly stated. This way of presenting the limits of uncertainty with data contains more information and is generally preferred for precision work.