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Science Connections
Measurement (Units, Uncertainty, and Dimensional Analysis)













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Advanced Placement Chemistry

Lesson Plan #2

2 class periods

Measurement (Units, Uncertainty, and Dimensional Analysis)

Units of Measurement

    • The units used for scientific measurements are those of the metric system
    • SI Units

    • Systme International d'Units
    • Has seven base units from which all other units are derived

    • Mass | kilogram | kg
    • Length | meter | m
    • Time | second | s or sec
    • Electric current | ampere | A
    • Temperature | Kelvin | K
    • Luminous intensity | candela | cd
    • Amount of substance | mole | mol

    • For now we will consider three of these units: those for length, mass, and temperature.

Length and Mass

    • The SI unit of length is the meter (m), a distance only slightly longer than a yard.
    • Mass is a measure of the amount of material in an object. The SI base unit of mass if the kilogram (kg), which is equal to about 2.2 pounds. This base unit is unusual because it used a prefix, kilo-, instead of the word gram alone.

Selected Prefixed Used in the Metric System (Table 1.5 pg.14)

Prefix

Abbreviation

Meaning

Example

Giga-

G

109

1 gigameter (Gm) = 1 X 109 m

Mega-

M

106

1 megameter (Mm) = 1 X 106 m

Kilo-

k

103

1 kilometer (km) = 1 X 103 m

Deci-

d

10-1

1 decimeter (dm) = 0.1 m

Centi-

c

10-2

1 centimeter (cm) = 0.01 m

Milli-

m

10-3

1 millimeter (mm) = 0.001 m

Micro-

10-6

1 micrometer ( m) = 1 X 10-6 m

Nano-

n

10-9

1 nanometer (nm) = 1 X 10-9 m

Pico-

p

10-12

1 picometer (pm) = 1 X 10-12 m

Femto-

f

10-15

1 femtometer (fm) = 1 X 10-15 m

Sample Exercise 1.1

What is the name given to the unit that equals (a) 10-9 gram; (b) 10-6 second; (c) 10-3 meter?

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.

Practice Exercise

What fraction of a second is a picosecond, ps? 10-12 second

Temperature

    • Temperature determines the direction of heat flow.
    • Heat always flows spontaneously from a substance at higher temperature to one at lower temperature.
    • Scales commonly employed in scientific studies are the Celsius and Kelvin scales.

    • Celsius scale widely used in chemistry.

    • Based on the assignment of 0 C to the freezing point of water
    • 100 C to the boiling point of water
    • C = 5/9( F - 32) or
    • F = 9/5( C) + 32

    • Kelvin scale is the SI temperature scale; (K)

    • Zero on this scale is the lowest attainable temperature, -273.15 C, a temperature referred to as absolute zero.
    • K = C + 273.15
    • Freezing point of water; 273.15K
    • Boiling point of water; 373.15K
    • Notice that we do not use a degree sign ( ) with temperatures on the Kelvin scale.

Sample Exercise 1.2

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?

Solution

    1. Using equation K = C + 273
    2. K = 31 + 273 = 304K

    3. Using equation F = 9/5( C) + 32

F = 9/5(31) + 32 = 56 + 32 = 88 F

Practice Exercise

Ethylene glycol, the major ingredient in antifreeze, freezes at -11.5 C. What is the freezing point in (a) K; (b) F?

Answers: (a) 261.7 K; (b) 11.3 F

Volume

    • The volume of a cube is given by its length cubed, (length)3. Thus, the basic SI unit of volume is the cubic meter, or m3.
    • Because this is a very large volume, smaller units are commonly used for most applications of chemistry. The cubic centimeter, cm3 (sometimes written as cc), is one such unit.
    • The cubic decimeter, dm3 is also used. This volume is more commonly known as the liter (L), and is slightly larger than a quart.

    • The liter is not a SI unit.
    • 1000 milliliters (mL) in a liter
    • milliliter is the same volume as a cubic centimeter
    • 1 mL = 1 cm3 = 1 cc
    • milliliter and cubic centimeter used interchangeably in expressing volume.

    • Syringes, burets, and pipets allow delivery of liquids with more accuracy than do graduated cylinders.
    • Volumetric flasks are used to contain specific volumes of liquid (see figure 1.20 on page 17)

Density

    • Widely used to characterize substances
    • Defined as the amount of mass in a unit volume of the substance
    • Density = mass/volume
    • Commonly expressed in units of grams per cubic centimeter (g/cm3)
    • Density and weight are sometimes confused. 1 kg of air has the same mass as 1 kg of iron, but the iron occupies a small volume, thereby giving it a higher density.

