General Chemistry Fundamentals: Significant Figures, Dimensional Analysis, Atomic Theory, and Classification of Matter

Mathematical Operations and Significant Figures

Rules for Significant Figures in Calculations

Significant figures (SFs) are crucial in chemistry for expressing the precision of measurements and calculations. The rules for handling significant figures depend on the type of mathematical operation:

• 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 be rounded to the last accurately known digit, which is determined by the least precise decimal place among the numbers involved.

• Rounding: If the digit to be dropped is 5 or greater, round up; if less than 5, round down.

Example: $14.79 \times 12.11 \times 5.05$ (Multiplication: answer should have 3 SFs, as 5.05 has the fewest SFs) $0.236 + 128.55 - 102.1$ (Addition/Subtraction: answer should be rounded to the tenths place, as 102.1 is least precise)

Practice Problems

• Calculate: $\frac{32.44 + 4.9 - 0.3}{82.94}$

• Calculate: $\left(1.302 \times 10^3\right) - \left(1.57 \times 10^2\right)$

These problems reinforce the application of significant figure rules in real calculations.

Dimensional Analysis Practice

Unit Conversions and Problem Solving

Dimensional analysis is a method for converting between units using conversion factors. It is essential for solving problems involving measurements in chemistry.

• Example: The Kentucky Derby is 10.0 furlongs. Given: 8 furlongs = 1 mile, 10 chains = 1 furlong, 100 links = 1 chain, 5280 feet = 1 mile, 12 inches = 1 foot. Task: Convert 10.0 furlongs to feet and links to inches.

• Density Conversion: The density of iron is 0.284 lb/in3. Convert to g/cm3 using appropriate conversion factors.

• Speed Conversion: Convert running pace from min/mile to m/s, and compare to swimming world record speed.

Atoms and the Mole: How Many Particles

Definition of Mole and Avogadro's Number

The mole is a fundamental unit in chemistry representing a specific number of particles (atoms, molecules, ions). Avogadro's number ($N_A$) is the number of particles in one mole:

• $N_A = 6.022 \times 10^{23}$ particles/mol

• One mole of carbon-12 contains $6.022 \times 10^{23}$ atoms and has a mass of 12.00 g.

Interconverting Moles, Mass, and Number of Particles

• To find number of particles: $\text{Number of particles} = \text{moles} \times N_A$

• To find mass from moles: $\text{Mass (g)} = \text{moles} \times \text{molar mass (g/mol)}$

• To find moles from mass: $\text{Moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}$

Example: How many grams are in 1.85 mol U? How many moles are in 29.127 g Na?

A Particulate View of the World: Structure Determination

Definitions

• Matter: Anything that has mass and occupies space.

• Atoms: The smallest units of matter that retain the properties of an element.

• Molecules: Groups of two or more atoms bonded together in specific geometric arrangements.

• Chemical Compounds: Substances composed of two or more different elements chemically bonded in fixed proportions.

Classifying Matter: A Particulate View

Classification by State and Composition

• States of Matter: Solid, liquid, gas.

• Composition:

  • Pure Substances: Elements and compounds with uniform composition.

  • Mixtures: Physical combinations of two or more substances.

    • Homogeneous Mixture: Uniform composition throughout (e.g., salt water).

    • Heterogeneous Mixture: Non-uniform composition (e.g., salad).

Modern Atomic Theory and the Laws that Led to It

Law of Conservation of Mass

The law of conservation of mass states that in a chemical reaction, mass is neither created nor destroyed.

• Example: $2KI \rightarrow I_2 + 2KNO_3$ The total mass of reactants equals the total mass of products.

Law of Definite Proportions

Different samples of a pure chemical compound always contain the same proportion of elements by mass.

• Example: Water is always 88.8% oxygen and 11.2% hydrogen by mass.

Law of Multiple Proportions

When two elements form more than one compound, the masses of one element that combine with a fixed mass of the other are in ratios of small whole numbers.

• Example: Nitrogen and oxygen can form NO and NO2; the ratio of oxygen masses that combine with a fixed mass of nitrogen is a small whole number.

Dalton’s Atomic Theory (Four Postulates)

1. All matter is made of tiny particles called atoms.

2. Atoms of a given element are identical in mass and properties; atoms of different elements differ.

3. Compounds are formed by the combination of atoms in simple whole-number ratios.

4. Chemical reactions involve rearrangement of atoms; atoms themselves are not changed.

The Discovery of the Electron

Thomson’s Cathode Ray Experiment

• J.J. Thomson discovered the electron by studying cathode rays, showing they are negatively charged particles.

• Mass-to-charge ratio: $-1.76 \times 10^8$ coulomb/g

Millikan’s Oil Drop Experiment

• Measured the charge of the electron: $-1.60 \times 10^{-19}$ coulomb

• Combined with Thomson’s data, determined the mass of the electron.

The Structure of the Atom

Rutherford’s Gold Foil Experiment

• Alpha particles were directed at thin gold foil; most passed through, but some were deflected.

• Led to the nuclear model: atoms have a small, dense, positively charged nucleus surrounded by electrons.

Basic Parts of the Atom

• Nucleus: Central core containing protons and neutrons.

• Proton: Positively charged particle in the nucleus.

• Neutron: Neutral particle in the nucleus.

• Electron: Negatively charged particle outside the nucleus.

Subatomic Particles: Protons, Neutrons, and Electrons

Definitions and Properties

Particle	Mass (kg)	Mass (amu)	Charge
Proton	1.6726 × 10-27	1.0073	+1
Neutron	1.6749 × 10-27	1.0087	0
Electron	9.1094 × 10-31	0.00055	-1

Atomic Number, Mass Number, and Chemical Symbol

• Atomic Number (Z): Number of protons in the nucleus; defines the element.

• Mass Number (A): Total number of protons and neutrons.

• Chemical Symbol: One- or two-letter abbreviation for the element.

Isotopes

• Atoms of the same element with different numbers of neutrons.

• Representation: $^{A}_{Z}X$ or X-A (e.g., Ne-20, Ne-21, Ne-22)

Ions: Cations and Anions

• Cation: Positively charged ion (loss of electrons).

• Anion: Negatively charged ion (gain of electrons).

Atomic Mass: The Average Mass of an Element’s Atoms

Calculating Atomic Mass

The atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes.

• Formula: $\text{Atomic mass} = \sum (\text{fraction of isotope n}) \times (\text{mass of isotope n})$

Example: Carbon: $\text{Atomic mass} = (0.9889 \times 12) + (0.01109 \times 13.00335)$

Isotope Abundance Practice

• Lithium has two isotopes: $^6$Li (6.01513 amu) and $^7$Li (7.01601 amu). The atomic mass is 6.941 amu. The more abundant isotope is $^7$Li.

Additional Info

• Shape and structure of molecules influence their properties and behavior (e.g., ball vs. football vs. tennis ball).

• Practice problems and objectives are designed to reinforce understanding of fundamental concepts in general chemistry.