Essential Concepts & Formulas in Physical Chemistry & Physics
Fundamental Principles & Equations in Science
Quantum Mechanics & Atomic Structure
Einstein’s Mass-Energy Equivalence
Binding Energy: E = mc² (Joules)
Magnetic Moment
μ = eh / (2m)
de Broglie Wavelength
λ = h / (mv)
Photon Energy & Frequency
Energy: E = hν
Frequency: ν = c / λ (Hertz)
Rydberg Formula (Wavenumber)
Wavenumber: ṽ = R_H (1/n₁² – 1/n₂²)
Heisenberg Uncertainty Principle
The product of the uncertainty in a particle’s momentum (Δp) and its position (Δx) is greater than or equal to the reduced Planck constant (ħ): ΔpΔx ≥ ħ.
Schrödinger Equation
Describes how the quantum state of a physical system changes over time: Ĥψ = Eψ. Solving the equation means finding the wave function (ψ) for a particle.
Wave-Particle Duality
There is a characteristic wavelength associated with each particle, as described by de Broglie’s Law.
Molecular Properties & Spectroscopy
Molar Refraction
R_m = [(n² – 1) / (n² + 2)] * (M / d)
Molar Polarization
Related to relative permittivity (ε).
Resonance Condition
ν = gμ_N B / h
Dipole Moment
p = αE
Nuclear Magnetic Resonance (NMR)
A powerful analytical technique used to determine the structure, reaction state, and chemical environment of molecules. It is based on the absorption of radiofrequency radiation by nuclei in a magnetic field.
Chemical Shift (NMR)
Shows the position of a spectral line with respect to a standard: (B_sample – B_ref) / B_ref × 10⁶ ppm.
Vibrational Frequency of Diatomic Molecules
Wavenumber: ṽ = (1 / 2πc) * √(k/μ)
Relationship between wavenumbers and reduced masses: ṽ_A / ṽ_B = √(μ_B / μ_A)
Reduced Mass
The effective mass of two atoms vibrating relative to each other in a diatomic molecule: μ = (M₁M₂) / (M₁ + M₂)
Raman Spectroscopy
Based on the inelastic scattering of light, involving a change in the polarizability of a molecule during its interaction with light.
Thermodynamics & Chemical Equilibrium
Chemical Potential of Gases
μ = μ° + RT Ln (P/P°)
For solutions: μ = μ° + RT Ln (a)
Chemical Potential of Multicomponent Mixtures
Gibbs Energy: G = n_Aμ_A + n_Bμ_B + …
Gibbs Energy of Mixing
ΔG_mix = nRT (X_A Ln X_A + X_B Ln X_B)
Temperature Dependence of Equilibrium Constant
Van ‘t Hoff Equation: d(Ln K) / dT = ΔrH° / (RT²)
Clausius-Clapeyron Equation
Ln (P₂/P₁) = ΔH_vap / R * (1/T₁ – 1/T₂)
Van ‘t Hoff Equation (for Equilibrium Constant)
Ln (K₂/K₁) = -ΔH° / R * (1/T₁ – 1/T₂)
Boyle’s Law
For a fixed amount of gas at constant temperature: P₁V₁ = P₂V₂
Gay-Lussac’s Law
For a fixed amount of gas at constant volume: P₁/T₁ = P₂/T₂
Expansion Work of Gas
W = -P_ext ΔV
Enthalpy
H = U + PV
Thermal Capacity
C = q / ΔT
Zeroth Law of Thermodynamics
If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other, meaning they are at the same temperature.
First Law of Thermodynamics
Conservation of Energy: Energy cannot be created or destroyed in an isolated system; it can only change forms. The total energy remains constant over time. ΔU = Q + W.
Second Law of Thermodynamics
Entropy (a measure of chaos/randomness in a system) tends to increase in any natural process over time. ΔS ≥ Q/T (for a reversible process, ΔS = Q/T).
Third Law of Thermodynamics
As the temperature of a system approaches absolute zero (0 K), the entropy of a perfect crystal approaches zero.
Chemical Equilibrium
A state where the concentrations of reactants and products remain constant over time.
Gibbs Energy
Indicates how much non-expansion work can be done by a reaction at constant temperature and pressure.
Equilibrium Constant
The ratio of product concentrations to reactant concentrations at equilibrium.
Le Chatelier’s Principle
If a system at equilibrium is subjected to a change in concentration, pressure, temperature, or volume, the system will respond by counteracting the effect of that change to restore equilibrium.
Raoult’s Law
Describes the vapor pressure of an ideal solution.
Dalton’s Law of Partial Pressures
Describes the total pressure of a mixture of non-reacting gases.
Temperature Dependence of Reaction Enthalpies
When temperature increases, the enthalpy of both products and reactants increases. The change in enthalpy depends on the heat capacities of the substances.
Colligative Properties & Solutions
Depression of Freezing Point
The solution freezes at a lower temperature than the pure solvent: ΔT_f = K_f * c (where K_f is the cryoscopic constant and c is molality).