Sample Exercise 1.3

  1. Calculate the density of mercury if 1.00 X 102 g occupies a volume of 7.36 cm3.
  2. Calculate the mass of 65.0 cm3 of mercury.

Solution

  1. density = mass/volume = 1.00 X 102g/7.36 cm3 = 13.6 g/cm3
  2. mass = volume X density = (65.0 cm3)(13.6 g/cm3) = 884 g

Practice Exercise

A student needs 15.0 g of ethanol (ethyl alcohol) for an experiment. If the density of the alcohol is 0.789 g/mL, how many milliliters of alcohol are needed?

Answer: 19.0 mL (15.0 g/0.789 g/mL)[volume = mass/density]

Uncertainty in Measurement

    • Numbers obtained by measurement are always inexact.
    • There are always inherent limitations in the equipment used to measure quantities (equipment error).
    • There are differences in how different people make the same measurement (human errors).
    • Uncertainties always exist in measured quantities.

Precision and Accuracy

    • Precision is a measure of how closely individual measurements agree with one another.
    • Accuracy refers to how closely individual measurements agree with the correct, or "true" value.
    • If ten students measure a object and all of them determine the length to be 12 cm, they are precise
    • But if the actual length of the object is 18 cm, they are inaccurate.
    • The goal of all scientists is to be both precise and accurate.

Significant Figures

    • The notation with the understanding that an uncertainty of at least one unit exists in the last digit of the measured quantity.

    • Measured quantities are generally reported in such a way that only the last digit is uncertain.

    • All digits, including the uncertain one, are called significant figures.
    • The number of significant figures indicates the exactness of a measurement.

Sample exercise 1.4

What is the difference between 4.0 g and 4.00 g?

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 implies that the second measurement is more precise. A mass of 4.0 indicates that the mass is between 3.9 and 4.1 g; the mass is 4.0 0.1 g. A measurement of 4.00 implies that the mass is between 3.99 and 4.01 g; the mass is 4.00 0.01 g,

We will be measuring mass to this level of precision in this class.

Practice Exercise

A balance has a precision of 0.001 g. A sample that weighs about 25 g is weighed on this balance. How many significant figures should be reported for this measurement?

Answer: 5 [25.000 0.001 g]

The following guidelines apply to determining the number of significant figures in a measured quantity:

    1. Nonzero digits are always significant - 457 cm (3 significant figures); 2.5 g (2 significant figures).
    2. Zeros between nonzero digits are always significant - 1005 kg (4 significant figures); 1.03 (3 significant figures).
    3. Zeros at the beginning of a number are never significant; they merely indicate the position of the decimal point - 0.02 (1 significant fig); 0.0026 cm (2 significant figures).
    4. Zeros that fall both at the end of a number and after the decimal point are always significant - 0.0200 (3 significant figures); 3.0 (2 significant figures).
    5. When a number ends in zero but contains no decimal point, the zeros may or may not be significant - 130 cm (2 or 3 significant figures); 10300 g (3, 4, or 5 significant figures). So how do we remove this ambiguity? By exponential notation.

The use of exponential notation avoids the potential ambiguity of whether the zeros at the end of a number are significant. For example , a mass of 10300 g can be written in exponential notation showing 3, 4, or 5 significant figures.

1.03 X 104 g [3 significant figures]

1.030 X 104 g [4 significant figures]

1.0300 X 104 g [5 significant figures]

Significant Figures in Calculations

    • In multiplication and division the result must be reported with the same number of significant figures as the measurement with the fewest significant figures.

    • When the result contains more than the correct number of significant figures, it must be rounded off.

    • In addition and subtraction the result cannot have more digits to the right of the decimal point than any of the original numbers.

This number limits 20.4 1 decimal place

the number of significant 1.322 3 decimal places

figures in the result 83____ 0 decimal places

104.722 round off to 105

(1 uncertain digit)

Dimensional Analysis

    • In dimensional analysis we carry units through all calculations.
    • Units are multiplied together, divided into each other, or "cancelled."
    • Dimensional analysis will help ensure that the solutions to problems yield the proper units.
    • Dimensional analysis provides a systematic way of solving many numerical problems and of checking solutions for possible errors.
    • The key to using dimensional analysis is the correct use of conversion factors to change one unit into another.
    • A conversion factor is a fraction whose numerator and denominator are the same quantity expressed in different units. For example 2.54 cm and 1 in are the same length, 2.54 cm = 1 in.

Given unit x desired unit/given unit = desired unit

3526 mL x 1 L/1000 mL = 3.526 L

Summary of Dimensional Analysis

    • In using dimensional analysis to solve problems, we will always ask three questions:

    • What data are we given in the problem?
    • What quantity do we wish to obtain in the problem?
    • What conversion factors do we have available to take us from the given quantity to the desired one?