Significant Figures and Measurement

Significant Figures: Rules and Operations

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. Proper use of significant figures is essential for reporting scientific data accurately.

• Counting Significant Figures: 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.

• Operations:

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

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

• Rounding: If the digit to be dropped is less than 5, leave the last digit unchanged; if 5 or greater, increase the last digit by one.

Unit Conversion and Dimensional Analysis

Dimensional analysis is a method to convert one unit to another using conversion factors.

• Conversion Factors: Ratios derived from the equality between two different units.

• Process: Multiply the original value by the conversion factor(s) so that units cancel appropriately.

• Example: To convert 10.0 inches to centimeters: 10.0 in × (2.54 cm / 1 in) = 25.4 cm

SI Units, Prefixes, and Derived Units

• Base SI Units: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd)

• Common Prefixes: kilo- (k, 103), centi- (c, 10-2), milli- (m, 10-3), micro- (μ, 10-6), nano- (n, 10-9)

• Derived Units: Units formed from base units, e.g., m3 for volume, kg/m3 for density.

Temperature Conversions

• Celsius to Kelvin:

• Celsius to Fahrenheit:

• Fahrenheit to Celsius:

Density Calculations

• Definition: Density is mass per unit volume.

• Formula:

• Units: Commonly g/cm3 or kg/m3

Atoms, Elements, and Isotopes

Atomic Number, Mass Number, and Chemical Symbols

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

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

• Annotated Symbol: where X is the element symbol, A is mass number, Z is atomic number.

Protons, Neutrons, and Electrons

• Protons: Equal to atomic number (Z).

• Neutrons:

• Electrons: For neutral atoms, equal to protons; for ions, adjust for charge.

• Example: has 11 protons, 12 neutrons, and 10 electrons.

Isotopes and Average Atomic Mass

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

• Average Atomic Mass: Weighted average of isotopic masses based on natural abundance.

• Formula:

• Atomic Mass Spectrum: Graphical representation of isotopic abundances.

Electromagnetic Radiation and Quantum Theory

Types of Electromagnetic Radiation

• Order by Wavelength (longest to shortest): Radio > Microwave > Infrared > Visible > Ultraviolet > X-ray > Gamma ray

• Order by Frequency (lowest to highest): Reverse of wavelength order.

Energy, Wavelength, and Frequency of Photons

• Relationship: where is speed of light, is wavelength, is frequency.

• Photon Energy: where is Planck's constant.

• Conversions: ,

Photoelectric Effect

• Concept: Emission of electrons from a metal surface when light of sufficient frequency shines on it.

• Equation: where is the work function, is the kinetic energy of the emitted electron.

Quantum Numbers

• Principal Quantum Number (n): Positive integer (), indicates energy level.

• Angular Momentum Quantum Number (l): Integer from 0 to , indicates subshell (s, p, d, f).

• Magnetic Quantum Number (m_l): Integer from to , indicates orbital orientation.

Heisenberg Uncertainty Principle

• Statement: It is impossible to simultaneously know both the exact position and momentum of a particle.

• Equation:

de Broglie Wavelength

• Formula:

• Application: All matter has wave-like properties; significant for small particles like electrons.

Hydrogen Atom: Energies and Spectra

• Rydberg Equation: ,

• Application: Used to calculate wavelengths of emitted/absorbed photons for hydrogen and hydrogen-like ions.

Atomic Orbitals: Shapes and Quantum Numbers

• s orbitals: Spherical shape,

• p orbitals: Dumbbell shape,

• d orbitals: Cloverleaf shape,

• Each orbital is defined by a unique set of quantum numbers (n, l, m_l).

Electron Configuration and Periodic Properties

Electron Configurations and Orbital Diagrams

• Aufbau Principle: Electrons fill orbitals from lowest to highest energy.

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.

• Hund’s Rule: Electrons occupy degenerate orbitals singly before pairing.

• Orbital Diagram: Visual representation using arrows for electrons in boxes for orbitals.

Valence and Core Electrons

• Valence Electrons: Electrons in the outermost shell, involved in bonding.

• Core Electrons: Electrons in inner shells, not involved in bonding.

• Identification: For main group elements, valence electrons = group number.

Quantum Mechanical Explanation of Periodic Law

• Periodic Law: Properties of elements recur periodically when arranged by increasing atomic number.

• Quantum Mechanics: Explains periodicity based on electron configurations and energy levels.

Irregular Electron Configurations

• Some transition metals: e.g., Cr ([Ar] 4s1 3d5) and Cu ([Ar] 4s1 3d10) have irregular configurations due to stability of half-filled or fully filled d subshells.

Periodic Trends and Explanations

• Atomic Radii: Decreases across a period, increases down a group. Explained by effective nuclear charge and quantum numbers.

• Ionic Radii: Cations are smaller, anions are larger than parent atoms.

• Magnetic Properties: Paramagnetic (unpaired electrons), diamagnetic (all electrons paired).

• Ionization Energy: Increases across a period, decreases down a group.

• Electron Affinity: Generally becomes more negative across a period.

• Metallic Character: Increases down a group, decreases across a period.

Chemical Bonding and Nomenclature

Types of Chemical Bonds

• Ionic Bonds: Transfer of electrons from metal to nonmetal.

• Covalent Bonds: Sharing of electrons between nonmetals.

• Metallic Bonds: Delocalized electrons among metal atoms.

Writing Formulas and Nomenclature

• Ionic Compounds: Name cation first, then anion (e.g., NaCl: sodium chloride).

• Molecular Compounds: Use prefixes to indicate number of atoms (e.g., CO2: carbon dioxide).

• Acids: Binary acids: hydro- + base name + -ic acid (e.g., HCl: hydrochloric acid); Oxyacids: based on polyatomic ion (e.g., H2SO4: sulfuric acid).

Useful Equations