Chapter 3: Measurements, Units, and Problem Solving in Chemistry

3.1 Measurements and Their Uncertainty

Definition and Components of a Measurement

In chemistry, a measurement is a quantitative observation that includes both a number (magnitude) and a unit (standard of comparison). Accurate measurements are essential for scientific analysis and communication.

• Number: Indicates the amount or magnitude.

• Unit: Specifies the standard used (e.g., grams, liters, meters).

Scientific Notation

Scientific notation expresses numbers as the product of a coefficient (a number between 1 and 10) and a power of ten. This format is useful for very large or very small numbers commonly encountered in chemistry.

• General form: $a \times 10^n$

• Coefficient: The first factor (1 ≤ a < 10)

• Exponent: Indicates the power of ten

Examples:

• Large number: 7,500 = $7.5 \times 10^3$

• Large number: 530,000 = $5.3 \times 10^5$

• Small number: 0.000042 = $4.2 \times 10^{-5}$

• Small number: 0.039 = $3.9 \times 10^{-2}$

Accuracy and Precision

• Accuracy: How close a measurement is to the true or accepted value.

• Precision: How close repeated measurements are to each other, regardless of their accuracy.

Example: If a student measures a reaction rate three times and gets 4 mol/s, 25 mol/s, and 1000 mol/s, the data is not precise (values are not close to each other).

Note: To determine accuracy, the theoretical (true) value must be known.

Percent Error

Percent error quantifies the accuracy of an experimental value compared to a theoretical value.

Formula:

$\text{Percent error} = \frac{|\text{theoretical} - \text{experimental}|}{\text{theoretical}} \times 100$

Example: Theoretical yield = 2 g, Experimental yield = 1.8 g:

$\text{Percent error} = \frac{|2 - 1.8|}{2} \times 100 = 10\%$

Significant Figures

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one estimated digit. They reflect the precision of the measuring instrument.

• Not all instruments measure the same number of significant figures.

• Different meter sticks or devices may yield measurements with different numbers of significant figures due to their precision.

3.2 The International System of Units (SI)

SI Base Units

The International System of Units (SI) is the standard system of measurement in science. It is based on seven base units, several of which are commonly used in chemistry.

Quantity	SI Base Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Electric current	ampere	A
Luminous intensity	candela	cd

Commonly used by chemists: meter, kilogram, second, kelvin, mole.

SI Prefixes and Derived Units

Prefixes are added in front of SI units to indicate multiples or fractions of units.

Prefix	Symbol	Factor
kilo-	k	103
centi-	c	10-2
milli-	m	10-3
micro-	μ	10-6
nano-	n	10-9

Note: A prefix goes in front of the unit.

• 1 millimeter = $1 \times 10^{-3}$ meters

• 1 kilometer = $1 \times 10^3$ meters

Common SI Units in Chemistry

• Volume: SI unit is the cubic meter (m3), but liters (L) and milliliters (mL) are commonly used. 1 L = 1 dm3; 1 mL = 1 cm3.

• Mass: SI unit is the kilogram (kg), but the gram (g) is often used in the laboratory (1 g = 1/1000 kg).

• Temperature: SI unit is the kelvin (K). The Celsius scale is also used: water freezes at 0°C and boils at 100°C. Absolute zero is 0 K.

• Energy: Common units are the joule (J) and the calorie (cal).

Temperature conversion:

$K = ^\circ C + 273.15$

Example: 30°C = $30 + 273.15 = 303.15$ K

Choosing Appropriate SI Units

• Diameter of a marble: millimeter (mm)

• Volume of a raindrop: microliter (μL) or milliliter (mL)

• Distance between cities: kilometer (km)

• Mass of a book: kilogram (kg) or gram (g)

• Temperature: degree Celsius (°C) or kelvin (K)

• Mass of a penny: gram (g)

• Height: meter (m) or centimeter (cm)

• Mass of a bowling ball: kilogram (kg)

• Length of a pencil: centimeter (cm)

• Duration of a class: minute (min) or second (s)

• Mass of a grain of rice: milligram (mg)

• Time to blink: millisecond (ms)

3.3 Conversion Problems

Conversion Factors and Dimensional Analysis

Conversion factor: A ratio that expresses how many of one unit are equal to another unit. It is used to convert measurements from one unit to another.

• Example: $1 \text{ dollar} = 4 \text{ quarters} = 10 \text{ dimes} = 20 \text{ nickels} = 100 \text{ pennies}$

• In a conversion factor, the numerator and denominator represent equivalent quantities in different units.

Dimensional analysis (also called the factor-label method) is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

Example: To convert 2.5 km to meters:

$2.5 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 2500 \text{ m}$

• When multiplying by a conversion factor, the actual quantity does not change, only the units.

• Sometimes, multiple conversion factors are needed for multi-step problems.

3.4 Density

Definition and Formula

Density is the ratio of the mass of an object to its volume. It is an intensive property, meaning it does not depend on the amount of substance present.

Formula:

$\text{Density} = \frac{\text{mass}}{\text{volume}}$

• Common units: g/cm3 (solids), g/mL (liquids), g/L (gases)

Example: Gold is denser than aluminum. (Gold: 19.3 g/cm3, Aluminum: 2.7 g/cm3)

• Density is independent of sample size but depends on composition and temperature.

• As temperature increases, density usually decreases (due to expansion).

Concept Map (Organizing Information)

Key terms to include in a concept map for this chapter:

• Measurement

• Accuracy

• Precision

• Significant Figures

• SI Units

• Conversion Factor

• Dimensional Analysis

• Density

These concepts are interconnected and form the foundation for quantitative work in chemistry.