�Before You Begin:
This material will make more sense if you have had some exposure to the metric system. To master this material you need to know basic algebra and scientific notation.
SI System of Measurement
Système International d’Unités, or SI, is the set of units for scientific measurements. This set of units is based on the metric system. Many students who are more comfortable with the English system really hate the metric system. This is largely because they have had to do too many English to metric conversions. The real value of the SI system is in converting units within the system. Calculating energy and pressure from SI base units is much easier than calculating these using base units from the English system.
SI Base Units
The base units are the fundamental units representing the most straightforward measurements.
The base units can be scaled up (or down) using multiples (or factors) of ten. The scale is indicated by prefixes familiar to those who have worked with the metric system.
SI and Metric Derived Units
Derived units are more complex units that require more than one dimension or type of measurement. Length is an example of a base unit. Volume is an example of a derived unit, because it is a three dimensional representation of length.
For more information about SI units see the National Institute of Standards and Technology web site.
Systematic and Random Error
In any measurement, chaos, instrumental limitations, and human failings prevent perfection. This lack of perfection in measurements is called error. Some errors are random errors, meaning that they cause some of the measurements to be a little high and some to be a little low. These are generally due to chaos, and, while they can be minimized, they can never be eliminated. Systematic errors are ones that cause readings to be always high or always low. These are generally due to mistakes, equipment malfunction, and poor laboratory technique. For example, reading the volume of a graduated cylinder from the top of the meniscus rather than the bottom will cause all of the volume measurements to be slightly high.
Concept Check: Students in chemistry lab are warned to use the same balance for all the mass measurements for an experiment. What kind of error might occur if a student uses several different balances during the course of a single experiment?
Answer: All balances are sensitive to their environments. Dozens of factors contribute to the accuracy of a balance: minor things like air currents and the slope of the countertop and major things like spilled chemicals on the pan. Any given balance will read slightly high or slightly low on any given day. However, these differences tend to be fairly repeatable. If a student uses the same balance throughout an experiment, the errors tend to "subtract out" when weighing by difference. If the student changes balances, some readings will by high and others low by random amounts. This will cause a random error.
Accuracy and Precision
Accuracy and precision are two indicators of the validity of measurements. Accuracy is the agreement of a measurement is with an actual value (how close it is to what it is supposed to be). This is the best indicator of how ‘good’ a measurement is. Unfortunately, if something is truly worth measuring, we don’t know its actual value. Accuracy is important in the chemistry lab during experiments in which the lab instructor knows what the value actually is and the students try to get a result as close as possible to this ‘true’ value. Scientists routinely check their instruments for accuracy by using them to measure a standard for which the values are well known. If the instrument’s readings match the accepted values for the standard, it is probably working correctly. Precision, on the other hand, is an indication of how close several measurements are to one another. Systematic errors tend to have an additive effect. If a student has poor laboratory techniques, his or her measurements will tend to fluctuate more than those of another student with good techniques. Laboratory instructors will tend to have students perform measurements several times then calculate an average with a standard deviation. If the instructor knows what the result should be, the student’s average indicates the accuracy and the standard deviation indicates the precision.
!Warning! I need to give students a word of warning regarding the terms accuracy and precision. In other disciplines, these terms have meanings very different from their meanings in chemistry.
4Concept Check: What will happen to the precision and accuracy of the results for the student above who changes balances during an experiment?
Answer: The precision should be poor if a student randomly changes balances when repeating the same procedures. Suppose balance A reads high by 0.1 gram and balance B reads low by 0.1 gram. If the student uses balance A to do the first trial and balance B to do the second trial, the two results should be different by roughly 0.2 gram. We can't predict how this will effect the accuracy of the student's result because we don't know how many other errors he or she might have made and what impact those errors had. All of the errors could cancel each other out and lead to a perfect result, but most of us aren't that lucky.
Because all measurements have errors, scientists have developed methods of representing numbers such that they can indicate a degree of trustworthiness. Measurements have uncertainty (the lowest decimal place that can be measured with estimation). If we use a ruler to measure length, we read the centimeters directly from the scale. We count the millimeters marks directly. We estimate the length to the tenth of a millimeter based on where it falls between marks, halfway is 0.5 mm, for instance. The uncertainty in that ruler is to the tenth of a millimeter. If we try to use that instrument to measure something very tiny, we can’t expect to get very good results. If we perform mathematical operations with the numbers that results from that measurement, we need to round our numbers so that they don’t imply less uncertainty than our instruments warrant.
Rules for significant figures:
The rules for assigning significant figures are based on deducing which digits could not have been the result of rounding off another number. When you round a number, you replace digits with zeroes, so zeroes are what we have to scrutinize. The following rules are a systematic scrutiny to determine if the digits in a number reflect actual measurements with their associated degrees of uncertainty.
4Concept Check: How many significant figures are there in the number in the title 1001 Arabian Nights?
Answer: There are four significant figures in the number 1001. the zeroes between non-zero digits count as significant figures because they cannot be the result of a rounding operation. The author understood the ideas behind the rules for assigning significant figures. '1000 nights' implies a pretty long time, but the number '1001nights' implies that we know how many nights to the nearest night.
