A-1. Systems of Measurement. Doing quantitative experiments involves measurement, the process of placing a numerical value on a property of a substance, in terms of some universally accepted standard value for that property. Properties which are frequently measured in science include length, mass, time, energy, and charge. A measurement involves both a number and a unit. For example, a volume might be 5 gallons; a mass might be 6.0 kilograms; a time might be 3.43 seconds. In specifying the result of a measurement, the number is useless without the unit.

Currently in the United States, 3 systems of units are in use: the English system for everyday transactions; the metric system (used by scientists and more and more by industry); and the SI system (recommended for use by scientists, but not yet universally adopted). This appendix is concerned with the metric and SI systems.

The metric system is very logically based on powers of 10. The basic unit of length is the meter (symbol, m), that of volume the liter (L), and that of mass the gram (g). Prefixes are added to these units to indicate powers of 10 larger or smaller. The common prefixes are summarized in Table A-1.

In the laboratory, the commonly used length, mass, and volume units are the cm, the g, and the L, respectively. Since volume has dimension of (length)³, volume units can be readily obtained from length units. Thus the cubic centimeter (cm³ or cc) is the volume of a cube with an edge length of 1 cm; the cubic decimeter (dm³⁾ is the volume of a cube 1 dm on a side. There are 1000 cm³ in 1 dm³. The liter and the dm³ are the same, so 1 L = 1000 cm³ = 1000 mL.

The unit of time for both the English and Metric systems is the second (symbol s). Metric prefixes may be used with the second.

In an attempt to standardize the use of metric units by scientists, a conference was held in 1960 to establish a universal system of units. The resulting system is called the International System of Units, or the SI system. This system has seven fundamental (base) units, from which all other units can be derived. These are given in Table A-2. Under the SI system there are several categories of units which include (1) the 7 base units, (2) supplementary units (for example, the radian measure of plane angle), (3) derived units (for example, 1 Joule = 1 kg-m²/s²), (4) non-SI units which are permitted for use with SI on a continuing basis, (5) non-SI units permitted for use with SI for a limited time (angstrom and atmosphere), and (6) obsolete metric units not permitted with SI (calorie and erg). Many chemists continue even now to use the forbidden units. It is therefore necessary to be familiar even with these.

Standards. A standard is an agreed-upon reference of measurement. For example, prior to 1960, the standard meter was the distance between two lines on a piece of platinum-iridium alloy, kept in a constant temperature, constant humidity room at the International Bureau of Weights and Measures in Paris. Since 1960, the meter has been defined as 1,650,763.73 times the wavelength of the orange-red radiation of the krypton-86 atom. The advantage of the new standard over the old is that it will be meaningful to life forms from other star systems, when and if we should establish communication. There are similar standards for quantities other than length, the kilogram being the only SI unit still defined in terms of an earthbound object. The important point is that standards are necessary so that everyone is speaking the same mathematical language.

Temperature. There are 3 temperature scales in use. They are the Fahrenheit scale, used in the US; the Celsius scale, used in most of the rest of the world; and the Kelvin, or Absolute, scale, preferred by scientists. The Celsius and Fahrenheit scales use the same two reference points: the boiling point and freezing point of pure water under 1 atmosphere (760 mm or 29.92 inches of mercury) of pressure. The Fahrenheit system fixes a value of 212 degrees to the boiling point and 32 degrees to the freezing point of water. The Celsius scale assigns the values 100 and 0 to the same two points. Each scale is then marked off into equal segments, each segment being one degree. It requires 180 Fahrenheit degrees to cover the same range as are covered by 100 Celsius degrees. The Celsius degree is therefore larger than the Fahrenheit degree. The Absolute temperature scale is related to the Celsius scale, in that the size of the degree is the same. However, on the Absolute scale, the boiling and freezing points of water have the values 373.15 and 273.15 K (K for Kelvin), respectively. Chapter 5 discusses the significance of these seemingly odd numbers. To interconvert ^oC and ^oF, we use equation A-1:

To interconvert ^oC and K, we use equation A-2:

