Matter, Measurement, and Problem Solving: Significant Figures, Scientific Approach, and Units

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Matter, Measurement & Problem Solving

Significant Figures

Significant figures are crucial in science for reporting measurements accurately. They reflect the precision of a measured or calculated value and help communicate how close multiple measurements are to one another.

Definition: Significant figures are the digits in a number that carry meaning contributing to its measurement accuracy. This includes all digits except leading and trailing zeros where they serve merely as placeholders.
Purpose: They indicate the reliability and precision of a measurement or calculation.
Application: Used to report values to the correct number of decimal places, reflecting the measurement's precision.

Rules for Identifying Significant Figures

All nonzero digits are significant. Example: 129.5 has four significant figures.
Zeros between nonzero digits are significant. Example: 2007 has four significant figures.
Leading zeros (to the left of the first nonzero digit) are not significant. Example: 0.0055 has two significant figures.
Trailing zeros in a whole number with a decimal point are significant. Example: 320.0 has four significant figures.
Trailing zeros to the right of the decimal point are significant. Example: 1200.0 has five significant figures.
Exact numbers (defined values, counting numbers) have infinite significant figures. Example: 1 inch = 2.54 cm (exact).
Scientific notation: Only the digits in the coefficient are significant. Example: has two significant figures.

Significant Figures in Calculations

Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places. Example: 143.2 + 20.1 = 163.3 (rounded to one decimal place).
Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures. Example: (rounded to two significant figures).
Order of Operations: In multi-step calculations, keep track of significant figures at each step and round only the final answer.

Rounding Rules

If the digit to be dropped is less than 5, round down.
If the digit to be dropped is 5 or greater, round up.
When converting to scientific notation, the number of significant figures does not change.

Examples of Calculations

Addition: 101.70 + 0.252 = 101.952 (rounded to 101.95)
Multiplication: 11.57 × 0.003825290964 = 0.0337 (rounded to three significant figures)
Scientific Notation: 4500 can be written as

Scientific Approach

The scientific approach is a systematic method for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge.

Steps:
1. Observation/Experiment
2. Hypothesis/Law
3. Test
4. If confirmed, develop a theory

States of Matter

Matter is anything that occupies space and has mass. It exists in different states, each with distinct properties.

Solid: Definite shape and volume; particles are closely packed.
Liquid: Definite volume but no definite shape; particles can move past each other.
Gas: No definite shape or volume; particles are far apart and move freely.

Classification of Matter

Pure Substance: Has a constant composition (e.g., helium, water).
Mixture: Composed of two or more substances; can be homogeneous (uniform) or heterogeneous (not uniform).

Physical and Chemical Properties

Physical Property: Can be observed without changing the composition of the substance (e.g., boiling point).
Chemical Property: Can only be observed by changing the substance's composition (e.g., rusting of metal).

Physical and Chemical Changes

Physical Change: Alters only the state or appearance, not the composition (e.g., melting).
Chemical Change: Alters the composition of matter (e.g., metal rusting).

Accuracy and Precision

Accuracy: How close a measurement is to the actual value.
Precision: How close a series of measurements are to one another.

Measurement and Units

SI Units and English System

Measurements in chemistry use standardized units to ensure consistency and accuracy.

Quantity	SI Unit	Symbol
Length	Meter	m
Mass	Kilogram	kg
Time	Second	s
Temperature	Kelvin	K
Amount of Substance	Mole	mol

Meter (m): SI unit of length, slightly longer than a yard (1 m ≈ 39.37 inches).
Kilogram (kg): SI unit of mass (1 kg ≈ 2.205 lbs).
Second (s): SI unit of time (1 min = 60 s).
Kelvin (K): SI unit of temperature, based on absolute zero.

Prefixes for SI Units

Prefix	Symbol	Multiplier
Kilo	k
Centi	c
Milli	m
Micro	μ
Nano	n

Units of Volume

Volume is a derived unit, calculated as length cubed. Common units include cubic meters (), cubic centimeters (), and liters (L).

1 m3 = 1,000,000 cm3
1 L = 1,000 mL = 1,000 cm3

Density

Density is an intensive property, independent of the amount of substance. It is calculated as:

Example: If 1 mL = 1 cm3, and water has a mass of 1 g per cm3, then the density of water is 1 g/cm3.

Extensive vs. Intensive Properties

Extensive Property: Depends on the amount of substance (e.g., mass, volume).
Intensive Property: Independent of the amount of substance (e.g., density).

Mathematical Operations and Functions

Order of Operations (PEMDAS)

When performing calculations, follow the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Keep track of significant figures and decimals at each step.
Round only the final answer to the correct number of significant figures.

Examples

Example 1: (rounded to two significant figures)
Example 2: (rounded to two significant figures)

Scientific Notation

Scientific notation is used to express very large or small numbers. The coefficient (argument) must be between 1 and 10.

Example:
When converting to scientific notation, the number of significant figures remains unchanged.

Rounding in Scientific Notation

When rounding, look at the digit to the right of the place you are rounding to. If it is 5 or greater, round up; otherwise, round down.

Summary Table: Significant Figures Rules

Rule	Example	Sig. Figs
Nonzero digits	123.45	5
Zeros between nonzero digits	2007	4
Leading zeros	0.0055	2
Trailing zeros (decimal)	320.0	4
Trailing zeros (whole number, no decimal)	1200	2
Exact numbers	1 inch = 2.54 cm	Infinite
Scientific notation	5.0 × 103	2

Additional Info

Irrational numbers: Numbers that cannot be expressed as a ratio of integers (e.g., π, √2) are considered exact for significant figures purposes.
In multi-step calculations, keep track of significant figures at each step and round only the final answer.