Essentials: Units, Measurement, and Problem Solving

Scientific Notation and Mathematical Operations

Scientific notation is a method for expressing very large or very small numbers in the form a × 10n, where a is a number between 1 and 10, and n is an integer.

• Adding/Subtracting: Ensure exponents are the same before performing the operation on the coefficients.

• Multiplying: Multiply the coefficients and add the exponents.

• Dividing: Divide the coefficients and subtract the exponents.

• Example:

Significant Figures

Significant figures (sig figs) reflect the precision of a measured quantity.

• Rules:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

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

• Calculations:

  • Addition/Subtraction: Result has the same number of decimal places as the measurement with the fewest decimal places.

  • Multiplication/Division: Result has the same number of significant figures as the measurement with the fewest significant figures.

Density

Density is defined as mass per unit volume.

• Formula:

• Units: Commonly g/cm3 or g/mL for liquids and solids; g/L for gases.

• Applications: Used to identify substances and solve for mass or volume when the other is known.

Temperature Scales and Conversions

There are three main temperature scales: Celsius (°C), Fahrenheit (°F), and Kelvin (K).

• Conversion Equations:

Dimensional Analysis (Unit Conversions)

Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

• Steps:

  1. Identify the starting and desired units.

  2. Set up conversion factors so units cancel appropriately.

  3. Multiply through to obtain the answer in the desired units.

• Example: Convert 5.0 km to meters:

Metric Prefixes

Metric prefixes are used to express multiples or fractions of base units.

Atoms

Law of Conservation of Mass

The Law of Conservation of Mass states that mass is neither created nor destroyed in a chemical reaction. The total mass of reactants equals the total mass of products.

Classification of Matter

Matter can be classified based on its composition:

• Pure Substances:

  • Elements: Cannot be broken down into simpler substances.

  • Compounds: Composed of two or more elements in fixed proportions.

• Mixtures:

  • Homogeneous: Uniform composition throughout (solutions).

  • Heterogeneous: Non-uniform composition.

Dalton's Atomic Theory (4 Main Ideas)

1. All matter is composed of tiny, indivisible particles called atoms.

2. All atoms of a given element are identical in mass and properties.

3. Compounds are formed by the combination of atoms of different elements in fixed ratios.

4. Chemical reactions involve rearrangement of atoms; atoms are not created or destroyed.

Rutherford's Nuclear Theory (3 Main Ideas)

1. Most of the atom's mass and all of its positive charge are contained in a small nucleus.

2. Most of the atom's volume is empty space, with electrons dispersed throughout.

3. The number of protons equals the number of electrons in a neutral atom.

Subatomic Particles: Protons, Neutrons, Electrons

Atomic Number (Z) and Mass Number (A)

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

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

• Number of Neutrons:

Isotopes and Average Atomic Mass

Isotopes are atoms of the same element (same Z) with different numbers of neutrons (different A).

• Average Atomic Mass: Weighted average of all naturally occurring isotopes.

• Formula:

Ions: Cations and Anions

• Cation: Positively charged ion (fewer electrons than protons).

• Anion: Negatively charged ion (more electrons than protons).

• Net Charge:

Mole and Molar Mass

• Mole (mol): Amount of substance containing entities (Avogadro's number).

• Molar Mass: Mass of one mole of a substance (g/mol).

• Conversions:

  • Mass ↔ Moles:

  • Moles ↔ Number of Atoms:

The Quantum-Mechanical Model of the Atom

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength and frequency.

• Order of Increasing Energy: Radio < Microwave < Infrared < Visible < Ultraviolet < X-rays < Gamma rays

• Example: Ultraviolet waves are more energetic than microwaves.

Wavelength (λ) and Frequency (ν)

• Wavelength (λ): Distance between successive crests of a wave (meters, m).

• Frequency (ν): Number of cycles per second (Hertz, Hz).

• Relationship: where is the speed of light ( m/s).

• Inversely Proportional: As wavelength increases, frequency decreases, and vice versa.

Energy of Electromagnetic Waves

• Planck's Equation: where J·s.

• Energy-Wavelength Relationship:

De Broglie Equation

The De Broglie equation relates the wavelength of a particle to its momentum, showing that matter exhibits wave-like properties.

• Equation:

• Importance: Demonstrates wave-particle duality, foundational for quantum mechanics.

Heisenberg's Uncertainty Principle

This principle states that it is impossible to simultaneously know both the exact position and momentum of a particle.

• Equation:

• Significance: Limits precision in measurements at the quantum scale.

Hydrogen Atom: Energy Transitions

When an electron in a hydrogen atom transitions between energy levels, it emits or absorbs a photon of specific energy.

• Energy Change Equation:

• J

• Where: = initial energy level, = final energy level

Quantum Numbers and Atomic Orbitals

• Principal Quantum Number (n): Indicates energy level (n = 1, 2, 3, ...).

• Angular Momentum Quantum Number (l): Indicates shape of orbital (l = 0 to n-1).

• Orbital Types:

  • s orbital: l = 0 (spherical)

  • p orbital: l = 1 (dumbbell-shaped)

  • d orbital: l = 2 (cloverleaf-shaped)

  • f orbital: l = 3 (complex shapes)