General Chemistry Study Guide: Significant Figures, Dimensional Analysis, Atomic Structure, Properties of Light, Quantum Numbers, and Electron Configurations

Significant Figures

Definition and Rules

Significant figures are the digits in a measurement that are known with certainty plus one digit that is estimated. They are important in scientific calculations to reflect the precision of measured quantities.

• Zeros between non-zero digits are significant (e.g., 205 has three significant figures).

• Leading zeros (zeros before the first non-zero digit) are not significant (e.g., 0.004 has one significant figure).

• All nonzero digits are significant.

• Trailing zeros are significant only if an explicit decimal point is present (e.g., 2.300 has four significant figures).

Example: The number 0.004460 has four significant figures.

Calculations with Significant Figures

When performing calculations, the result should be reported with the correct number of significant figures based on the operation:

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

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

Example Calculation:

$(97.300 \times 0.0270514) + (44.0096 + 41.47389) = 1.00000$ (rounded to the correct number of significant figures)

Dimensional Analysis

SI Prefix Multipliers

SI prefixes are used to express multiples or fractions of units. The following table summarizes common SI prefixes:

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

Unit Conversions

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

• Density Conversion: To convert density from $kg/m^3$ to $g/cm^3$, use the conversion $1~kg/m^3 = 0.001~g/cm^3$.

• Temperature Conversion: To convert Celsius to Kelvin: $K = ^ ext{o}C + 273.15$

Example: $36.7^ ext{o}C = 36.7 + 273.15 = 309.85~K$

Atomic Structure and Isotopes

Atomic Symbols and Isotopes

An atomic symbol includes the element's symbol, mass number (A), atomic number (Z), and charge if applicable:

• Example: $^{37}_{17}Cl^{-}$ (Chlorine with 17 protons, 20 neutrons, 18 electrons)

Isotopes are atoms of the same element with different numbers of neutrons.

• Ion: An atom or molecule with a net electric charge due to the loss or gain of electrons.

• Cation: A positively charged ion.

• Anion: A negatively charged ion.

Atomic Mass Calculation

The atomic mass of an element is calculated as the weighted average of the masses of its isotopes:

$\text{Atomic Mass} = \sum (\text{Fractional Abundance} \times \text{Isotope Mass})$

Example: For Neon:

Isotope	Natural Abundance (%)	Atomic Mass (amu)
$^{20}$Ne	90.5	19.992
$^{21}$Ne	0.27	20.993
$^{22}$Ne	9.23	21.991

Subatomic Particles

• Protons: Positively charged particles in the nucleus.

• Neutrons: Neutral particles in the nucleus.

• Electrons: Negatively charged particles outside the nucleus.

Most of the mass of the atom is in the nucleus; most of the volume is occupied by the electron cloud.

Properties of Light and Wave Nature of Matter

Electromagnetic Radiation

Light exhibits both wave-like and particle-like properties. The energy of light is related to its frequency and wavelength:

• Wavelength ($\lambda$): The distance between successive peaks of a wave.

• Frequency ($\nu$): The number of wave cycles per second.

• Energy ($E$): $E = h\nu$ where $h$ is Planck's constant.

• Speed of light ($c$): $c = \lambda \nu$

Example: To find the wavelength of a radio wave with frequency $91.5~\text{MHz}$:

$\lambda = \frac{c}{\nu}$

Where $c = 3.00 \times 10^8~\text{m/s}$

Photoelectric Effect

The photoelectric effect is the emission of electrons from a material when light of sufficient frequency shines on it. It provided evidence for the particle nature of light and led to the concept of wave-particle duality.

• Threshold frequency: The minimum frequency required to eject electrons.

• Einstein's equation: $E = h\nu - \phi$ where $\phi$ is the work function.

de Broglie Wavelength

Particles such as electrons have wave-like properties, described by the de Broglie wavelength:

$\lambda = \frac{h}{mv}$

Where $h$ is Planck's constant, $m$ is mass, and $v$ is velocity.

Line Spectra and Quantum Numbers

Atomic Emission and Energy Levels

Electrons in atoms occupy discrete energy levels. When an electron transitions between levels, it emits or absorbs light of specific wavelengths.

• Shortest wavelength: Emitted during the largest energy transition (e.g., $n=4 \rightarrow n=1$).

Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and electrons:

• Principal quantum number ($n$): Indicates the energy level (possible values: 1, 2, 3, ...).

• Angular momentum quantum number ($l$): Indicates the shape of the orbital (possible values: 0 to $n-1$).

• Magnetic quantum number ($m_l$): Indicates the orientation (possible values: $-l$ to $+l$).

• Spin quantum number ($m_s$): Indicates the spin ($+1/2$ or $-1/2$).

Example: For a 4p orbital: $n=4$, $l=1$, $m_l=-1,0,1$, $m_s=\pm1/2$

Electron Configurations

Noble Gas Notation

Electron configurations can be abbreviated using the noble gas notation, which uses the symbol of the previous noble gas in brackets.

• K: [Ar]$4s^1$

• Kr: [Ar]$4d^{10}5s^2 5p^6$

• Ne: [He]$2s^2 2p^6$

• Na: [Ne]$3s^1$

Orbital Diagrams and Hund's Rule

Hund's rule states that electrons will fill degenerate orbitals singly before pairing up, and with parallel spins.

• Example: The ground-state orbital diagram for Fluorine shows seven electrons filling the $1s$, $2s$, and $2p$ orbitals.

Valence Electrons

Valence electrons are the electrons in the outermost shell, important for chemical bonding.

• Boron: Has three valence electrons.

Electron Sublevels

The first two columns of the periodic table correspond to the $s$ sublevel.

• Sublevels: $s$, $p$, $d$, $f$

Additional info:

• Some questions reference images or diagrams; these have been described in text for clarity.

• All equations are provided in LaTeX format for clarity and academic rigor.