2.1: Measurement and Significant Figures Lab Procedure

Learning Objectives

• To use a metric ruler to measure the dimensions of regular geometric shapes, and to use these measurements to determine the areas of the shapes.
• To measure the volume of a sample of water using a graduated cylinder and a beaker in order to compare their precision.
• To measure the mass of an item using a triple-beam balance and an analytical (electronic) balance in order to compare their precision; also, to determine the mass of a powder by weighing by difference.

 

Background

Our knowledge of chemistry and chemical processes largely depends on our ability to obtain correct information about matter. Often this information is quantitative, in the form of measurements. In this lab, students will be introduced to some common measuring instruments so that they can practice making measurements, and to learn about instrument precision. In Part A of this lab, a metric ruler will be used to measure length in centimeters (cm). In Part B, a beaker and a graduated cylinder will be used to measure liquid volume in milliliters (mL). In Part C, an electronic balance will be used to measure mass in grams (g).

Since all measuring devices are subject to some error, it is impossible to make exact measurements. Scientists record all the digits of a measurement that are known exactly, plus the first one that is uncertain. These digits are collectively referred to as significant digits. Digital instruments, such as an electronic balance, are designed to limit themselves to the correct number of significant digits, and their readings are properly recorded as given. However, when using analog instruments such as rulers and thermometers, the experimentalist is responsible for determining the correct number of significant figures. These instruments are properly read to one place beyond the graduations of the scale.

The system of measurement used in science and the healthcare field is the metric system. Within the metric system, a variety of units can be employed to measure the same fundamental quantity; for example, energy could be expressed within the metric system in units of ergs, electron-volts, joules, and two kinds of calories. Therefore, a more basic set of units the Systeme Internationale (SI) units that are now recognized as the standard for science and for technology of all kinds. In principle, any physical quantity can be expressed in terms of base units (Table 1), with each base unit defined by a standard described in the NIST Web site.

Table 1: SI Units

Property	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Volume	liter	L
Time	second	s
Temperature (absolute)	kelvin	k
Amount of substance	mole	mol

Owing to the wide range of values that quantities can have, it has long been the practice to employ prefixes such as milli and kilo to indicate decimal fractions and multiples of metric units. As part of the SI standard, this system has been extended and formalized (Table 2).

 

Table 2: Prefixes used to scale up or down base units

Prefix	Abbreviation	Multiplier
giga	G	10^9
mega	M	10^6
kilo	k	10^3
deci	d	10^-1
centi	c	10^-2
milli	m	10^-3
micro	µ	10^-6
nano	n	10^-9
pico	p	10^-12

 

Measured and Exact Numbers

Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number. If we count eggs in a carton, we know exactly how many eggs the carton contains. The numbers of defined quantities are also exact. By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.

 

Significant Figures in Measurement

The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid, Figure 1.

Figure \(\PageIndex{1}\): To measure the volume of liquid in this graduated cylinder, you must mentally subdivide the distance between the 21 and 22 mL marks into tenths of a milliliter, and then make a reading (estimate) at the bottom of the meniscus.

Refer to the illustration in Figure 1. The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.

This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram. If we weigh the quarter on a more sensitive balance, we may find that its mass is 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some uncertainty, which depends on the device used (and the user’s ability). All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows “120,” then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.

Whenever you make a measurement properly, all the digits in the result are significant. But what if you were analyzing a reported value and trying to determine what is significant and what is not? Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought. We will use the terms “leading,” “trailing,” and “captive” for the zeros and will consider how to deal with them.

Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point

Captive zeros result from measurement and are therefore always significant. Leading zeros, however, are never significant—they merely tell us where the decimal point is located.

The leading zeros in this example are not significant. We could use exponential notation (as described in Appendix B) and express the number as 8.32407 ×10−3 ; then the number 8.32407 contains all of the significant figures, and 10−3 locates the decimal point. The number of significant figures is uncertain in a number that ends with a zero to the left of the decimal point location. The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located. The ambiguity can be resolved with the use of exponential notation: 1.3 ×103 (two significant figures), 1.30 ×103 (three significant figures, if the tens place was measured), or 1.300 ×103 (four significant figures, if the ones place was also measured). In cases where only the decimal-formatted number is available, it is prudent to assume that all trailing zeros are not significant.