Molar Mass from Freezing Point Depression
M_B = (K_f * m_solute) / (ΔT_f * m_solvent)
Elevation of Boiling Point
An increase in the boiling point of a solvent when a non-volatile solute is added to it: ΔT_b = K_b * c (where K_b is the ebullioscopic constant and c is molality).
Osmotic Pressure
The pressure that must be applied to a solution to stop the flow of solvent across a semipermeable membrane.
Isotonic Solutions
Two solutions with different concentrations are mixed to create a solution with equal osmotic pressure (or concentration, in a specific context).
Chemical Kinetics & Catalysis
Acid-Base Catalysis
Rate constant: k_c
Michaelis-Menten Equation
Enzyme-substrate complex concentration: [ES] = (k₁[E]₀[S]) / (k₋₁ + k₂ + k₁[S])
Reaction Orders
- Zero-Order Reaction: c = c₀ – kt
- First-Order Reaction: c = c₀e⁻ᵏᵗ
- Second-Order Reaction: 1/c = 1/c₀ + kt
Rate Constant Units (Second-Order)
k: dm³/mol·min
Half-Life (Second-Order)
t₁/₂ = 1 / (kC₀)
Reaction Order
Influences how changes in concentration affect the reaction rate; it’s determined by studying the rate of reaction.
Law of Mass Action
The rate of a chemical reaction is directly proportional to the product of the concentrations of reactants, each raised to its stoichiometric coefficient.
Rate Law
Describes how the rate of a reaction depends on the concentrations of reactants.
Activation Energy
The minimum kinetic energy of reactants required for the formation of products. It determines the speed of a reaction: the higher the activation energy, the slower the reaction.
Collision Theory
Molecules must collide with sufficient energy to overcome the activation energy barrier.
Transition State Theory (Activated Complex Theory)
Reactions with lower activation energies tend to proceed faster.
Homogeneous Catalysis
A catalyst participates with reagents in elementary reactions that proceed much faster.
Electrochemistry & Acid-Base Chemistry
Electrical and Molar Conductivity
Electrical Conductivity: κ = 1/ρ (Siemens/meter)
Molar Conductivity: Λ = κ/c (Siemens·meter²/mol)
Molar Conductivity of Strong Electrolytes
Molar conductivity of a strong electrolyte changes with concentration, often decreasing due to interionic interactions at higher concentrations, as described by the Debye-Hückel limiting law at low concentrations.
Henderson-Hasselbalch Equation
pH = pK_a + log([A⁻]/[HA])
pH of Weak Acid
pH = 1/2 (pK_a – log[HA])
pK_a Definition
pK_a = -log K_a
pH Definition
A measure of acidity or alkalinity, defined as the negative logarithm of the concentration of hydrogen ions in a solution.
Brønsted-Lowry Theory
Defines acids and bases as conjugate pairs (proton donors and acceptors).
Spectroscopy & Photochemistry
Bathochromic Shift (Red Shift)
A change of a spectral band position in the absorption of a molecule to a longer wavelength.
Hypsochromic Shift (Blue Shift)
A change of a spectral band position in the absorption of a molecule to a shorter wavelength.
Hyperchromic Effect
An increase in absorbance.
Hypochromic Effect
A decrease in absorbance.
Fluorescence
The emission of light by a substance that has absorbed light or other radiation of a different wavelength.
Phosphorescence
Energy absorbed by a substance is released slowly in the form of light.
Stokes Shift
The difference (in wavelength or frequency units) between the positions of the band maxima of the absorption and emission spectra.
Fluorescence Spectroscopy
- Emission Spectrum: Different wavelengths of fluorescent light emitted by a sample are measured, holding the excitation light at a constant wavelength.
- Excitation Spectrum: Emission light is held at a constant wavelength, and the excitation light is scanned through many different wavelengths.
Spectral Bands
Result from changes in vibrational and rotational energy levels of molecules.
Nuclear Chemistry & Stability
Belt of Stability
A region on the graph of nuclides where stable isotopes are located, characterized by a balance between the numbers of protons and neutrons in the nucleus.
Activity of Radioactive Nuclei
Indicates how many atomic nuclei decay per second, measured in Becquerel (Bq) or Curie (Ci).
Half-Life
The time it takes for half of the radioactive nuclei in a sample to decay into more stable forms.
General Concepts & Processes
Partial Molar Quantities
The change in a property (volume, energy, or entropy) of a solution when a small amount of a component is added or removed while keeping the temperature, pressure, and composition of other components constant.
Distillation
A process to separate components of a mixture based on differences in boiling points. For two completely miscible liquids, distillation involves the vaporization of the liquid mixture and condensation of the vapor to enrich one component over the other.
Partition Equilibrium
The distribution of a solute between two immiscible phases, typically a liquid phase and a solid phase, or between two liquid phases.