For work in the laboratory you will need to become familiar with the correct ways to express the data that you collect. For any experimental data, one of the most important features of the numbers collected are the units that accompany them. In publications, SI units are most often used as they are known and accepted internationally. In the laboratories you will complete we will normally use metric units but not necessarily the SI units so you need to become familiar with both. Several useful tables containing these units are found in the Appendices of this lab manual. There are also tables of conversion factors, physical constants and metric prefixes.
Measurement
When measurements are taken, there is always some degree of uncertainty. The size and type of uncertainty (and therefore possible errors) depend upon the care with which a measurement is made. These errors typically take two forms: errors arising from the imperfections of the instrument used to make the measurement (mis-calibration) and errors arising from the skill or technique of the experimenter. In the first case, the errors are systematic (all in the same direction). In the second case, the errors will tend to be random (sometimes in one direction, sometimes in another). The two types of errors affect measurements differently. Systematic errors affect accuracy (closeness to the “true” value). Random errors affect precision (how well a set of data agree). In taking measurements in the laboratory, we are concerned with both accuracy and precision of measurements. One way to minimize the impact of any error is to take multiple measurements of the same value and use a statistical treatment of the set of values obtained, such as average and/or standard deviation to determine the value. While it is seldom possible to determine accuracy (the “true” value is not often known), the size of the standard deviation does give some indication of the precision of a series of measurements.
Significant Figures
In the chemistry laboratory, a number of different tools and instruments are used to make measurements. Most of these instruments are very finely machined or calibrated to produce accurate and precise values. For instance, which do you think would produce a better value for the length of a room, a measurement in notebook lengths, or a measurement produced with use of a small ruler? No matter how well calibrated a given instrument may be, there is still some degree of uncertainty in the measurement made.
Look at the smallest delineating mark on a metric ruler. The numbered marks are centimeters and the smallest mark is a millimeter (0.001 m). If an object is measured, it may fall between the 14.1 cm mark and the 14.2 cm mark. An estimate of the length is then made in the next decimal place (one place past the marks), and the object’s length is determined to be 14.13 cm. In this example, the uncertainty is ± 0.01 cm, because we have estimated the final decimal place. Although the last digit is not certain, it is still reasonably reliable and should be reported. Any digits that are considered reliable are important for calculations involving a measurement, and therefore, these digits are known as significant digits or significant figures. In using any measuring device, an estimate of one decimal place past the marks should be made, thus the uncertainty in any measurement is in the last digit. On some instruments such as the barometer, there is a built-in estimating device called a Vernier scale, which allows the final decimal to be estimated more quickly. In reading various digital displays, the last digit is also assumed to be an estimate, unless the instrument states a different uncertainty. Often, very sensitive devices will state explicitly what the error in the measurement is calibrated to be.
It stands to reason that any time a series of measured values is combined mathematically, the value which is least precisely known determines the precision of the whole set. The following set of rules will help to clarify how to determine the number of significant digits in a measurement and how any combination of measured values should be rounded to show an appropriate number of significant figures in the end result.
Rules for determining significant figures:
1. All nonzero digits are significant regardless of their position relative to a decimal point.
2. Any zeros surrounded by nonzero digits are significant:
1.101 has four significant figures
750.25 has five significant figures
303 has three significant figures
3. Zeros to the left of all nonzero digits are not considered significant because they serve only as decimal place holders.
0.08206 has four significant figures (start with the 8)
0.0003 has only one significant figure
4. Zeros to the right of the last nonzero digit may or may not be considered significant.
If there is a decimal point to the left of any of this type of zero, then they are to be considered significant.
0.03750 has four significant figures (start with the 3)
69.0 has three significant figures
10.000 has five significant figures
5. If the decimal point is to the right of this type of zero, then it is necessary to know what the number represents in order to determine the amount of significant figures.
Example: The number 100 could have one or three significant figures depending on its context. If it represented an approximation of the volume of water in a glass (about 100 mL), then it would only be considered to have one significant figure. However, if the water had been measured in a graduated cylinder, then all three digits would be significant.
Often, this kind of guessing can be avoided if the measurement is recorded in scientific notation. For instance, in the above example, the number could be written as 1.00 * 102 mL, indicating that there are 3 significant digits rather than one in the number.
Rounding Numbers
The number of significant figures in a measurement becomes very important when doing calculations with collected data. For example, the density of an object should not be reported to four significant figures if its weight or volume can only be recorded to three significant figures. Similarly, the density should not be reported to only two significant figures, because useful information would be discarded. Generally, the calculated result is only as reliable as the least precisely measured value used in the calculation. (In the Examples: below, all numbers are rounded to the hundredth place for simplicity. The least number of significant figures present in the calculation would determine the actual number to be reported.)
Rules for rounding
1. If the first digit being dropped is lower than 5, then the previous digit is not changed.
Examples: 1.234 rounds to 1.23
0.091 rounds to 0.09
2. If the first digit being dropped is higher than 5, then the previous digit is increased by 1.
Examples: 2.3154 rounds to 2.32
78.987 rounds to 78.99
3. If the digit to be rounded is exactly 5 (or 5 followed by zeroes), then the number should be rounded to be even.
Examples: 3.72500 rounds to 3.72
0.975 rounds to 0.98
Rounding during addition or subtraction
Rounding in addition and subtraction is determined by the least precisely measured value, not just the count of significant figures. It is the placement of the decimal relative to the significant figures which determines how many significant figures the answer should have.
Examples: 45.5609 + 0.975 + 34.9 + 56.43 = 137.8659, but the answer should be rounded to 137.9, because 34.9 is the least precisely known value, therefore, the answer should be recorded to only one decimal.