When determining significant figures, be sure to pay attention to reported values and think about the measurement and significant figures in terms of what is reasonable or likely when evaluating whether the value makes sense. For example, the official January 2014 census reported the resident population of the US as 317,297,725. Do you think the US population was correctly determined to the reported nine significant figures, that is, to the exact number of people? People are constantly being born, dying, or moving into or out of the country, and assumptions are made to account for the large number of people who are not actually counted. Because of these uncertainties, it might be more reasonable to expect that we know the population to within perhaps a million or so, in which case the population should be reported as 3.17×10⁸ people.

 

Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to know both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value. These characterizations can be extended to other contexts, such as the results of an archery competition (Figure 2).

Figure \(\PageIndex{2}\): (a) These arrows are close to both the bull’s eye and one another, so they are both accurate and precise. (b) These arrows are close to one another but not on target, so they are precise but not accurate. (c) These arrows are neither on target nor close to one another, so they are neither accurate nor precise.

 

Part A: Measuring Length

The ruler markings are every 0.1-centimeter. The correct reading is 1.67 cm. The first 2 digits 1.67 are known exactly. The last digit 1.67 is uncertain. You may have instead estimated it as 1.68 cm.

 

Part B: Measuring Volume of a Liquid

When measuring liquid volumes, the graduated scale must be read from the lowest point of the curved surface of the liquid – the liquid meniscus.

The graduated cylinder markings are every 1-milliliter. The correct reading is 30.0 mL. The first 2 digits 30.0 are known exactly. The last digit 30.0 is uncertain. Even though it is a zero, it is significant and must be recorded.

 

Part C: Measuring Mass

The mass of an object is the amount of matter present in the object, whereas weight is the force of gravity on an object. We must use mass for accurate measurements of how much matter we have. We still call the process of obtaining an accurate mass "weighing". Accurate weighing is usually done with a single pan balance. The accuracy of a measured value, such as a melting point, may be evaluated by a calculation of percent error. Percent error is a common way of reporting how close a measured experimental value (EV) is to the true value (TV):

\[{Percent\; Error} = \frac{\left | EV - TV \right |}{TV} \times 100\]

Accurate measurements will typically have low percent errors of <5%.

 

Experimental Procedure

Materials and Equipment

Metric ruler*, shape sheet, electronic balance, large test tube, 100-mL beaker, 100-mL graduated cylinder, triple-beam balance, 250-mL Erlenmeyer flask, electronic balance, sugar, small watch glass.

 

Safety

Always wear your safety goggles!

 

Part A: Measuring the Dimensions of Regular Geometric Shapes

1. Obtain a ruler from the instructor’s desk.
2. Obtain a “three shape object” from the fume hood, and then use the ruler to measure the dimensions in centimeter to three decimal place. Measure the length and width of the rectangle, and the diameter of the circle. Record these measurements on your report form.
3. When finished, return the ruler.
4. Use your measurements to calculate the areas of the assigned geometrical shapes.

\[Area\; of\; a\; rectangle = length \times width \\
Area\; of\; a\; circle = \pi r^{2} \\
(where\; r = radius = 1/2 \times diameter)\]

 

Part B: Measuring the Volume of a Sample of Water

1. Obtain a large test tube. Fill this test-tube to the brim with tap water, then carefully transfer it to a 50-mL beaker (obtain from your bin). Note that if your 50-mL beaker has no scale markings on it, you will need to get a replacement from the stockroom or your instructor and swap it for one that does. Measure and record the volume of water in the beaker.
2. Again, fill the same test-tube to the brim with tap water, then carefully transfer it to a 50- mL or 100-mL graduated cylinder (obtain from your bin). Measure and record the volume of water in the graduated cylinder. Do these measured volumes have the same number of significant figures?

 

Part C: Measuring the Mass of Solids

Comparing the Precision of two types of Balances

1. Use a triple-beam balance to obtain the mass of a 250-mL Erlenmeyer flask (obtain from your bin). Now use an electronic balance to obtain the mass of the same Erlenmeyer flask. Do these measured masses have the same number of significant figures? Be sure to record your measured masses on your report form.

Weighing by Difference

1. Using the electronic balance again, obtain the mass of a 50-mL beaker. If you already used this same beaker in Part B, make sure that you carefully dry it before weighing it. Add two spoonfuls of salt to this beaker, using a scoopula. Do not do this over the balance! Then obtain the new combined mass of both the beaker and the salt. Be sure to use the same electronic balance as before.
2. When finished, dispose of the salt used in the waste container under the fume hood. Use your two measurements to determine the mass of salt (only) weighed out