Scientific Notation, Significant Figures, and Unit Conversions in Chemistry

Scientific Notation and Significant Figures

Introduction to Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It is commonly used in chemistry to handle measurements that span many orders of magnitude.

• Scientific Notation: A number is written as the product of a coefficient (between 1 and 10) and a power of ten.

• Example: 70,000,000 can be written as $7.0 \times 10^7$.

• Application: Used for reporting measurements such as Avogadro's number or atomic radii.

Converting Between Scientific and Standard Notation

• To convert a number to scientific notation, move the decimal point so that only one nonzero digit remains to its left, and count the number of places moved to determine the exponent.

• To convert from scientific to standard notation, move the decimal point to the right (for positive exponents) or left (for negative exponents) as indicated by the exponent.

• Example: $3.5 \times 10^4 = 35,000$

Significant Figures (Sig Figs)

Significant figures are the digits in a measurement that are known with certainty plus one digit that is estimated. They reflect the precision of a measured quantity.

• Definition: All nonzero digits are significant; zeros may or may not be significant depending on their position.

• Example: 45.119 g has five significant figures.

Rules for Determining Significant Figures

1. All nonzero digits are significant. Example: 123 has three significant figures.

2. Zeros between nonzero digits are significant. Example: 1002 has four significant figures.

3. Leading zeros (zeros to the left of the first nonzero digit) are not significant. Example: 0.041 has two significant figures.

4. Trailing zeros in a number containing a decimal point are significant. Example: 2.300 has four significant figures.

5. Trailing zeros in a whole number with no decimal shown are not significant. Example: 400 has one significant figure (unless otherwise indicated).

6. Exact numbers (from counting or defined quantities) have an infinite number of significant figures. Example: 12 eggs.

Rounding to Significant Figures

• When rounding, if the digit to be dropped is less than 5, the preceding digit remains unchanged; if it is 5 or greater, the preceding digit increases by one.

• Example: Rounding 2.3456 to three significant figures gives 2.35.

Significant Figures in Calculations

• Multiplication and Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Example: $3.11 \times 2.0 = 6.2$ (two significant figures)

Unit Conversions and SI Prefixes

Introduction to Unit Conversions

Unit conversions are essential in chemistry for expressing measurements in different units, often using SI (International System of Units) prefixes.

• SI Prefixes: Used to indicate multiples or fractions of base units (e.g., kilo-, centi-, milli-).

• Example: 1 kilometer (km) = 1,000 meters (m).

Converting Units

• Use conversion factors to change from one unit to another.

• Set up the conversion so that units cancel appropriately.

• Example: To convert 15 miles to kilometers, use the conversion factor $1 \text{ mile} = 1.609 \text{ km}$.

Compound Units and Dimensional Analysis

• When converting compound units (e.g., mi/hr to km/s), convert each unit separately.

• Dimensional analysis (factor-label method) is a systematic approach to unit conversions.

• Example: $60 \text{ mi/hr} \times \frac{1.609 \text{ km}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 0.0268 \text{ km/s}$

Density and Its Applications

Definition and Formula

Density is a physical property defined as the mass of a substance per unit volume. It is commonly used to identify substances and solve problems involving mass and volume.

• Formula:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Units: Commonly expressed in g/mL or g/cm3 for liquids and solids.

• Example: If a liquid has a mass of 541 g and a volume of 700.0 mL, its density is $\frac{541 \text{ g}}{700.0 \text{ mL}} = 0.773 \text{ g/mL}$.

Using Density as a Conversion Factor

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

• Example: To find the mass of 300. cm3 of copper with a density of 8.96 g/cm3:

$\text{Mass} = \text{Density} \times \text{Volume} = 8.96 \text{ g/cm}^3 \times 300. \text{ cm}^3 = 2,688 \text{ g}$

Summary Table: SI Prefixes (Selected)

Prefix	Symbol	Factor
kilo-	k	103
centi-	c	10-2
milli-	m	10-3
micro-	μ	10-6

Practice Problems (Overview)

• Convert numbers between scientific and standard notation.

• Identify certain and estimated digits in measurements.

• Apply rules for significant figures in various contexts.

• Round numbers to a specified number of significant figures.

• Perform calculations and report answers with correct significant figures.

• Convert between units, including compound and SI prefix units.

• Calculate density and use it as a conversion factor.

Additional info: The above notes expand on the brief points and practice problems in the original file, providing definitions, rules, and examples for each key concept relevant to introductory chemistry.