Introduction to Chemistry: Scientific Notation, Unit Conversions, and Significant Figures

Scientific Notation

Understanding 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.

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

• Example: $2.3 \times 10^8$ represents 230,000,000.

Writing Numbers in Scientific Notation

• To convert a standard number to scientific notation:

  1. Move the decimal point so that only one nonzero digit remains to its left.

  2. Count the number of places the decimal was moved; this becomes the exponent of 10.

  3. If the decimal is moved to the left, the exponent is positive; if to the right, it is negative.

• Example: 1,300,000 becomes $1.3 \times 10^6$.

Converting to Standard Notation

• To convert from scientific to standard notation:

  1. Multiply the coefficient by 10 raised to the given exponent.

• Example: $5.19 \times 10^3 = 5,190$.

Multiplying and Dividing in Scientific Notation

• Multiplication: Multiply the coefficients and add the exponents.

• Division: Divide the coefficients and subtract the exponents.

• Formula:

$(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$

$(a \times 10^m) \div (b \times 10^n) = (a \div b) \times 10^{m-n}$

Unit Conversions

Understanding Units and Prefixes

Units are standardized quantities used to measure physical properties. Prefixes indicate multiples or fractions of base units.

• Common Prefixes:

  • kilo- (k): $10^3$ times the base unit

  • centi- (c): $10^{-2}$ times the base unit

  • milli- (m): $10^{-3}$ times the base unit

  • micro- (μ): $10^{-6}$ times the base unit

  • nano- (n): $10^{-9}$ times the base unit

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

Comparing Units

• To compare units with different prefixes: The unit with the larger prefix represents a larger quantity.

• Example: 1 kilometer (km) is larger than 1 centimeter (cm).

Filling in Unit Conversions

• 1 microliter (μL) = $1 \times 10^{-6}$ liters (L)

• 1 gram (g) = 1,000 milligrams (mg)

• 1 millimeter (mm) = $1 \times 10^{-3}$ meters (m)

• 1 kilogram (kg) = 1,000 grams (g)

Significant Figures

Definition and Importance

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one estimated digit. They reflect the precision of a measurement.

• Rules for Counting Significant Figures:

  1. All nonzero digits are significant.

  2. Zeros between nonzero digits are significant.

  3. Leading zeros are not significant.

  4. Trailing zeros are significant only if there is a decimal point.

• Example: 0.00562 has 3 significant figures.

Exact Numbers

• Definition: Exact numbers are values known with complete certainty, often from counting or defined relationships.

• Examples: 12 eggs in a dozen, 1 meter = 100 centimeters.

• Significant Figures in Exact Numbers: Exact numbers have an infinite number of significant figures.

Reporting Measurements and Calculations

• When recording measurements: Use the correct number of significant figures based on the instrument's precision.

• When performing calculations:

  • Multiplication/Division: The result should have as many significant figures as the measurement with the fewest significant figures.

  • Addition/Subtraction: The result should have as many decimal places as the measurement with the fewest decimal places.

• Example: $2.43 \times 1.2 = 2.9$ (rounded to 2 significant figures)

Table: Common Prefixes and Their Multipliers

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

Summary

• Scientific notation simplifies working with very large or small numbers.

• Unit conversions require understanding prefixes and their relationships to base units.

• Significant figures communicate the precision of measurements and must be considered in calculations.

• Exact numbers are counted or defined values with infinite significant figures.

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard introductory chemistry curricula.