A-2. Uncertainty, Precision, and Error in Experimental Results. Measurement of quantities such as length, volume, mass, temperature, light absorbance, and time is the heart of experimental science. Ultimately, the most elegant theory rests on numbers obtained in the laboratory by a scientist using a measuring device or instrument. No device or instrument is capable of producing an exact result--a result with no uncertainty. When a scientist reports the result of a single measurement, such as a length or volume, s/he also reports the uncertainty in the result, a number indicating the reliability of the measuring device used. For example, the mass of a sample obtained using a balance which weighs to the nearest 0.1 g may be 32.0 g. The uncertainty in this number is 1 unit in the tenths decimal place, and indicates that the balance reading is uncertain by 1 unit in this place. The mass is reported as 32.0 ± 0.1g, and the result of the weighing is known to 1 part in 320. Scientist sometimes communicate uncertainty implicitly rather than explicitly. For example, the mass above might be reported as 32.0 g. Reporting the tenths decimal place implies an uncertainty of 1 digit in this place. The mass is known to 3 significant figures, which is the number of digits written (except in numbers like 0.0032, where leading zeros are place holders, not significant figures). Weighing this sample on a more sensitive balance might give the result 32.0492g. Now the uncertainty is 1 in the ten-thousandths decimal place, and the mass is known to 6 significant figures. The result of this weighing is known to 1 part in about 320000, much better than with the less sensitive balance.

Conversion Between Scales. Most quantities can be expressed in more than one system of units. For example, length can be expressed in English units or metric units; temperature can be expressed in Celsius degrees, Fahrenheit degrees, or Kelvins. Generally, uncertainties are related by the same conversion factors as relate the quantities in two scales. If a length measured in inches has an uncertainty of 1 inch, then the same length expressed in cm has an uncertainty of 2.54 cm, because 1 in = 2.54 cm. If a temperature on the Celsius scale has an uncertainty of 0.1^oC, the same temperature on the Fahrenheit scale will have an uncertainty of 0.18^oF because there are 1.8 Fahrenheit degrees per Celsius degree. Some examples will make this clear.

Example. A man is 74 inches tall. What is his height in meters?

The uncertainty in this number is ±2.54 cm. Since the uncertainty occurs first in the 1's place, we report the height as 188 ± 3 cm. Height in meters is 1.88 ± 0.03. The usual significant figure rule would force us to report this as 1.9 m, because the height in inches is given to 2 sig figs. But this would imply that we know the man's height to only 1 part in 19 in meter units, whereas we know it to 1 part in 74 in inches. Reporting the height as 1.9 m would mean a loss of information!

Example. What is 9^oC on the Fahrenheit scale? The relationship between the two scales is

Substituting 9 for ^oC and solving for ^oF gives 48.2^oF. How many significant figures should we report? The significant figure rule tells us to report 50^o, with 1 significant figure. But according to this rule, 7^oC would be 40^oF, while 8, 9 and 10^oC would all be 50^oF! Clearly this significant figure rule is not working well. The key here, as above, is to realize that 1 centrigrade degree is equivalent to 1.8 Fahrenheit degrees. An uncertainty of 1 degree in Centrigrade temperature will translate to an uncertainty of 1.8 degrees in Fahrenheit temperature. Since uncertainty occurs first in the 1's place, the correct Fahrenheit temperature above is 48^o.

Reproducibility and Accuracy. The precision in a result indicates the reproducibility of the result--how well we can expect to reproduce it when we make repeated measurements. The higher the precision, the better the reproducibility. Precision is different from uncertainty, which has meaning only for a single measurement, as explained above.

Example. 3 drivers, A, B, and C, make the trip along Route 9 from Spags in Shrewsbury to Pizzerria Uno in Natick. From his initial and final odometer readings, driver A reports the distance as 22.4 miles. Drivers B and C report the distance as 23.6 and 21.0 miles, respectively. Find a) the uncertainty in driver A's measurement; b) the best estimate of distance from Spags to Uno, based on the efforts of these three drivers; c) the precision (reproducibility) of this best estimate.

a) The uncertainty in driver A's result is a measure of how well he can read his odometer. Most odometers read to 0.1 mile, meaning we can estimate a single odometer reading to ±0.05 mile. A distance, the difference between two odometer readings, is probably good to only 0.1 mile. The uncertainty in driver A's distance is ± 0.1 mile; he should report his result as 22.4 ± 0.1 mile. Similarly, drivers B and C should report their results as 23.6 ± 0.1 miles and 21.0 ± 0.1 miles.

b) The best estimate of the distance is the average or mean of the results of the three drivers.

c) The precision of this number is a measure of how close any driver could expect to come to 22.3 miles, driving his own car. The precision is estimated as 1/2 the difference between the maximum and minimum distances obtained by our 3 drivers, because the result obtained by any particular driver would probably fall within this range.

