Significant Figures, Units, Dimensional Analysis, Temperature, and Density: A General Chemistry Study Guide

Significant Figures in Calculations, Units, Dimensional Analysis, Temperature, and Density

Significant Figures: Concepts and Application

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. They communicate the precision of a measurement and are essential in reporting scientific data accurately.

• Definition: Significant figures include all nonzero digits, zeros between nonzero digits, and trailing zeros in the decimal portion.

• Purpose: To reflect the precision of measured quantities and ensure that calculated results do not imply greater certainty than the measurements allow.

• Exact Numbers: Numbers that are counted or defined (e.g., 12 eggs in a dozen, 1 inch = 2.54 cm) have infinite significant figures and do not limit the precision of calculations.

• Scientific Notation: Used to express very large or small numbers, showing clearly the number of significant figures (e.g., $1.5 \times 10^2$ has two significant figures).

Accuracy vs. Precision

Accuracy refers to how close a measurement is to the true value, while precision refers to how close repeated measurements are to each other.

• Accuracy: Closeness to the actual or accepted value.

• Precision: Closeness of a set of measurements to each other, regardless of their accuracy.

Significant Figures in Calculations

When performing calculations, the number of significant figures in the result must reflect the precision of the least certain measurement.

• Addition/Subtraction: The result is rounded to the least precise decimal place among the numbers used.

• Multiplication/Division: The result is rounded to the same number of significant figures as the measurement with the fewest significant figures.

Examples:

• Addition: $37.68 + 108.428 + 6.71862 = 152.82662$ (calculator result). The correct answer, rounded to the least precise decimal place (hundredths), is 152.83.

• Subtraction: $190 - 41.47 = 148.53$ should be reported as 150 or $1.5 \times 10^2$ (rounded to the tens place).

• Multiplication: $0.80 \times 63.8 = 51.04$ (calculator result). The correct answer, rounded to two significant figures, is 51.

Practice Table: Significant Figures in Addition/Subtraction

Question	Answer
9.01 + 8.4	17.4
8.7 - 8.20	0.5
48.7 + 39.81 + 150	239
945.26 + 295 - 0.7	1240
8.82 × 10-3 + 6.50 × 10-2	7.38 × 10-2

Additional info: Answers inferred based on significant figure rules and typical practice problems.

Units and Measurement Systems

Units: English vs. Metric Systems

Units are standardized quantities used to specify measurements. The two main systems are the English (Imperial) and Metric (SI) systems.

• English System: Uses units such as inches, feet, pounds, and gallons. Conversions are not always decimal-based (e.g., 12 inches = 1 foot).

• Metric System: Based on powers of ten, making conversions straightforward (e.g., 1 meter = 100 centimeters).

Common Metric Prefixes

Prefix	Symbol	Factor
Giga	G	$10^9$
Mega	M	$10^6$
Kilo	k	$10^3$
Deci	d	$10^{-1}$
Centi	c	$10^{-2}$
Milli	m	$10^{-3}$
Micro	μ	$10^{-6}$
Nano	n	$10^{-9}$
Pico	p	$10^{-12}$

Basic Metric Units

• Mass: gram (g)

• Length: meter (m)

• Volume: liter (L)

• Time: second (s)

Dimensional Analysis (Unit Conversions)

Dimensional Analysis

Dimensional analysis is a systematic method for converting between units using conversion factors. It ensures that units cancel appropriately, leaving the desired unit.

• Conversion Factor: A ratio derived from the equality between two units (e.g., $1 \text{ kg} = 1000 \text{ g}$ gives $\frac{1000\ \text{g}}{1\ \text{kg}}$ or $\frac{1\ \text{kg}}{1000\ \text{g}}$).

• Process: Multiply the original value by conversion factors so that unwanted units cancel and only the desired unit remains.

