Measurement and Problem Solving – Introductory Chemistry Study Notes

Chapter 2: Measurement and Problem Solving

Introduction

This chapter covers the foundational concepts of measurement, scientific notation, significant figures, unit conversions, and density calculations, all of which are essential for problem solving in introductory chemistry. Mastery of these topics is crucial for accurate data collection, analysis, and reporting in scientific experiments.

Scientific Notation

Definition and Purpose

• Scientific notation is a method for expressing very large or very small numbers in the form a × 10n, where 1 ≤ a < 10 and n is an integer.

• This notation simplifies calculations and clearly indicates the precision of measurements.

Examples

• 332,000,000 people = $3.32 \times 10^8$ people

• 0.000000000070 m = $7.0 \times 10^{-11}$ m

• 28,430,000,000,000 = $2.843 \times 10^{13}$

Significant Figures

Definition and Rules

• Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated.

• Rules for counting significant figures:

  • All nonzero digits are significant.

  • Interior zeros (between nonzero digits) are significant.

  • Trailing zeros after a decimal point are significant.

  • Leading zeros are not significant; they only indicate the position of the decimal point.

  • Trailing zeros before a decimal point may or may not be significant (ambiguous unless specified).

  • Exact numbers (from definitions or counting) have unlimited significant figures.

Examples

• 0.0035 (2 sig figs), 1.080 (4 sig figs), 2371 (4 sig figs), 100.00 (5 sig figs), 100,000 (ambiguous)

Reporting Measurements

• Measurements should be reported to one digit beyond the smallest scale division.

• The last digit is always an estimate and may vary slightly between observers.

Rounding

• When rounding, if the digit to be dropped is less than 5, leave the last retained digit unchanged; if 5 or greater, increase the last retained digit by one.

Significant Figures in Calculations

Multiplication and Division

• The result should have the same number of significant figures as the factor with the fewest significant figures.

• Example: $0.12 \times 96 = 12$ (2 sig figs)

Addition and Subtraction

• The result should have the same number of decimal places as the quantity with the fewest decimal places.

• Example: $125.1 + 2.03 = 127.1$ (rounded to one decimal place)

Mixed Operations

• Follow the order of operations, applying the appropriate significant figure rule at each step.

• Intermediate results should be rounded according to the operation performed before proceeding to the next step.

Unit Conversions

Conversion Factors and Solution Maps

• Unit conversion involves multiplying by conversion factors to change from one unit to another.

• Solution maps visually organize the steps and conversion factors needed to solve a problem.

Common Conversion Factors

Quantity	Unit 1	Unit 2
Length	1 km	0.6214 mi
Length	1 m	39.37 in.
Length	1 in.	2.54 cm (exact)
Mass	1 kg	2.205 lb
Volume	1 L	1.057 qt
Volume	1 gal	3.785 L

SI Prefix Multipliers

Prefix	Symbol	Multiplier
kilo	k	$10^3$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	μ	$10^{-6}$
nano	n	$10^{-9}$

Multistep Conversions

• Some problems require multiple conversion steps, especially when converting between units raised to a power (e.g., cm2 to m2).

• Each conversion factor must be raised to the appropriate power.

Density

Definition and Formula

• Density is the ratio of mass to volume and is a physical property used to identify substances.

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

Applications

• Density can be used as a conversion factor between mass and volume.

• It is useful for determining the identity of a substance or checking the purity of a sample.

Example Calculation

• A ring with mass 5.84 g and volume 0.614 cm3 has density $= \frac{5.84\ \text{g}}{0.614\ \text{cm}^3} = 9.51\ \text{g/cm}^3$

• If the density does not match that of platinum (21.4 g/cm3), the ring is not platinum.

Table: Densities of Some Common Substances

Substance	Density (g/cm3)
Water	1.00
Platinum	21.4
Silver	10.5
Gold	19.3
Gasoline	0.70–0.77
Ethanol	0.789
Air	0.0012
Ice	0.92
Mercury	13.6
Iron	7.87

Summary Table: Key Concepts in Measurement and Problem Solving

Concept	Key Points
Scientific Notation	Expresses large/small numbers; clarifies significant figures
Significant Figures	Indicate precision; rules for counting and rounding
Unit Conversion	Uses conversion factors; solution maps for multistep problems
Density	Mass/volume; used for identification and conversions

Practice and Application

• Express numbers in scientific notation and identify significant figures.

• Perform calculations with correct significant figures.

• Convert between units using appropriate conversion factors and solution maps.

• Calculate density and use it as a conversion factor between mass and volume.