Chapter 2: Standards for Measurement Notes

Standards for Measurement

Chapter Outline

• 2.1 Mass and Weight
• 2.2 Measurement and Significant Figures
• 2.3 Rounding Off Numbers
• 2.4 Scientific Notation of Numbers
• 2.5 Significant Figures in Calculations
• 2.6 The Metric System
• 2.7 Measurement of Length
• 2.8 Problem Solving
• 2.9 Measurement of Mass
• 2.10 Measurement of Volume
• 2.11 Measurement of Temperature
• 2.12 Density

Introduction

• Chemistry is an experimental science, and measurements are crucial for determining experimental results.
• Measurements are vital in various scenarios, such as cooking, car repairs, and medication preparation.
• The metric system and appropriate units are fundamental for measuring mass, length, volume, and temperature in chemistry.

2.1 Mass and Weight

• In science, mass is defined as the amount of matter in a body, whereas weight is the measure of Earth's gravitational attraction for that object.
• Mass is a fixed quantity independent of location and measured using a balance.
• Weight is measured using a scale and varies with the object's position relative to Earth.
• Example: An astronaut's mass remains constant at 70.0 kg regardless of their location, but their weight changes as they move into space, eventually reaching a state of weightlessness.

2.2 Measurement and Significant Figures

• Measurements involve numerical values and units and are never exact due to limitations of instruments and human skill.
• Significant figures indicate the reliability (precision) of a measurement.
• A measurement should include all known digits plus one estimated digit, which introduces uncertainty.

Determining Significant Figures

• Consider measuring temperature using thermometers with different calibrations (Figure 2.2).
  • Thermometer A Calibration: Degrees.
    • Temperature lies between 21 and 22 degrees.
    • Estimate: Mercury is two-tenths the distance between 21 and 22.
    • Recorded Temperature: 21.2 degrees (three significant figures).
  • Thermometer B Calibration: Degrees.
    • Temperature exactly on 22.
    • Recorded Temperature: 22.0 degrees (three significant figures, the zero indicates estimation to one-tenth degree).
  • Thermometer C Calibration: Tenths of a degree.
    • Recorded Temperature: 22.11°C (four significant figures, hundredths place is estimated).
• Exact numbers: They have an infinite number of significant figures, occur in counting (e.g., 25 dollars), or are defined numbers (e.g., 12 inches in 1 foot, 100 cm in 1 meter).

Rules for Evaluating Zero

• Nonzero numbers are always significant.
• Zero is significant when it is:
  1. Between nonzero digits (e.g., 205 has three significant figures).
  2. At the end of a number that includes a decimal point (e.g., 0.500 has three significant figures).
• Zero is not significant when it is:
  1. Before the first nonzero digit (e.g., 0.0025 has two significant figures).
  2. At the end of a number without a decimal point (e.g., 1000 has one significant figure).
• To indicate that zeros at the end of a number are significant, use a decimal point and a power of 10 (scientific notation) to clarify the number of significant figures.
  • Example: If 1000 has four significant figures, write it as $1.000 \times 10^3$
  • Example: If 590 has two significant figures, write it as $5.9 \times 10^2$

2.3 Rounding Off Numbers

• Excess digits resulting from calculations must be dropped to express the answer with the correct number of significant figures.
• Value of the last digit retained is determined by rounding off.
• Rule 1: When the first digit after those you want to retain is 4 or less, drop that digit and all others to its right. The last digit retained is not changed.
  • Example: Round 74.693 to four digits: 74.69 (3 is dropped)
  • Example: Round 1.00629 to four digits: 1.006 (29 are dropped)
• Rule 2: When the first digit after those you want to retain is 5 or greater, drop that digit and all others to the right and increase the last digit retained by one.
  • Example: Round 1.026868 to four digits: 1.027 (6868 are dropped, 6 is increased to 7)
  • Example: Round 12.899 to four digits: 12.90 (9 is dropped, 89 are changed to 90)
  • Example: Round 18.02500 to four digits: 18.03 (500 are dropped, 2 is changed to 3)

2.4 Scientific Notation of Numbers

• Scientific notation simplifies the writing of very large and very small numbers using powers of 10.
• To write a number in scientific notation:
  1. Move the decimal point so that it is located after the first nonzero digit.
  2. Multiply this new number by 10 raised to the power equal to the number of places the decimal point has been moved.
     • If the decimal is moved to the left, the power of 10 is positive.
     • If the decimal is moved to the right, the power of 10 is negative.
• General form: A number between 1 and 10 multiplied by 10 raised to a power.
  • Example: $2468 = 2.468 \times 10^3$

2.5 Significant Figures in Calculations

• The result of a calculation cannot be more precise than the least precise measurement used in the calculation.