Rounding the result of addition/subtraction:
When you add and subtract using pencil and paper (rather than a calculator), you line up the decimals so that you are adding digits from the same decimal place to one another. Decimal places shared by all of the numbers contribute to the result. Higher powers of ten contribute, even if they aren’t shared by all of the numbers. Lower powers of ten don’t have much impact on the result. The number that results from addition or subtraction should be rounded at the lowest decimal place that the original numbers have in common. If we add the numbers 2.2 + 41.066 + 19.11 we get the result 62.376. This implies that we measured to the thousandths place, but one of the numbers was only measured to the tenths place. The sum should be rounded to 62.4 at the tenths place, which is the lowest decimal place that all three numbers share.
Rounding the result of multiplication/division:
When you multiply or divide numbers, you change their scale. The decimal point may shift as a result, so it does not give us an indication of the uncertainty of the original measurements. The number that results from multiplication or division should be rounded so that it has the same number of significant figures as the original number with the fewest number of significant figures. For example, if we multiply the numbers 22 * 30.1 * 1.4 we get the result 927.08. The lowest digit is the one with uncertainty, so this implies that we made measurements that had only one in ten thousand parts uncertainty! We need to round the result to 930, which has two significant figures, since the original numbers had two and three significant figures.
4Concept Check: Translate the title 20,000 Leagues Under the Sea into metric units.
Answer: The metric title would be 100,000 Kilometers Under the Sea. One league is three miles, so 20,000 leagues is 60,000 miles. 1.000 mile is 1.609 km, so 60,000 miles is 96,540 km which rounds to 100,000 km. The number 20,000 has one significant figure by the rules. Context can always over-turn the rules, but, in this case, the context is "really deep" rather than a measurement to the nearest league.
Rounding more complex calculations:
When calculations involve different kinds of operations, apply the rules sequentially. In this example, a student measures the volume and mass of a sample of metal pieces. She measures the mass on a balance using a beaker to contain the pieces. She measures the volume by adding the pieces to some water in a graduated cylinder and reading the volume displaced. She then uses the data to calculate the density of the metal using the formula D = M/V.
The result of subtracting the mass of the beaker from the beaker + metal is rounded to the third decimal place by the addition/subtraction rule. The result of subtracting the volume of water from the volume of water + metal is rounded to the first decimal place by the subtraction rule. This gives 17.0 with three significant figures. The number is written in scientific notation to make it clear that the zero is significant. The density is the mass of the metal divided by the volume of the metal. When we divide a number with five significant figures by one with three significant figures, the answer is correctly rounded to three significant figures.
Unit analysis or dimensional analysis is a method for converting from one system of measure to another. In this technique, all measurements are written with their units and conversion factors are written as ratios such that units cancel, top to bottom, like common factors in fractions. For example, to convert my height from English inches to metric meters, I need a conversion factor for English to metric length. The only one I know is 1 inch = 2.54 cm. But I also know that 100 cm = 1m. My height is 67 inches, so I start with that. Then I choose a conversion factor for inches, and write it as a ratio with the inch unit in the denominator. That gives centimeters in the numerator, so I choose another conversion factor with centimeters, and write it as a ratio with centimeters in the denominator. The inches and centimeters cancel out leaving me with the meters as the units. Multiply the numerators and divide by the denominators.
In this case, the answer is rounded to two significant figures because the original measurement had two significant figures. The other values are exact numbers. The conversion between inches and centimeters is an unusual example of a definition between two totally different measurement systems.
After the first chapter review, chemistry students will seldom convert English to metric system. However, some topics require students to convert metric and SI derived units. For example, the there are two values for the gas law constant, R: 0.08206 L atm/mol K and 8.314 J/mol K. These two values are the same quantity but use different units. We can show this to be the case using unit analysis and some information about the derived units from the previous section. We know that a joule can be reduced to SI base units. That means we need to convert liters and atmospheres into base units; the moles and degrees Kelvin are fine. One milliliter is the same as a cubic centimeter. A cubic meter is a cube with each side 100 centimeters long, or 1x106 cm3. Setting up conversion factors as ratios such that the units cancel top to bottom gives us:
So we have a new conversion factor: 1L = 0.001 m3. If we convert atmospheres to pascals, we can find SI base units. Using the conversion for atmospheres to kilopascals, knowing that there are 1000 pascals in a kilopascal, and using the definitions of the pascal and the joule, we get:
So, 1 atm = 1.01325x105 J/m3. Note that the meters units do not cancel. We have cubic meters in the denominator. We can use these conversions to convert the gas law constant:
This is close enough to the accepted value to prove the point. Now we need to think a bit about the significant figures. The original constant, 0.08206, has four significant figures. The atmosphere to kilopascals conversion has six significant figures. All of the other conversions are exact, because they are based on metric system definitions rather than measurements. The result is rounded to four significant figures by the multiplication/division rule.
© Copyright 2005, Kelley Whitley, ChemProfessor. All rights reserved.
This site was last updated 05/15/06