Measurement, Unit Conversion, and Density – Chapter 2 Study Notes

Measurement and Problem Solving

Introduction

This chapter covers essential skills in measurement, unit conversion, and the concept of density, which are foundational for success in General Chemistry. Mastery of these topics enables students to solve quantitative problems and interpret scientific data accurately.

Unit Conversion in Both the Numerator and Denominator

Understanding Compound Units

• Compound units are measurements that involve two different units, such as speed (miles per hour, mi/h) or concentration (milligrams per milliliter, mg/mL).

• To convert compound units, convert both the numerator and denominator units separately.

Example: Speed Conversion

• Convert 65 mi/h to m/s, given 1 km = 0.6214 mi.

• Step 1: Convert miles to kilometers: $65 \text{ mi} \times \frac{1 \text{ km}}{0.6214 \text{ mi}}$

• Step 2: Convert kilometers to meters: $\text{result} \times 1000 \text{ m}/1 \text{ km}$

• Step 3: Convert hours to seconds: $1 \text{ h} = 3600 \text{ s}$

• Final calculation: $\text{Speed (m/s)} = \frac{65 \times 1000}{0.6214 \times 3600}$

Example: Concentration Conversion

• Convert 32 mg/mL to g/L.

• Step 1: Convert mg to g: $1 \text{ mg} = 10^{-3} \text{ g}$

• Step 2: Convert mL to L: $1 \text{ mL} = 10^{-3} \text{ L}$

• Calculation: $32 \text{ mg/mL} \times \frac{10^{-3} \text{ g}}{1 \text{ mg}} \times \frac{1 \text{ mL}}{10^{-3} \text{ L}} = 32 \text{ g/L}$

Units Raised to a Power

Converting Squared and Cubed Units

• When converting units raised to a power (e.g., cm2, cm3), the conversion factor must also be raised to that power.

• This is common in area and volume conversions.

Example: Volume Conversion

• Convert 1255 cm3 to in3.

• Step 1: Know the linear relationship: $1 \text{ in} = 2.54 \text{ cm}$

• Step 2: Cube both sides: $1 \text{ in}^3 = (2.54 \text{ cm})^3 = 16.387 \text{ cm}^3$

• Step 3: Set up the conversion: $1255 \text{ cm}^3 \times \frac{1 \text{ in}^3}{16.387 \text{ cm}^3} = 76.59 \text{ in}^3$ (rounded to 4 significant figures)

Example: Area Conversion

• To convert 28.2 cm2 to km2, use the squared conversion factor between cm and km.

• Step 1: $1 \text{ km} = 100,000 \text{ cm}$

• Step 2: $1 \text{ km}^2 = (100,000 \text{ cm})^2 = 1 \times 10^{10} \text{ cm}^2$

• Step 3: $28.2 \text{ cm}^2 \times \frac{1 \text{ km}^2}{1 \times 10^{10} \text{ cm}^2}$

Density

Definition and Units

• Density is the mass of a substance per unit volume.

• Formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Common units: g/cm3 (solids), g/mL (liquids), kg/m3 (SI unit)

Table: Densities of Some Common Substances

Substance	Density (g/cm3)
Water	1.00
Glass	2.7
Iron	7.86
Copper	8.96
Platinum	21.4
Titanium	4.51
Acetone	0.7857
Aluminum	2.70
Lead	11.3
Gold	19.3
Silver	10.5
Nickel	8.90
Additional info: Some values inferred for completeness.

Calculating Density

• Given mass and volume, calculate density using $d = \frac{m}{V}$.

• Example: A liquid has a mass of 22.7 g and a volume of 21.4 mL. $d = \frac{22.7 \text{ g}}{21.4 \text{ mL}} = 1.06 \text{ g/mL}$ (rounded to 3 significant figures)

Using Density as a Conversion Factor

• Density can be used to convert between mass and volume.

• Example: Acetone has a density of 0.7857 g/mL. What is the volume of 7.22 g of acetone?

• $\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{7.22 \text{ g}}{0.7857 \text{ g/mL}} = 9.19 \text{ mL}$

Identifying Substances by Density

• Compare calculated density to known values to identify a substance.

• Example: A vase has a mass of 157 g and displaces 18.65 mL of water. $d = \frac{157 \text{ g}}{18.65 \text{ mL}} = 8.42 \text{ g/mL}$, which is less than the density of platinum (21.4 g/cm3), so the vase is not solid platinum.

Practice Problems

Unit Conversion with Powers

• Convert 1.0 m2 to yd2 (1 m = 1.094 yd): $1.0 \text{ m}^2 \times (1.094)^2 = 1.2 \text{ yd}^2$

• Convert 1.0 yd2 to ft2 (1 yd = 3 ft): $1.0 \text{ yd}^2 \times (3)^2 = 9.0 \text{ ft}^2$

• Convert 68.3 cm3 to m3 (1 m = 100 cm): $68.3 \text{ cm}^3 \times \frac{1 \text{ m}^3}{(100)^3 \text{ cm}^3} = 6.83 \times 10^{-5} \text{ m}^3$

Density Practice

• A golf club head made of titanium has a volume of 114 cm3 and a mass of 514.1 g. $d = \frac{514.1 \text{ g}}{114 \text{ cm}^3} = 4.51 \text{ g/cm}^3$

• A penny has a mass of 2.49 g and a volume of 0.349 cm3. $d = \frac{2.49 \text{ g}}{0.349 \text{ cm}^3} = 7.13 \text{ g/cm}^3$ (less than copper, so not pure copper)

• 46 mg of acetone (density 0.788 g/cm3): $46 \text{ mg} \times \frac{1 \text{ g}}{1000 \text{ mg}} = 0.046 \text{ g}$; $\text{Volume} = \frac{0.046 \text{ g}}{0.788 \text{ g/cm}^3} = 0.0584 \text{ cm}^3$

Additional info: Significant figures are important in all calculations. Always round your final answer to the correct number of significant figures based on the data provided.