Chapter 1: Units of Measurement & Significant Figures Sections 4 & 5

Units of Measurement

SI Units Système International d’Unités A different base unit is used for each quantity.

Metric System Prefixes convert the base units into units that are appropriate for the item being measured.

Sample Exercise 1.2 Using Metric Prefixes What is the name given to the unit that equals (a) gram, (b) second, (c) meter? (a) What decimal fraction of a second is a picosecond, ps? (b) Express the measurement 6.0  10 3 m using a prefix to replace the power of ten. (c) Use exponential notation to express 3.76 mg in grams. Answer: (a) second, (b) 6.0 km, (c) 3.76  10 –3 g Practice Exercise Solution In each case we can refer to Table 1.5, finding the prefix related to each of the decimal fractions: (a) nanogram, ng, (b) microsecond,  s, (c) millimeter, mm.

Volume The most commonly used metric units for volume are the liter (L) and the milliliter (mL). –A liter is a cube 1 dm long on each side. –A milliliter is a cube 1 cm long on each side.

Uncertainty in Measurements Different measuring devices have different uses and different degrees of accuracy.

Temperature By definition temperature is a measure of the average kinetic energy of the particles in a sample.

Temperature In scientific measurements, the Celsius and Kelvin scales are most often used. The Celsius scale is based on the properties of water. –0  C is the freezing point of water. –100  C is the boiling point of water.

Temperature The Kelvin is the SI unit of temperature. It is based on the properties of gases. There are no negative Kelvin temperatures. K =  C

Temperature The Fahrenheit scale is not used in scientific measurements.  F = 9/5(  C) + 32  C = 5/9(  F − 32)

Sample Exercise 1.3 Converting Units of Temperature If a weather forecaster predicts that the temperature for the day will reach 31 °C, what is the predicted temperature (a) in K, (b) in °F? Ethylene glycol, the major ingredient in antifreeze, freezes at –11.5ºF. What is the freezing point in (a) K, (b) °F? Answer: (a) K, (b) 11.3 °F Practice Exercise Solution

Density Density is a physical property of a substance. d = mVmV

(a) Calculate the density of mercury if 1.00  10 2 g occupies a volume of 7.36 cm 3. (b) Calculate the volume of 65.0 g of the liquid methanol (wood alcohol) if its density is g/mL. (c) What is the mass in grams of a cube of gold (density = g/cm 3 ) if the length of the cube is 2.00 cm? Sample Exercise 1.4 Determining Density and Using Density to Determine Volume or Mass Solution

(a)Calculate the density of a g sample of copper if it has a volume of 41.8 cm 3. (b) A student needs 15.0 g of ethanol for an experiment. If the density of ethanol is g/mL, how many milliliters of ethanol are needed? Practice Exercise

Answers: (a) 8.96 g/cm 3, (b) 19.0 mL

Uncertainty in Measurement

Significant Figures The term significant figures refers to digits that were measured. When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers.

Sample Exercise 1.5 Relating Significant Figures to the Uncertainty of a Measurement What difference exists between the measured values 4.0 g and 4.00 g? A balance has a precision of ± g. A sample that has a mass of about 25 g is placed on this balance. How many significant figures should be reported for this measurement? Answer: five, as in the measurement g, the uncertainty being in the third decimal place Practice Exercise Solution Many people would say there is no difference, but a scientist would note the difference in the number of significant figures in the two measurements. The value 4.0 has two significant figures, while 4.00 has three. This difference implies that the first measurement has more uncertainty. Amass of 4.0 g indicates that the uncertainty is in the first decimal place of the measurement. Thus, the mass might be anything between 3.9 and 4.1 g, which we can represent as 4.0 ± 0.1 g. A measurement of 4.00 g implies that the uncertainty is in the second decimal place. Thus, the mass might be anything between 3.99 and 4.01 g, which we can represent as 4.00 ± 0.01 g. Without further information, we cannot be sure whether the difference in uncertainties of the two measurements reflects the precision or accuracy of the measurement.

Significant Figures 1.All nonzero digits are significant. 2.Zeroes between two significant figures are themselves significant. 3.Zeroes at the beginning of a number are never significant. 4.Zeroes at the end of a number are significant if a decimal point is written in the number.

Sample Exercise 1.6 Determining the Number of Significant Figures in a Measurement How many significant figures are in each of the following numbers (assume that each number is a measured quantity): (a) 4.003, (b)  10 23, (c) 5000? Solution (a) Four; the zeros are significant figures. (b) Four; the exponential term does not add to the number of significant figures. (c) One. We assume that the zeros are not significant when there is no decimal point shown. If the number has more significant figures, a decimal point should be employed or the number written in exponential notation. Thus, has four significant figures, whereas 5.00  10 3 has three.

How many significant figures are in each of the following measurements: (a) g, (b) 2.3  10 4 cm, (c) m 3 ? Practice Exercise

Answer: (a) four, (b) two, (c) three

Significant Figures When addition or subtraction is performed, answers are rounded to the least significant decimal place. When multiplication or division is performed, answers are rounded to the number of digits that corresponds to the least number of significant figures in any of the numbers used in the calculation.

Sample Exercise 1.7 Determining the Number of Significant figures in a Calculated Quantity The width, length, and height of a small box are 15.5 cm, 27.3 cm, and 5.4 cm, respectively. Calculate the volume of the box, using the correct number of significant figures in your answer. Solution When we use a calculator to do this calculation, the display shows , which we must round off to two significant figures. Because the resulting number is 2300, it is best reported in exponential notation, 2.3 × 10 3, to clearly indicate two significant figures.

It takes 10.5 s for a sprinter to run m. Calculate the average speed of the sprinter in meters per second, and express the result to the correct number of significant figures. Practice Exercise

Answer: 9.52 m/s (three significant figures)

Accuracy versus Precision Accuracy refers to the proximity of a measurement to the true value of a quantity. Precision refers to the proximity of several measurements to each other.