The best estimate of the distance from Spags to Uno from our data is 22 ± 1 miles. We write it this way, rather than as 22.3 ± 1.3, because the precision tells us that distances measured by various drivers will reproduce to no better than 1.3 miles. There is no point in reporting the average to any better than the 1's place! Occasionally, 1/2 the range of a series of measurements will be smaller than the uncertainty in a single measurement. In these cases, one would use the uncertainty in a single measurement rather than 1/2 the range to represent the precision, as reproducibility cannot be better than the uncertainty in a particular measurement.

The error in a measured result is the difference between the measured value and the true value of the quantity being measured. Error indicates the accuracy of the measured value. Error is to accuracy what precision is to reproducibility. An extension of the example above shows the meanings of these terms clearly.

Example. An accurate road survey gives the distance between Spags and Uno as 23.9 miles. What is the error in our estimate of this distance?

Error = measured value - true value = 22.3 - 23.9 = -1.6 miles. The negative sign indicates that the measured value is smaller than the true value.

In summary,

Precision--the reproducibility of a measurement, indicated by writing the measured value ± range/2;

Accuracy--the difference between the measured value of a quantity and the true value. Indicated by writing the error (measured value - true value).

Uncertainties in Calculated Results. The uncertainty in a measured number carries into calculations made using the number. A calculated result is expressed consistently with the uncertainties in the numbers used in the calculation. How this is done depends on the arithmetic operation performed.

Addition and Subtraction. The uncertainty in the result of adding or subtracting two numbers is the sum of their uncertainties.

Suppose that a buret is filled with solution to a volume reading of 2.64 mL. Solution is delivered from the buret to a flask. The final buret reading is 27.48 mL. What volume of solution was delivered? The uncertainty in a single buret reading is ±0.01 mL. The volume delivered is the difference between final and initial readings, and has an uncertainty which is the sum of the uncertainties in the two readings:

Multiplication and Division. The relative uncertainty in a product or quotient is the same as that of the least-well-known factor. The relative uncertainty is expressed using the "x parts in y" approach introduced above. If a number known to 1 part in 100 is multiplied by a number known to 1 part in 1000, the product can be known to at best 1 part in 100.

For example, suppose a 10.00 mL sample of water (delivered from a 10-mL volumetric pipet) has a mass of 9.9643g (measured with an analytical balance). What is its density? Since density is mass per unit volume, we calculate

This must be reported to a number of significant figures consistent with the uncertainties in mass and volume:

The volume is less certain than the mass. The density can be known with no less uncertainty than the less-well-known of mass and volume. At best, density is known to about 1 part in 1000. We should report density as 0.996 ±0.001 g/mL. (The usual significant figure rule would allow us to report density to 4 sig figs, as 0.9964 g/mL, since volume is known to 4 sig figs. However this implies that we know density to 1 part in 9964! How can this be if we only know volume to 1 part in 1000??) A common-sense approach to signicant figures using the "x parts in y" method avoids the pitfalls of the textbook rules.

Precisions carry through calculations according to the same rules.

Logarithms. Frequently we need to take the logarithm of a measured number to make a plot. How do we express the uncertainty in the logarithm of a measured value?

Suppose we need the logarithm of (1.25 ± 0.01)x10^-4. We first find the log of 1.25 x 10^-4 to be -3.9031. To find the uncertainty in this, we compute logs of the maximum and minimum values our measured value could have, based on its uncertainty:

Comparing with 3.9031, the variation is in the 3rd decimal place. Uncertainty in the log is 1/2 of 0.007, or 0.004. The result is 3.903 ± 0.004. The number of digits following the decimal point in the log is the same as the number of sig figs in the original measured value. This will always be true.