Example: Metric-to-Metric Conversion

• Convert 45.8 Mg to grams:

$45.8\ \text{Mg} \times \frac{1 \times 10^6\ \text{g}}{1\ \text{Mg}} = 4.58 \times 10^7\ \text{g}$

Example: Multi-Step Conversion

• Convert 5.8 kL to centiliters:

$5.8\ \text{kL} \times \frac{1 \times 10^3\ \text{L}}{1\ \text{kL}} \times \frac{1\ \text{cL}}{1 \times 10^{-2}\ \text{L}} = 5.8 \times 10^5\ \text{cL}$

Common English Unit Conversions

Equality	Conversion Factor
1 pound (lb) = 16 ounces (oz)	$\frac{16\ \text{oz}}{1\ \text{lb}}$
1 ton = 2000 lb	$\frac{2000\ \text{lb}}{1\ \text{ton}}$
1 foot (ft) = 12 inches (in)	$\frac{12\ \text{in}}{1\ \text{ft}}$
1 yard (yd) = 3 ft	$\frac{3\ \text{ft}}{1\ \text{yd}}$
1 mile = 5280 ft	$\frac{5280\ \text{ft}}{1\ \text{mile}}$
1 gallon (gal) = 4 quarts (qt)	$\frac{4\ \text{qt}}{1\ \text{gal}}$
1 qt = 2 pints (pt)	$\frac{2\ \text{pt}}{1\ \text{qt}}$
1 qt = 32 fluid oz	$\frac{32\ \text{fl oz}}{1\ \text{qt}}$

Bridging English and Metric Units

• Common bridging conversion factors:

Equality	Purpose
1 lb = 454 g	Mass
1 in = 2.54 cm	Length
1 qt = 0.946 L	Volume

Additional info: These bridging factors are exact and do not limit significant figures.

Temperature Scales and Conversions

Temperature Scales

Temperature is a measure of the average kinetic energy of particles in a substance. The three common temperature scales are Celsius (°C), Fahrenheit (°F), and Kelvin (K).

• Celsius (°C): Based on the freezing (0°C) and boiling (100°C) points of water.

• Fahrenheit (°F): Used primarily in the United States; water freezes at 32°F and boils at 212°F.

• Kelvin (K): The SI unit for temperature; absolute zero is 0 K, and water freezes at 273.15 K.

Temperature Conversion Formulas

• Fahrenheit to Celsius: $^{\circ}\text{C} = \frac{5}{9}(^{\circ}\text{F} - 32)$

• Celsius to Fahrenheit: $^{\circ}\text{F} = \frac{9}{5} \, ^{\circ}\text{C} + 32$

• Celsius to Kelvin: $K = ^{\circ}\text{C} + 273.15$

• Kelvin to Celsius: $^{\circ}\text{C} = K - 273.15$

Example:

• Convert 124°F to Celsius:

$^{\circ}\text{C} = \frac{5}{9}(124 - 32) = \frac{5}{9}(92) = 51.1^{\circ}\text{C}$

Density

Definition and Formula

Density is a physical property defined as the mass of a substance per unit volume. It is an intensive property, meaning it does not depend on the amount of substance.

• Formula: $D = \frac{m}{V}$, where $D$ is density, $m$ is mass, and $V$ is volume.

• Common Units: g/mL, g/cm3 (1 mL = 1 cm3).

Example:

• Calculate the density of lead if a sample has a mass of 220.9 g and a volume of 19.5 mL:

$D = \frac{220.9\ \text{g}}{19.5\ \text{mL}} = 11.3\ \text{g/mL}$

Table: Densities of Some Common Materials

Substance	Density (g/cm3)
Human fat	0.94
Human muscle	1.06
Cork	0.22
Balsa wood	0.12
Earth (average)	5.54

Applications:

• Density is used to identify substances and to calculate mass or volume when the other is known.

Summary of Key Concepts

• Significant figures in calculations (addition/subtraction, multiplication/division)

• Units and unit conversions (metric and English systems)

• Dimensional analysis for systematic unit conversion

• Temperature scales and conversions (Celsius, Fahrenheit, Kelvin)

• Density and its calculation and applications