Introductory Chemistry: A Foundation - Zumdahl S.S., DeCoste D.J. 2019

Measurements and Calculations
Measurements of Length, Volume, and Mass


· To understand the metric system for measuring length, volume, and mass.

The fundamental SI unit of length is the meter , which is a little longer than a yard . In the metric system, fractions of a meter or multiples of a meter can be expressed by powers of , as summarized in Table 2.3.

Table 2.3. The Metric System for Measuring Length



Meter Equivalent



m or m






m or m



m or m



m or m


m or m



m or m

The English and metric systems are compared on the ruler shown in Fig. 2.1. Note that

Figure 2.1.An illustration shows a small portion of a ruler with lengths measured in two different units. The top of the ruler is measured as inches and the bottom of the ruler is measured as centimeters. Inches have 4 graduations, corresponding to values of 1, 2, 3, and 4. Centimeters have 11 graduations, corresponding to values of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Vertical line is marked at 1 on inches on the ruler that indicates 1 inch. A vertical dashed line is marked between 2 and 3 centimeters on the ruler. The point at which the dashed line touches the ruler indicates 2.54 centimeters.

Comparison of English and metric units for length on a ruler.

Other English—metric equivalencies are given in Section 2.6.

Volume is the amount of three-dimensional space occupied by a substance. The fundamental unit of volume in the SI system is based on the volume of a cube that measures meter in each of the three directions. That is, each edge of the cube is meter in length. The volume of this cube is

or, in words, cubic meter.

In Fig. 2.2 this cube is divided into smaller cubes. Each of these small cubes represents a volume of , which is commonly called the liter (rhymes with “meter” and is slightly larger than a quart) and abbreviated L.

Figure 2.2.An illustration shows three cubes. The largest cube is divided into 1000 smaller cubes ten to each side, has a volume of 1 cubic meter. One of the smaller cubes has a volume of 1 cubic decimeter, which is equal to 1 liter. From this cube, an arrow points to its magnified image, another cube divided into 1000 cubes. One of these smaller cubes has sides of 1 centimeter in length, and has a volume of 1 cubic centimeter, which is equal to 1 milliliter.

The largest drawing represents a cube that has sides m in length and a volume of . The middle-size cube has sides dm in length and a volume of , or L. The smallest cube has sides cm in length and a volume of , or mL.

Chemistry in Focus Measurement: Past, Present, and Future

Measurement lies at the heart of doing science. We obtain the data for formulating laws and testing theories by doing measurements. Measurements also have very practical importance; they tell us if our drinking water is safe, whether we are anemic, and the exact amount of gasoline we put in our cars at the filling station.

Although the fundamental measuring devices we consider in this chapter are still widely used, new measuring techniques are being developed every day to meet the challenges of our increasingly sophisticated world. For example, engines in modern automobiles have oxygen sensors that analyze the oxygen content in the exhaust gases. This information is sent to the computer that controls the engine functions so that instantaneous adjustments can be made in spark timing and air—fuel mixtures to provide efficient power with minimum air pollution.

A recent area of research involves the development of paper-based measuring devices. For example, the nonprofit organization Diagnostics for All (DFA) based in Cambridge, Massachusetts, has invented a paper-based device to detect proper liver function. In this test, a drop of the patient’s blood is placed on the paper, and the resulting color that develops can be used to determine whether the person’s liver function is normal, worrisome, or requires immediate action. Other types of paper-based measuring devices include one used to detect counterfeit pharmaceuticals and another used to detect whether someone has been immunized against a specific disease. Because these paper-based devices are cheap, disposable, and easily transportable, they are especially useful in developing countries.

Scientists are also examining the natural world to find supersensitive detectors because many organisms are sensitive to tiny amounts of chemicals in their environments—recall, for example, the sensitive noses of bloodhounds. One of these natural measuring devices uses the sensory hairs from Hawaiian red swimming crabs, which are connected to electrical analyzers and used to detect hormones down to levels of g/L. Likewise, tissues from pineapple cores can be used to detect tiny amounts of hydrogen peroxide.

These types of advances in measuring devices have led to an unexpected problem: detecting all kinds of substances in our food and drinking water scares us. Although these substances were always there, we didn’t worry so much when we couldn’t detect them. Now that we know they are present, what should we do about them? How can we assess whether these trace substances are harmful or benign? Risk assessment has become much more complicated as our sophistication in taking measurements has increased.

See Problems 2.157 and 2.158

The cube with a volume of ( liter) can in turn be broken into smaller cubes, each representing a volume of . This means that each liter contains . One cubic centimeter is called a milliliter (abbreviated mL), a unit of volume used very commonly in chemistry. This relationship is summarized in Table 2.4.

Table 2.4. The Relationship of the Liter and Milliliter








The graduated cylinder (Fig. 2.3), commonly used in chemical laboratories for measuring the volume of liquids, is marked off in convenient units of volume (usually milliliters). The graduated cylinder is filled to the desired volume with the liquid, which then can be poured out.

Figure 2.3.An illustration shows a 100-milliliter graduated cylinder with a liquid filled up to the 100 milliliter mark.

A -mL graduated cylinder.

Another important measurable quantity is mass , which can be defined as the quantity of matter present in an object. The fundamental SI unit of mass is the kilogram .

Because the metric system, which existed before the SI system, used the gram as the fundamental unit, the prefixes for the various mass units are based on the gram , as shown in Table 2.5.

Table 2.5. The Most Commonly Used Metric Units for Mass



Gram Equivalent








In the laboratory we determine the mass of an object by using a balance. A balance compares the mass of the object to a set of standard masses (“weights”). For example, the mass of an object can be determined by using a single-pan balance (Fig. 2.4).

Figure 2.4.A photo shows a stoppered round-bottomed flask with a blue solution being weighed in an electronic analytical balance. The display of the balance reads 58.7298 grams.


An electronic analytical balance used in chemistry labs.

To help you get a feeling for the common units of length, volume, and mass, some familiar objects are described in Table 2.6.

Table 2.6. Some Examples of Commonly Used Units


A dime is mm thick.

A quarter is cm in diameter.

The average height of an adult man is m.


A nickel has a mass of about g.

A -lb woman has a mass of about kg.


A -oz can of soda has a volume of about mL.

A half gallon of milk is equal to about L of milk.