8.674 - 3.09 = 5.584, but the answer should be rounded to 5.58, because 3.09 is significant to the hundredths place and the answer should be the same.
Note that addition or subtraction occurs first and then rounding follows as the final step.
Rounding during multiplication and division
As stated earlier, a calculated result can only be as reliable as its least precisely measured value. In multiplication and division, this means that the result of a calculation should have the same number of significant figures as the number with the fewest significant figures used. If this is not done, the precision of the calculated result is overestimated.
Examples: 108.14 * 0.0015 = 0.16221, is rounded to 0.16 because 0.0015 only has two significant figures.
457.2 / 625 = 0.73152, is rounded to 0.732 because 625 contains three significant figures.
The role of conversion factors in rounding
Numbers in conversion factors can be either exact or inexact. Exact numbers are used when conversions are made from one set of units in the metric system to another set also in the metric system. The prefixes in the metric system are similar to definitions. There are exactly 100 centimeters in 1 meter, because of this both 100 and 1 are exact numbers. Exact numbers are considered to have an infinite number of significant figures. Therefore, when doing this type of conversion, the conversion factors do not limit the number of significant figures the answer can have. Conversions from the metric system to the English system (and vice versa) involve inexact numbers. There are approximately 30.5 cm in 1 foot. The 1 foot is the quantity being defined, so it is an exact number. However, 30.5 is a rounded value, which makes it an inexact number. Inexact numbers do affect the number of significant figures that can be present in the answer.
Examples: Exact: 355 mL * (1 L/1000 mL) = 0.355 L
Inexact: 27.9654 cm * (1 ft/30.5 cm) = 0.916893607 ft., but this is rounded to 0.917 ft. because of the inexact number in the conversion.
Scientific Notation
One way to express very large or very small quantities is to use exponential or “scientific” notation. Using powers of 10 is a reasonable way to specify the precision of a measurement. Very small numbers have a negative exponent, and very large numbers have a positive exponent. The rules on rounding remain the same. Remember in addition and subtraction to adjust the values to have the same exponent before performing the operation.
Examples: 1760 = 1.760 * 103, showing 4 significant figures
0.000000530 = 5.30 * 10–7, showing 3 significant figures
Addition/Subtraction
2.761 * 102 + 1.32 * 101 =
2.761* 102 + 0.132 * 102 = 2.893* 102
1.1941 * 10–2 - 8.62 * 10–3 =
11.941 * 10–3 - 8.62 * 10–3 =
3.321* 10–3 = 3.32 * 10–3
Multiplication/Division
(2.761 * 102) * (1.32 * 101) =
3.64452 * 103 = 3.64 * 103
(1.1941 * 10–2) / (8.62 * 10–3) = 1.38366 * 101 = 1.38* 101
Significant figures using logarithms
Logarithms have two parts: 1) The characteristic is the portion of a logarithm to the left of the decimal point, and reflects the exponent. 2) The mantissa is the portion to the right of the decimal and reflects the value of the measured quantity.
Example: log 559 = 2.747, 2 is the characteristic and 747 is the mantissa
The rule for determining the number of significant figures in a logarithm is that the mantissa should have the same number of significant figures as the measured quantity.
Examples: log 7 = 0.8
log 7.0 = 0.85
log 7.00 = 0.845
Likewise, the same rule applies when the operation is reversed (antilog).
Examples: antilog 0.60 = 4.0
antilog 0.602 = 4.00
antilog 0.6021 = 4.000
Graphing
Using graphs to illustrate a set of data and make predictions is a typical experimental approach. In many experiments, one parameter is varied systematically while observing what changes there are in a second parameter, such as observing the volume of a gas when pressure is increased or decreased. If a linear relationship can be found between two parameters, then predictions are possible. Thus graphs can be used both to visually inspect a set of data for linear behavior and predict other information. In order to make the best use of graphical data, a few points on the best way to make them are in order.
1. Use a whole sheet of graph paper for each graph. If the goal is to read information from a graph, then shouldn’t the graph be as large as possible?
2. Decide which of the two parameters on the graph is independent (usually this is the one which is varied at regular intervals) and which is dependent (the one which is being observed.) The independent variable is plotted on the x axis, the dependent variable on the y axis.
3. The best graph of a set of data does not always have the origin, (0, 0), on the graph. Look at the highest and lowest data points of the variables to be plotted. Choose convenient numbers slightly higher and lower than those data points and make them the extremes on the graph. Again, the goal is to spread the data out on the graph as much as possible.
4. Label axes clearly. This should include regularly spaced marks and the units of measurement for each axis. A graph will not be of much use if the axis is not linear.
5. Put a title on the graph that will explain something, not just “graph of y vs. x”. For instance, “Volume of argon gas as a function of pressure at 25 oC.”
6. Use a ruler or curve to draw the best line through the plotted points. One way of showing the points is putting a circle around each one. Alternatively, draw error bars, showing the precision of the measurement taken.
7. If a graph is not linear, don’t try to draw a straight line through the points, draw a curve. To get a straight line, try a mathematical manipulation of one or both of the variables, such as inversion (1/y) or logarithm (log y) and do another plot. It may take two or three attempts to find something that works to yield a straight line.
8. Once a straight line is obtained, the slope of the line can be found using two conveniently read points on the line, which should not be previously plotted data points. Remember from algebra that slope, m, is Dy/Dx.
9. Using the equation for a straight line, y = mx + b, also from algebra, the y intercept can be found by substituting any (x, y) point and the slope, m, and solving for b.
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