Exponentiation and Trigonometric Functions. The same approach used for logarithms can be used when exponentiating a number or finding the values of trigonometric functions of a number:

For example, exponentiate the number -2.4 ± 0.2 in base 10.

The value of the function is reported (4 ± 2) * 10^-3. Note how an uncertainty of 1 part in 12 in the original value translates to uncertainty of 2 parts in 4 in the exponential. Exponentiation magnifies uncertainty!

Reading Graduated Measuring Devices. Many laboratory measurement devices are graduated--marked with equally spaced lines corresponding to incremental values of the quantity measured. For example, a meter stick is 1 meter long, with large lines for centimeter subdivisions, and smaller lines for millimeter subdivisions. It is graduated in millimeters. A 100-mL graduated cylinder is marked with large lines every 10 mL and smaller lines every mL. It is graduated in mL. Similarly, burets are graduated in 0.1 mL, 10-mL measuring pipets to 0.1 mL, and standard laboratory thermometers to 1^oC.

There is a standard procedure for reading a value from a graduated device, summarized as follows: estimate the value to one decimal place more than the level of graduation. When reading a temperature with a thermometer graduated in degrees, estimate to tenths of a degree. Record a buret reading to the nearest hundredth of a mL (if the liquid level falls exactly on a major graduation, say at 32.00 mL, write 32.00, not 32). Record a volume measured with a 100-mL graduate to the nearest 0.1 mL. Please adhere strictly to this procedure. If you do not, you will introduce into your measurement more uncertainty than actually exists.

A-3. Dimensional Analysis. Dimensional analysis is a problem solving technique based on the recognition that the units associated with measured or exact numbers combine and cancel during algebraic manipulations in the same way as do pure numbers and algebraic variables. To illustrate this, consider the cancellations and combinations below:

This property of units can guide us to the correct answer to a problem if we know what the units of the answer should be.

Consider the relation between feet and yards: 1 yard = 3 feet. 1 yard and 3 feet are two equivalent ways of expressing the same length; in this sense they are equal. 1 yard is equivalent to or equal to 3 feet. This is an example of an equivalency. Other examples are

The last of these indicates that a 0.998 g mass of H₂O has a volume of 1 mL at 21^oC. The mass and volume are two ways of representing the same amount of water, so are equivalent. From any equivalency, we can develop two conversion factors, each equal to 1. From the equivalency 1 yard = 3 feet we develop the conversion factors

A conversion factor is a ratio of two equivalent quantities, each in a different set of units.

An example will show how these ideas can be used to solve problems. Suppose a consumer buys 50 liters of gasoline at a filling station in Canada. How many gallons did s/he buy? The question being asked can be framed as

The right side of this equation can be multiplied by 1 without changing its value. From the equivalency 1 L = 0.265 gal, the conversion factor 0.265 gal/1L can be developed. This conversion factor is equal to 1, so the right side of the equation can be multiplied by it without changing the value:

This conversion factor cancels the L unit and replaces it with gal. This is exactly what we wanted to do. Multiplying out the right side of the equation gives the answer:

Since the units of the answer are correct, it is likely that the numerical part is correct as well. Knowing that the units of the answer must be gallons has enabled us to readily solve the problem.

Dimensional Analysis is essentially a 3-step process:

Example. The earth revolves once around the sun in 365 days. How many seconds is this?

Solution. The desired time is ? sec. The given time is 365 days.

Note how conversion factors are strung together to cancel unwanted units until the correct units are obtained. Note also that some thought about what conversions are needed is wise before beginning.

Example. The density of mercury (Hg) at 25^oC is 13.6 g/mL. What is the mass of a 96.0-mL volume of mercury? (Mercury is one of 3 elements which are liquid at room temperature.)

Solution. The density is used as a conversion factor:

Example. In the United States, fish sells for $2.45 a pound. Assuming an exchange rate of $1.00 = 225 yen, calculate the price of fish in Japan in units of yen/kg.

Solution. The basic question is

Two conversions are required: dollars to yen, and pounds to kilograms. We should do one at a time:

Since 1 dollar = 225 yen and 1 lb = 0.454 kg, we obtain