Taking Measurements

Chemists measure the properties of matter and express these measurements as quantities. A quantity is the amount of something and consists of a number and a unit. The number tells us how many (or how much), and the unit tells us what the unit of measurement is. For example, when a distance is stated as “5 km”, we know that the quantity is expressed in units of kilometers and the number of kilometers is 5. If you ask a friend how far he walks from home to school. , and the friend answers “12” without specifying a unit, you don’t know if your friend walks – for example, 12 miles, 12 kilometers, 12 furlongs, or 12 yards. To express a quantity correctly, it is necessary to include both a number and a unit.

Significant Figures: Writing Numbers to Reflect Precision

The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the ruler illustration below, the bottom ruler gave a length with 2 significant figures, while the top ruler gave a length with 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. With either ruler, it would not be possible to report the length at 2.553cm2.553cm as there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported.

Scientific Notation: Writing Large and Small Numbers

Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes. In scientific notation, these numbers are expressed in the form

N×10n(4.2.1)(4.2.1)N×10n

where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (10⁰ = 1). The number 10 is called the base because it is this number that is raised to the power nn. Although a base number may have values other than 10, the base number in scientific notation is always 10.

A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows:

If the decimal point is moved to the right n places, n is negative.

If the decimal point is moved to the left n places, n is positive.

Measurement Uncertainty

Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest 0.1g0.1g, other highly sensitive balances are capable of measuring to the nearest 0.001g0.001g or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement.

With either ruler, it is clear that the length of the object is between 22 and 3cm3cm. The bottom ruler contains no millimeter markings. With that ruler, the tenths digit can be estimated and the length may be reported as 2.5cm2.5cm. However, another person may judge that the measurement is 2.4cm2.4cm or perhaps 2.6cm2.6cm. While the 2 is known for certain, the value of the tenths digit is uncertain.

The top ruler contains marks for tenths of a centimeter (millimeters). Now the same object may be measured as 2.55cm2.55cm. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be 2.54cm2.54cm or 2.56cm2.56cm. In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.

Significant Figures in Calculations

Rounding

Before dealing with the specifics of the rules for determining the significant figures in a calculated result, we need to be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1