Multiplication or Division

• The answer must contain the same number of significant figures as the measurement with the fewest significant figures.
  • $(190.6)(2.3) = 438.38$
    • The calculator value is 438.38.(2.3) has two significant figures has the fewest number of significant figures, so the answer should have two significant figures.
    • The correct answer is $440$ or $4.4 \times 10^2$
  • $\frac{(13.59)(6.3)}{12} = 7.13475$
    • The calculator value is 7.13475. The numbers 6.3, 12 each have two significant figures, therefore the answer should contain two significant figures.
    • The correct answer is $7.1$

Addition or Subtraction

• The answer must be expressed to the same precision as the least precise measurement (i.e., the value with the fewest decimal places).

2.6 The Metric System

• The metric system, or International System (SI), is a decimal system of units for measurement.
• It uses factors of 10 to express larger or smaller numbers of units.
• Prefixes are added to the names of the units to represent multiples of 10.
• Examples:
  • 1 kilometer = 1000 meters.
  • 1 kilogram = 1000 grams.
  • 1 millimeter = 0.001 meter.
  • 1 microsecond = 0.000001 second.
• The seven standard units in the International System, their abbreviations, and the quantities they measure are given in Table 2.2.

2.7 Measurement of Length

• The standard unit of length in the metric system is the meter (m).
• Historical context: Initially, the meter was defined as one ten-millionth of the distance from the equator to the North Pole.
• Modern definition: The meter is now defined as the distance that light travels in a vacuum during 1/299,792,458 of a second.
• $1 m = 100 cm = 1000 mm$
• Relationships between length units:
  • $1 m = 10^{10} A$
  • $10 mm = 0.01 m$
  • $2.54 cm = 1 in.$

2.8 Problem Solving

• A systematic approach is essential for solving numerical problems in chemistry.
• Dimensional analysis method:
  1. Read the problem carefully and identify what needs to be solved.
  2. Tabulate the given data with proper units.
  3. Determine the principles and unit relationships needed.
  4. Set up the problem so that unwanted units cancel.
  5. Perform the necessary mathematical operations and ensure the answer has the correct number of significant figures.
  6. Check if the answer is reasonable.
• Dimensional analysis converts one unit to another using conversion factors:
  • $unit<em>1 \times conversion factor = unit</em>2$
• Conversion Factors Example $1 m = 1000 mm$
  • $\frac{1 m}{1000 mm} = 1$
  • $\frac{1000 mm}{1 m} = 1$

2.9 Measurement of Mass

• The gram is a unit of mass measurement.
• The standard unit of mass in the SI system is the kilogram (kg), equal to 1000 g.
• The kilogram is defined as the mass of a platinum-iridium cylinder kept in a vault at Sèvres, France.
• Comparing this unit of mass to 1 lb (16 oz), we find that 1 kg is equal to 2.205 lb. A pound is equal to 453.6 g (0.4536 kg).

Mass Relationships:

• $1 g = 1000 mg$
• $1 kg = 1000 g$
• $1 kg = 2.205 lb$
• $1 lb = 453.6 g$

2.10 Measurement of Volume

• Volume is the amount of space occupied by matter. The SI unit of volume is the cubic meter ($m^3$).
• The liter (L) and milliliter (mL) are standard units of volume used in chemistry laboratories.
• Calibrated glassware such as graduated cylinders, volumetric flasks, burets, pipets, and syringes are used to measure liquids (see Figure 2.5).
• Common Volume Relationships:
  • $1 L = 1000 mL = 1000 cm^3$
  • $1 mL = 1 cm^3$
  • $1 L = 1.057 qt$
  • $946.1 mL = 1 qt$

2.11 Measurement of Temperature

• Heat is a form of energy associated with the motion of small particles of matter.
• Heat refers to the quantity of energy within a system or to a quantity of energy added to or taken away from a system.
• Temperature measures the intensity of heat; temperature is a measure of how hot a system is, regardless of its size.
• The SI unit of temperature is the kelvin (K).
• Temperature Scales are expressed in the following ways:
  • Degrees Celsius = °C
  • Kelvin (absolute) = K
  • Degrees Fahrenheit = °F
• $0°C = 273 K = 32°F$

Temperature Conversion Formulas:

• $K = °C + 273.15$
• $°F = (1.8 \times °C) + 32$
• $°C = \frac{°F - 32}{1.8}$

2.12 Density

• Density (d) is the ratio of the mass of a substance to the volume occupied by that mass:
  • $d = \frac{mass}{volume}$
• Units used to express density include grams per milliliter (g/mL) or grams per cubic centimeter (g/cm³).

Calculating Volume with Density:

• Density can also be used to calculate the volume of a material given its mass:
  $Volume = \frac{Mass}{Density}$

Specific Gravity

• The specific gravity (sp gr) of a substance is the ratio of its density to the density of a reference substance, usually water at 4°C:
  $sp \ gr = \frac{Density \ of \ substance}{Density \ of \ water}$
• Specific gravity is dimensionless (no units) because the density units cancel out.