Chapter 1: Matter, Measurement, and Problem Solving – Study Notes

Chapter 1: Matter, Measurement, and Problem Solving

Overview

This chapter introduces foundational concepts in general chemistry, focusing on the nature of matter, measurement techniques, and systematic approaches to solving chemical problems. Key topics include derived units, density, intensive and extensive properties, and strategies for unit conversion and dimensional analysis.

Derived Units

Definition and Examples

Derived units are combinations of base units used to express quantities that cannot be described by a single base unit.

• Derived unit: A unit formed by combining two or more base units.

• Examples:

  • Speed: miles per hour (mi/hr)

  • Density: grams per milliliter (g/mL)

Derived units are essential for expressing physical properties such as velocity, acceleration, and density.

Density

Definition and Formula

Density is a physical property that describes the mass of a substance per unit volume. It is commonly used to identify substances and predict their behavior in mixtures.

• Density (d): The ratio of mass (m) to volume (V).

Formula:

$d = \frac{m}{V}$

• Units: Typically expressed in g/cm3 or g/mL.

• Density is a physical property that is temperature dependent.

• Density determines whether a substance will sink or float in another substance (e.g., less dense substances float).

Intensive vs. Extensive Properties

Intensive Properties

Intensive properties are characteristics of matter that do not depend on the amount of substance present.

• Intensive property: Independent of the quantity of substance.

• Examples: Density, boiling point, color.

• Density is an example: The density of aluminum is the same whether you have 1 g or 1 kg.

• Intensive properties are often used to identify substances because they depend only on the type of substance.

Example: The density of pure gold is 19.3 g/cm3. To determine if a substance is pure gold, measure its density and compare to this value.

Extensive Properties

Extensive properties depend on the amount of substance present.

• Extensive property: Varies with the quantity of substance.

• Examples: Mass, volume, length.

• Knowing only the mass of a substance will not allow you to identify it.

Calculating Density

To calculate the density of a substance, divide its mass by its volume.

• Step 1: Measure the mass (m) of the substance.

• Step 2: Measure the volume (V) of the substance.

• Step 3: Use the formula $d = \frac{m}{V}$ to calculate density.

Example: A nugget suspected to be gold has a mass of 22.5 g and a volume of 2.38 cm3. Its density is:

$d=\frac{22.5\ g}{2.38\ cm^3}=9.45\ \left(\frac{g}{\operatorname{cm}}\right)^3$

Since pure gold has a density of 19.3 g/cm3, the nugget is not pure gold.

Unit Conversion and Dimensional Analysis

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another using conversion factors.

• Conversion factor: A ratio that expresses how many of one unit are equal to another unit (e.g., 2.54 cm = 1 in).

• Conversion factors can be inverted as needed to cancel units.

Dimensional Analysis

Dimensional analysis is a systematic method for solving problems by tracking units throughout calculations.

• Units are treated as algebraic quantities that can be multiplied, divided, and canceled.

• General format: Information given × Conversion factor(s) = Information sought

Example: Convert 1.76 yards to centimeters.

• Given: 1.76 yards

• Find: centimeters

• Strategy: Use conversion factors (1 yard = 0.9144 meters, 1 meter = 100 centimeters)

$1.76\ \text{yd} \times \frac{0.9144\ \text{m}}{1\ \text{yd}} \times \frac{100\ \text{cm}}{1\ \text{m}} = 161\ \text{cm}$

Units Raised to a Power

When converting units raised to a power (e.g., area, volume), both the numerical value and the unit must be raised to the appropriate power.

• Example: $2.54\ \text{cm} = 1\ \text{in}$

• For area: $(2.54\ \text{cm})^2 = 6.45\ \text{cm}^2 = 1\ \text{in}^2$

Problem-Solving Strategy

General Steps

Effective problem solving in chemistry involves a systematic approach:

1. Sort: Identify the given information and what is to be found.

2. Strategize: Develop a conceptual plan for solving the problem.

3. Solve: Carry out the calculations according to the plan.

4. Check: Verify that the answer makes sense, units are correct, and significant figures are appropriate.

Conceptual Plan

A conceptual plan is a visual outline that maps the flow of information from the starting point to the desired result, often using arrows to indicate conversion steps.

• Example: Convert 12.5 in to cm using the conversion factor 2.54 cm/in.

• in → cm

Worked Problems: Density and Unit Conversion

Density as a Conversion Factor

Density can be used as a conversion factor to relate mass and volume.

• Example: A jet is fueled with 173,231 L of jet fuel (density = 0.768 g/cm3). Find the mass in kg.

Steps:

• Convert volume from liters to cm3 ($1\ \text{L} = 1000\ \text{cm}^3$)

• Multiply by density to get mass in grams

• Convert grams to kilograms ($1\ \text{kg} = 1000\ \text{g}$)

Formula:

$\text{mass (kg)} = \text{volume (L)} \times \frac{1000\ \text{cm}^3}{1\ \text{L}} \times \frac{0.768\ \text{g}}{1\ \text{cm}^3} \times \frac{1\ \text{kg}}{1000\ \text{g}}$

Problems Involving Equations

Some problems require using equations to relate variables.

• Example 1: Find the radius (r) of a sphere with volume $V = 0.058\ \text{cm}^3$.

Formula:

$V = \frac{4}{3} \pi r^3$

Solve for r:

$r = \left( \frac{3V}{4\pi} \right)^{1/3}$

• Example 2: Find the density of a metal cylinder with mass m, length l, and radius r.

Formula for volume of a cylinder:

$V = \pi r^2 l$

Density:

$d = \frac{m}{V}$

Comparison Table: Intensive vs. Extensive Properties

Property Type	Definition	Examples	Depends on Amount?
Intensive	Independent of quantity	Density, boiling point, color	No
Extensive	Dependent on quantity	Mass, volume, length	Yes

Summary

• Understanding the difference between intensive and extensive properties is crucial for identifying substances.

• Density is a key intensive property used in chemical identification and calculations.

• Unit conversion and dimensional analysis are essential skills for solving chemical problems.

• Systematic problem-solving strategies improve accuracy and efficiency in chemistry.

Additional info: These notes expand on the brief points in the provided materials, adding definitions, formulas, and examples for clarity and completeness.