Chemistry and Measurements: Equalities and Conversion Factors

Chemistry and Measurements

Introduction

This section introduces the foundational concepts of measurements in chemistry, focusing on how different units are related through equalities and conversion factors. Understanding these relationships is essential for solving problems in General, Organic, and Biological (GOB) Chemistry.

Equalities

Definition and Purpose

• Equality is a statement that two different units describe the same measured amount.

• Equalities are used to relate units within the metric system, within the U.S. customary system, or between metric and U.S. units.

• Examples of equalities include:

  • 1 m = 1000 mm

  • 1 lb = 16 oz

  • 2.20 lb = 1 kg

Example: The maximum amount of mercury allowed in tuna is 1.0 ppm (parts per million). This is an example of a measurement that may require conversion between units.

Conversion Factors

Writing Conversion Factors

Each equality can be written as two fractions called conversion factors, which are used to convert from one unit to another.

• For the equality 60 min = 1 h:

  • $\frac{60\ \text{min}}{1\ \text{h}}$

  • $\frac{1\ \text{h}}{60\ \text{min}}$

• For the metric equality 1 m = 100 cm:

  • $\frac{100\ \text{cm}}{1\ \text{m}}$

  • $\frac{1\ \text{m}}{100\ \text{cm}}$

Conversion factors are used in dimensional analysis to convert between units in calculations.

Exact Numbers and Significant Figures

• Equalities between two metric units or two U.S. system units are defined by definition and are considered exact numbers.

• Exact numbers do not affect the determination of significant figures in calculations.

• Equalities between metric and U.S. units usually contain one measured value, which is not exact and does affect significant figures.

• Exception: The equality 1 in. = 2.54 cm is defined as an exact relationship; both numbers are exact.

Metric Conversion Factors

Examples and Application

• Metric conversion factors are derived from metric equalities.

• For 1 m = 100 cm, the conversion factors are:

  • $\frac{100\ \text{cm}}{1\ \text{m}}$

  • $\frac{1\ \text{m}}{100\ \text{cm}}$

• Both 100 and 1 are exact numbers in this context.

Metric-U.S. System Conversion Factors

Examples and Application

• Conversion factors between metric and U.S. units are commonly used in everyday life and science.

• For 1 kg = 2.20 lb, the conversion factors are:

  • $\frac{2.20\ \text{lb}}{1\ \text{kg}}$

  • $\frac{1\ \text{kg}}{2.20\ \text{lb}}$

• In these conversions, 2.20 is a measured value and is not exact, so it affects significant figures.

Example: Food packaging in the United States often lists contents in both metric and U.S. units, requiring conversion factors for interpretation.

Conversion Factors in Problem Solving

Application in Real-World Problems

• Equalities may be stated within a problem, such as a car traveling at 85 km/h or the price of onions being $1.24 per pound.

• Measured values (like 85 km/h or $1.24/lb) are not exact and have significant figures.

Conversion Factors in Dosage Problems

Medical Applications

• Dosage problems in medicine often use conversion factors derived from medication strengths.

• Example: Cephalexin is available in 250-mg capsules. The conversion factor is:

  • $\frac{250\ \text{mg}}{1\ \text{capsule}}$

  • $\frac{1\ \text{capsule}}{250\ \text{mg}}$

• Measured values (like 250 mg) have significant figures.

Percentage, ppm, and ppb as Conversion Factors

Definitions and Use

• Percentage (%) expresses a ratio of parts per hundred and can be used as a conversion factor.

• Parts per million (ppm) and parts per billion (ppb) are used for very small ratios, such as concentrations of contaminants.

• For example, 18% body fat by mass means:

  • $\frac{18\ \text{kg body fat}}{100\ \text{kg body mass}}$

• ppm is often expressed as micrograms per gram ($\mu$g/g) or milligrams per kilogram (mg/kg).

Summary Table: Common Equalities and Conversion Factors

Equality	Conversion Factor 1	Conversion Factor 2	Exact or Measured?
1 m = 100 cm	$\frac{100\ \text{cm}}{1\ \text{m}}$	$\frac{1\ \text{m}}{100\ \text{cm}}$	Exact
1 lb = 16 oz	$\frac{16\ \text{oz}}{1\ \text{lb}}$	$\frac{1\ \text{lb}}{16\ \text{oz}}$	Exact
1 kg = 2.20 lb	$\frac{2.20\ \text{lb}}{1\ \text{kg}}$	$\frac{1\ \text{kg}}{2.20\ \text{lb}}$	Measured
1 in. = 2.54 cm	$\frac{2.54\ \text{cm}}{1\ \text{in.}}$	$\frac{1\ \text{in.}}{2.54\ \text{cm}}$	Exact
18% body fat	$\frac{18\ \text{kg fat}}{100\ \text{kg body mass}}$	$\frac{100\ \text{kg body mass}}{18\ \text{kg fat}}$	Measured

Additional info:

• Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

• Always pay attention to significant figures when using measured values in calculations.