Every measurement has an uncertainty. When you give directions to your house, you might give distances with a certain amount of give and take. For example, you might tell a person to turn "five miles after the stoplight." Do you mean five miles plus or minus an inch? Or maybe five miles plus or minus a quarter mile?
In chemistry (like in most other fields!) we concern ourself with the size of that plus/minus interval. Just how big it is will determine how many significant digits there are in a result.
For example, when someone writes 4.5332 g they are implying that the number of grams is known to +/- 0.0001 gram. If that mass were to be compared to another mass of say 43.23 g (which as written is implied to be known to +/- 0.01 g) we would say that the uncertainty in the second mass is larger than in the first. So there would be no point in reporting the sum of the two masses to any smaller uncertainty than +/- 0.01 g. So claiming that the sum was 47.7632 g (+/- 0.01 g) would be silly since the last two digits obviously have no meaning. The correct way of reporting the sum would be to round the result to the correct uncertainty of +/- 0.01 g.
| Rule: Addition and Subtraction |
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| When adding or subtracting values, always retain the largest absolute uncertainty in the result. |
| Example |
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| Question: A beaker is weighed and found to have a mass of 78.45 g. A sample of sodium chloride is added to the beaker and the beaker is weighed again using a different balance. The NaCl plus the beaker are found to have a mass of 79.6743 g. What is the mass of the sodium chloride? |
| Solution: First, take the difference to get the mass of the sodium chloride:
79.7643 g (mass of beaker and NaCl)
- 78.45 g - (mass of beaker )
--------- -------------------------
3.3143 g (mass of NaCl)
Next, round to the correct number of significant digits:
3.3143 g ==> 3.31 g
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| Note: We can retain only those decimal places that the uncertainty allows. In this case, +/- 0.01 g for the mass of the beaker meant we needed to round to the nearest hundredth of a gram. |
Counting significant digits is as simple as 1-2-3. Just follow the rules!
| Counting Significant Digits | |
|---|---|
| Rule 1 | All non-zero digits and "trapped zeros" (zeros which occur between significant digits) are significant digits. |
| Rule 2 | ALeading zeros are never significant digits. All they do is hold the decimal place. |
| Rule 3 | Trailing zeros to the right of the decimal are always significant digits. If they weren't no one would include them! |
| Rule 4 | railing zeros to the left of the decimal place may or may not be significant. It is ambiguous in these cases. To help remove the ambiguity, people often use scientific notation. Then if the zeros are significant digits, it is clear because the occur as trailing zeros to the right of the decimal place. |
| Rule 5 | Conversion factors which are based on definitions (such as 1000 mL per L or 3600 seconds per hour) have an infinite number of significant digits (because they are exact.) Exact numbers also occur in things you can count with zero uncertainty (such as 12 eggs or 37 railroad cars.) |
| Rule 6 | Some physical constants are so well determined that the number of significant digits can be chosen. An example of this is pi which can be used to 3 significant digits (3.14) or 8 (3.1415927) when really high prescision is required. |
| Some Examples | ||
|---|---|---|
| 12.34 g | 4 significant digits | Rule 1 |
| 0.000234 L | 3 significant digits | Rules 1, 2 |
| 102.0 m | 4 significant digits | Rules 1, 3 |
| 1200 km/sec | 2, 3, or 4 significant digits | Rules 1, 4 |
| 6.02 x 1023 atoms | 3 significant digits | Rule 1 |
| 1.400 x 10-3 mol | 4 significant digits | Rules 1, 3 |
| 0.002320 coulomb | 4 significant digits | Rules 1, 2, and 3 |
| 5280 ft/mile | infinite number of significant digits | Rule 5 |
| c (speed of light in a vaccum) | generally, as many as you need | Rule 6 |
The relative uncertainty in a reported value is given by 1 in the last decimal place reported (or the last significant digit) divided by the value of the number. The number of significant figits in a reported measurement is related to the relative uncertainty of the number. For example, the value 3.2950 g would be said to be know to within 1 part in 32950. Some more examples can be seen in the following table.
| Value | Relative Uncertainty | Absolute Uncertainty |
|---|---|---|
| 125 mL | 1 part in 125 | ± 1 mL |
| 9.700 km | 1 part in 9700 | ± 0.001 km |
| 6.02 x 1023 atoms | 1 part in 602 | ± 0.01 x 1023 atoms |
| 0.0024 g | 1 part in 24 | ± 0.0001 g |
| 347.34 min | 1 part in 34734 | ± 0.01 min |
It is convenient (and produces reasonable results) to approximate the relative uncertainty based on the number of significant digits in a value. In this way, the approximate relative uncertainty depends only on the number of significant digits in the value. Examples follow.
| Value | Approximate Relative Uncertainty | Absolute Uncertainty |
|---|---|---|
| 125 mL | 1 part in 1000 | ± 1 mL |
| 9.700 km | 1 part in 10000 | ± 0.001 km |
| 6.02 x 1023 atoms | 1 part in 1000 | ± 0.01 x 1023 atoms |
| 0.0024 g | 1 part in 100 | ± 0.0001 g |
| 347.34 min | 1 part in 100000 | ± 0.01 min |
This is important because in multiplication and division, we rertain the largest relative uncertainty. That means that the number of significant digits reported in the answer will mtch that of the component value with the fewest significant digits. Important note: This rule applies to multiplication and division only - not to addition and subtrction!
| Example |
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| Question: How many moles are contained in 0.025 L of a 0.1104 mol/L solution? |
Solution:
0.025 L 0.1104 mol
------- x ---------- = 0.0027600 mol ==> 0.0028 mol
L
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| Since the value 0.025 L only contains two significant digits, we round the result to two significant digits. |
| Example |
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| Question: How many Joules are needed to raise the temperature of 15 g of H2O 24.2 K? |
Solution:
15 g 24.2 K 4.184 J
---- x ------ x ------- = 1518.7920 J ==> 1500 J
g K
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| Since the value 15 g only contains two significant digits, we round the result to two significant digits. |
| Example |
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| Question: What is the frequency of the sodium "D" line (used in many street lamps) given it's wavelength of 588.995 nm? |
Solution:
1 109 nm 2.9979256 x 108 m
---------- x ------ x ----------------- = 1.7657632 x 1020 s-1
588.995 nm m s
==> 1.76576 x 1020 s-1
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| Since the value of the wavelength 588.995 nm is known to six significant digits, the final result should be determined to six significant digits. Thus, the value of the speed of light must have more than six significant digits so it does not limit the number of significant digits in the result. |