Chapt 2. Measurement and Problem Solving: Scientific Notation, Units, Significant Figures, and Dimensional Analysis

Scientific Notation

Definition and Purpose

Scientific notation is a method used to express very large or very small numbers efficiently by using powers of ten. This notation is essential in chemistry for handling measurements that span many orders of magnitude.

• Format: A number is written as coefficient × 10exponent.

• Example: $2.14 \times 10^{-3}$, where 2.14 is the coefficient and -3 is the exponent.

• Exponent meaning: The exponent indicates the number of places the decimal point is moved.

• Number of zeros: For positive exponents, the number of zeros after the coefficient equals the exponent. For negative exponents, the number of decimal places before the coefficient equals the absolute value of the exponent.

Base Units and Unit Modifiers

SI Base Units

The International System of Units (SI) defines standard base units for fundamental physical quantities.

Measurement	Unit
Mass	kilogram (kg)
Length	meter (m)
Time	second (s)
Temperature	kelvin (K)
Amount	mole (mol)

Prefixes for Larger and Smaller Units

Prefixes are used to modify base units to represent quantities that are much larger or smaller than the base unit.

Prefix	Abbreviation	Relation to Base Unit
Tera-	T	1 T = $10^{12}$
Giga-	G	1 G = $10^{9}$
Mega-	M	1 M = $10^{6}$
Kilo-	k	1 k = $10^{3}$
Centi-	c	1 c = $10^{-2}$
Milli-	m	1 m = $10^{-3}$
Micro-	μ	1 μ = $10^{-6}$
Nano-	n	1 n = $10^{-9}$
Pico-	p	1 p = $10^{-12}$

English and Metric Unit Relationships

Common Conversions

Understanding the relationships between English and metric units is crucial for solving chemistry problems involving measurements.

Measurement	Metric Unit	English Unit	Relationship
Length	meter (m)	foot (ft)	1 m = 3.280 ft
Length	kilometer (km)	mile (mi)	1 km = 0.621 mi
Mass or Weight	kilogram (kg)	pound (lb)	1 kg = 2.205 lb
Volume	liter (L)	gallon (gal)	1 L = 0.264 gal

Significant Figures

Definition and Importance

Significant digits (or significant figures) indicate the precision of a measured value. The more significant digits, the more precise the measurement.

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros (before the first nonzero digit) are not significant.

• Trailing zeros after a decimal point are significant.

• Trailing zeros in a whole number without a decimal point are not significant.

Example: 47.1 mL (3 significant digits, more precise than 48 mL, which has 2 significant digits).

Rules for Identifying Significant Figures

• "Placeholder zeros" are not significant (e.g., 0.0000160 has 3 significant figures).

• "Precision zeros" are significant (e.g., 4670. has 4 significant figures).

• All other numbers are significant.

Example: 0.0820573660 has 9 significant figures.

Exact Numbers

Exact numbers have no uncertainty and do not affect significant figures in calculations. They include:

• Counted values (e.g., exactly 7 pennies)

• Defined values (e.g., 1,000 mg = 1 g, 3 feet = 1 yard)

Calculations with Significant Figures

Multiplication and Division

When multiplying or dividing, the result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $\frac{315.3\ \text{miles}}{5.2\ \text{hours}} = 61\ \text{miles/hour}$ (rounded to 2 significant figures)

Addition and Subtraction

When adding or subtracting, the result should be rounded to the last decimal place of the least precise measurement.

• Example: $0.432\ \text{mi} + 18.1\ \text{mi} = 18.5\ \text{mi}$ (rounded to the tenths place)

Scientific Notation in Calculations

Calculator Use

• Multiplication: $3.1 \times 10^4 \times 2.0 \times 10^2 = 6.2 \times 10^6$

• Division: $\frac{8.4 \times 10^7}{2.0 \times 10^3} = 4.2 \times 10^4$

Dimensional Analysis (Unit Conversions)

Introduction and Steps

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

1. Write the starting value and unit.

2. Write the goal unit.

3. Brainstorm relevant conversions.

4. Write each conversion as a fraction to cancel the undesired unit.

5. Repeat as necessary.

6. Check your answer for reasonableness.

7. Round to the appropriate significant figures.

Example: Convert 0.281 kg to grams. $0.281\ \text{kg} \times \frac{1000\ \text{g}}{1\ \text{kg}} = 281\ \text{g}$

Multiple Conversions

Some problems require more than one conversion factor. Follow the same steps, chaining conversion factors as needed.

• Example: Convert 78,000 mm to km: $78,000\ \text{mm} \times \frac{1\ \text{m}}{1000\ \text{mm}} \times \frac{1\ \text{km}}{1000\ \text{m}} = 0.078\ \text{km}$

Density and Its Use in Conversions

Definition and Formula

Density is the mass of a substance per unit volume. It is commonly used to convert between mass and volume.

Formula:

$\text{Density} = \frac{\text{mass (g)}}{\text{volume (mL or cm}^3)}$

Example Calculation

• A saltwater solution has a mass of 11.29 g and a volume of 10.4 mL. Its density is $\frac{11.29\ \text{g}}{10.4\ \text{mL}} = 1.09\ \text{g/mL}$

Densities of Common Materials

Material	Density (g/cm3)
Aluminum	2.70
Titanium	4.51
Iron	7.87
Copper	8.96
Lead	11.34
Gold	19.31
Water	1.00
Seawater	1.02
Air	0.001

Application Example

• A section of railroad track has a volume of 88,000 cm3. If the steel has a density of 7.80 g/cm3, its mass is $88,000\ \text{cm}^3 \times 7.80\ \text{g/cm}^3 = 686,400\ \text{g}$, or $686,400\ \text{g} \times \frac{1\ \text{kg}}{1000\ \text{g}} = 686.4\ \text{kg}$, and in pounds: $686.4\ \text{kg} \times 2.205\ \text{lb/kg} = 1,514\ \text{lb}$.

Practice and Exam-Level Questions

• Determine the number of significant digits in various values.

• Perform calculations with correct significant figures.

• Convert between units using dimensional analysis.

• Apply density in mass-volume conversions.

Example Exam Question: You own a 50.0 gallon salt-water fish tank. How many grams does the salt water weigh? (Density = 1.03 g/mL)

Example Exam Question: Your friend has 5.0 kg of ethanol (0.79 g/mL). How many gallons is this?

Additional info: This chapter covers foundational measurement and problem-solving skills essential for all subsequent chemistry topics, including scientific notation, unit conversions, significant figures, and density calculations.